mirror of
https://github.com/NotXia/unibo-ai-notes.git
synced 2025-12-16 19:32:21 +01:00
Moved IPCV1 in year1
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|
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|
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|
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|
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@ -0,0 +1,22 @@
|
||||
\documentclass[11pt]{ainotes}
|
||||
\usepackage{biblatex}
|
||||
\addbibresource{./references.bib}
|
||||
|
||||
\title{Image Processing and Computer Vision\\(Module 1)}
|
||||
\date{2023 -- 2024}
|
||||
\def\lastupdate{{PLACEHOLDER-LAST-UPDATE}}
|
||||
|
||||
\begin{document}
|
||||
|
||||
\makenotesfront
|
||||
|
||||
\input{./sections/_image_acquisition.tex}
|
||||
\input{./sections/_spatial_filtering.tex}
|
||||
\input{./sections/_edge_detection.tex}
|
||||
\input{./sections/_local_features.tex}
|
||||
\input{./sections/_instance_obj_detection.tex}
|
||||
\eoc
|
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|
||||
\printbibliography[heading=bibintoc]
|
||||
|
||||
\end{document}
|
||||
@ -0,0 +1,49 @@
|
||||
@misc{ wiki:1d_convolution,
|
||||
title = "Convolution --- {Wikipedia}{,} The Free Encyclopedia",
|
||||
author = "{Wikipedia contributors}",
|
||||
year = "2024",
|
||||
howpublished = "\url{https://en.wikipedia.org/w/index.php?title=Convolution&oldid=1212399231}"
|
||||
}
|
||||
|
||||
@misc{ wiki:dirac,
|
||||
title = "Dirac delta function --- {Wikipedia}{,} The Free Encyclopedia",
|
||||
author = "{Wikipedia contributors}",
|
||||
year = "2024",
|
||||
howpublished = "\url{https://en.wikipedia.org/w/index.php?title=Dirac_delta_function&oldid=1198785224}"
|
||||
}
|
||||
|
||||
@misc{ wiki:kronecker,
|
||||
title = "Kronecker delta --- {Wikipedia}{,} The Free Encyclopedia",
|
||||
author = "{Wikipedia contributors}",
|
||||
year = "2023",
|
||||
howpublished = "\url{https://en.wikipedia.org/w/index.php?title=Kronecker_delta&oldid=1192529815}"
|
||||
}
|
||||
|
||||
@book{ book:sonka,
|
||||
title={Image Processing, Analysis, and Machine Vision},
|
||||
author={Sonka, M. and Hlavac, V. and Boyle, R.},
|
||||
isbn={978-1-133-59360-7},
|
||||
year={2015},
|
||||
publisher={Cengage Learning}
|
||||
}
|
||||
|
||||
@misc{ slides:filters,
|
||||
title = {Filters},
|
||||
author={Ramani Duraiswami},
|
||||
howpublished = {\url{http://users.umiacs.umd.edu/~ramani/cmsc828d_audio/Filters.pdf}},
|
||||
year = {2006}
|
||||
}
|
||||
|
||||
@misc{ wiki:crosscorrelation,
|
||||
title = "Cross-correlation --- {Wikipedia}{,} The Free Encyclopedia",
|
||||
author = "{Wikipedia contributors}",
|
||||
year = "2024",
|
||||
howpublished = "\url{https://en.wikipedia.org/w/index.php?title=Cross-correlation&oldid=1193503271}"
|
||||
}
|
||||
|
||||
@misc{ slides:scale_normalized_log,
|
||||
title = {Blob features},
|
||||
author={Trym Vegard Haavardsholm},
|
||||
howpublished = {\url{https://www.uio.no/studier/emner/matnat/its/TEK5030/v20/forelesninger/lecture_3_2_2_blob_features.pdf}},
|
||||
year = {2020}
|
||||
}
|
||||
@ -0,0 +1,391 @@
|
||||
\chapter{Edge detection}
|
||||
|
||||
|
||||
\begin{description}
|
||||
\item[Edge] \marginnote{Edge}
|
||||
Pixel lying in between regions of the image with different intensities.
|
||||
\end{description}
|
||||
|
||||
|
||||
|
||||
\section{Gradient thresholding}
|
||||
|
||||
\subsection{1D step-edge}
|
||||
\marginnote{1D step-edge}
|
||||
|
||||
In the transition region, the absolute value of the first derivative grows (the absolute value is used as the polarity is not relevant).
|
||||
By fixing a threshold, an edge can be detected.
|
||||
|
||||
\begin{figure}[H]
|
||||
\begin{subfigure}{0.4\linewidth}
|
||||
\centering
|
||||
\includegraphics[width=0.5\linewidth]{./img/1d_step_edge_example1.png}
|
||||
\caption{Input signal}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{0.4\linewidth}
|
||||
\centering
|
||||
\includegraphics[width=0.5\linewidth]{./img/1d_step_edge_example2.png}
|
||||
\caption{Derivative of the signal}
|
||||
\end{subfigure}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\subsection{2D step-edge}
|
||||
\marginnote{2D step-edge}
|
||||
|
||||
In a 2D signal (e.g. an image), the gradient allows to determine the magnitude and the direction of the edge.
|
||||
\[
|
||||
\nabla I(x, y) =
|
||||
\begin{pmatrix} \frac{\partial I(x, y)}{\partial x} & \frac{\partial I(x, y)}{\partial y} \end{pmatrix} =
|
||||
\begin{pmatrix} \partial_x I & \partial_y I \end{pmatrix}
|
||||
\]
|
||||
\[
|
||||
\begin{split}
|
||||
\text{Magnitude: } & \Vert \nabla I(x, y) \Vert \\
|
||||
\text{Direction: } & \arctan\left(\frac{\partial_y I}{\partial_x I}\right) \in [-\frac{\pi}{2}, \frac{\pi}{2}] \\
|
||||
\text{Direction and sign: } & \arctan2(\partial_x I, \partial_y I) \in [0, 2\pi] \\
|
||||
\end{split}
|
||||
\]
|
||||
|
||||
\begin{description}
|
||||
\item[Discrete gradient approximation] \marginnote{Discrete gradient approximation}
|
||||
Approximation of the partial derivatives as a difference.
|
||||
\begin{description}
|
||||
\item[Backward difference] \marginnote{Backward difference}
|
||||
\[ \partial_x I(i, j) \approx I(i, j) - I(i, j-1) \hspace{3em} \partial_y I(i, j) \approx I(i, j) - I(i-1, j) \]
|
||||
|
||||
|
||||
\item[Forward difference] \marginnote{Forward difference}
|
||||
\[ \partial_x I(i, j) \approx I(i, j+1) - I(i, j) \hspace{3em} \partial_y I(i, j) \approx I(i+1, j) - I(i, j) \]
|
||||
|
||||
\begin{remark}
|
||||
Forward and backward differences are equivalent to applying two cross-correlations with kernels
|
||||
$\begin{pmatrix} -1 & 1 \end{pmatrix}$ and $\begin{pmatrix} -1 \\ 1 \end{pmatrix}$.
|
||||
\end{remark}
|
||||
|
||||
\item[Central difference] \marginnote{Central difference}
|
||||
\[ \partial_x I(i, j) \approx I(i, j+1) - I(i, j-1) \hspace{3em} \partial_y I(i, j) \approx I(i+1, j) - I(i-1, j) \]
|
||||
|
||||
\begin{remark}
|
||||
Central difference is equivalent to applying two cross-correlations with kernels
|
||||
$\begin{pmatrix} -1 & 0 & 1 \end{pmatrix}$ and $\begin{pmatrix} -1 \\ 0 \\ 1 \end{pmatrix}$.
|
||||
\end{remark}
|
||||
\end{description}
|
||||
|
||||
\item[Discete magnitude approximation] \marginnote{Discete magnitude approximation}
|
||||
The gradient magnitude can be approximated using the approximated partial derivatives:
|
||||
\[
|
||||
\Vert \nabla I \Vert = \sqrt{(\partial_x I)^2 + (\partial_y I)^2} \hspace{1.5em}
|
||||
\Vert \nabla I \Vert_+ = \vert \partial_x I \vert + \vert \partial_y I \vert \hspace{1.5em}
|
||||
\Vert \nabla I \Vert_\text{max} = \max(\vert \partial_x I \vert, \vert \partial_y I \vert)
|
||||
\]
|
||||
|
||||
Among all, $\Vert \nabla I \Vert_\text{max}$ is the most isotropic (i.e. gives a more consistent response).
|
||||
\begin{example}
|
||||
Given the following images:
|
||||
\[
|
||||
E_v = \begin{pmatrix}
|
||||
0 & 0 & h & h \\
|
||||
0 & 0 & h & h \\
|
||||
0 & 0 & h & h \\
|
||||
0 & 0 & h & h \\
|
||||
\end{pmatrix}
|
||||
\,\,\,
|
||||
E_h = \begin{pmatrix}
|
||||
0 & 0 & 0 & 0 \\
|
||||
0 & 0 & 0 & 0 \\
|
||||
h & h & h & h \\
|
||||
h & h & h & h \\
|
||||
\end{pmatrix}
|
||||
\,\,\,
|
||||
E_d = \begin{pmatrix}
|
||||
0 & 0 & 0 & 0 & h \\
|
||||
0 & 0 & 0 & h & h \\
|
||||
0 & 0 & h & h & h \\
|
||||
0 & h & h & h & h \\
|
||||
\end{pmatrix}
|
||||
\]
|
||||
The magnitudes are:
|
||||
\begin{center}
|
||||
\begin{tabular}{c|ccc}
|
||||
\toprule
|
||||
& $\Vert \nabla I \Vert$ & $\Vert \nabla I \Vert_+$ & $\Vert \nabla I \Vert_\text{max}$ \\
|
||||
\midrule
|
||||
$E_h$ & $h$ & $h$ & $h$ \\
|
||||
$E_v$ & $h$ & $h$ & $h$ \\
|
||||
$E_d$ & $\sqrt{2}h$ & $2h$ & $h$ \\
|
||||
\bottomrule
|
||||
\end{tabular}
|
||||
\end{center}
|
||||
\end{example}
|
||||
\end{description}
|
||||
|
||||
\begin{remark}
|
||||
In practice, the signal of an image is not always smooth due to noise.
|
||||
Derivatives amplify noise and are therefore unable to recognize edges.
|
||||
|
||||
Smoothing the signal before computing the derivative allows to reduce the noise but also blurs the edges making it more difficult to localize them.
|
||||
|
||||
A solution is to smooth and differentiate in a single operation by approximating the gradient as a difference of averages.
|
||||
\end{remark}
|
||||
|
||||
\begin{description}
|
||||
\item[Smooth derivative] \marginnote{Smooth derivative}
|
||||
Compute the approximation of a partial derivative as the difference of the pixels in a given window.
|
||||
For instance, considering a window of 3 pixels, the cross-correlation kernels are:
|
||||
\[ \frac{1}{3} \begin{pmatrix} -1 & 1 \\ -1 & 1 \\ -1 & 1 \end{pmatrix} \hspace{3em} \frac{1}{3} \begin{pmatrix} -1 & -1 & -1 \\ 1 & 1 & 1 \end{pmatrix} \]
|
||||
|
||||
\begin{description}
|
||||
\item[Prewitt operator] \marginnote{Prewitt operator}
|
||||
Derivative approximation using central differences.
|
||||
The cross-correlation kernels are:
|
||||
\[
|
||||
\frac{1}{3} \begin{pmatrix} -1 & 0 & 1 \\ -1 & 0 & 1 \\ -1 & 0 & 1 \end{pmatrix}
|
||||
\hspace{3em}
|
||||
\frac{1}{3} \begin{pmatrix} -1 & -1 & -1 \\ 0 & 0 & 0 \\ 1 & 1 & 1 \end{pmatrix}
|
||||
\]
|
||||
|
||||
\item[Sobel operator] \marginnote{Sobel operator}
|
||||
Prewitt operator where the central pixels have a higher weight.
|
||||
The cross-correlation kernels are:
|
||||
\[
|
||||
\frac{1}{4} \begin{pmatrix} -1 & 0 & 1 \\ -2 & 0 & 2 \\ -1 & 0 & 1 \end{pmatrix}
|
||||
\hspace{3em}
|
||||
\frac{1}{4} \begin{pmatrix} -1 & -2 & -1 \\ 0 & 0 & 0 \\ 1 & 2 & 1 \end{pmatrix}
|
||||
\]
|
||||
\end{description}
|
||||
\end{description}
|
||||
|
||||
\begin{remark}
|
||||
Thresholding is inaccurate as choosing the threshold is not straightforward.
|
||||
An image has strong and weak edges. Trying to detect one type might lead to poor detection of the other.
|
||||
|
||||
A better solution is to find a local maxima of the absolute value of the derivatives.
|
||||
\end{remark}
|
||||
|
||||
|
||||
|
||||
\section{Non-maxima suppression (NMS)}
|
||||
\marginnote{Non-maxima suppression (NMS)}
|
||||
|
||||
Algorithm that looks for local maxima of the absolute value of the gradient along the gradient direction.
|
||||
|
||||
The algorithm works as follows:
|
||||
\begin{enumerate}
|
||||
\item Given a pixel at coordinates $(i, j)$,
|
||||
estimate the magnitude $G = \Vert \nabla I(i, j) \Vert$ and the direction $\theta$ of the gradient.
|
||||
\item Consider two points $A$ and $B$ along the direction $\theta$ passing through $(i, j)$
|
||||
and compute their gradient magnitudes $G_A$ and $G_B$.
|
||||
\item Substitute the pixel $(i, j)$ as follows:
|
||||
\[ \texttt{NMS}(i, j) = \begin{cases}
|
||||
1 & (G > G_A) \land (G > G_B) \text{ (i.e. local maximum)} \\
|
||||
0 & \text{otherwise} \\
|
||||
\end{cases} \]
|
||||
\end{enumerate}
|
||||
|
||||
\begin{remark}
|
||||
After applying NMS, the resulting signal (now composed of 0s and 1s) is converted back to the original gradient magnitudes
|
||||
in such a way that pixels 0ed by NMS remain 0 and pixels set at 1 by NMS return to their original value.
|
||||
|
||||
After the conversion, a thresholding step might be applied to filter out unwanted edges that are either due to noise or not relevant.
|
||||
\end{remark}
|
||||
|
||||
|
||||
\subsection{Linear interpolation}
|
||||
|
||||
As there might not be two points $A$ and $B$ along the direction $\theta$ belonging the the discrete pixel grid
|
||||
and because one doesn't want to consider two points too far from $(i, j)$,
|
||||
it is possible to use linear interpolation to estimate the gradients of $A$ and $B$ even if off-grid by an offset $\Delta x$:\\
|
||||
\begin{minipage}{0.6\linewidth}
|
||||
\[
|
||||
\begin{split}
|
||||
G_1 &= \Vert \nabla I(i-1, j) \Vert \hspace{2em} G_2 = \Vert \nabla I(i-1, j+1) \Vert \\
|
||||
G_3 &= \Vert \nabla I(i+1, j) \Vert \hspace{2em} G_4 = \Vert \nabla I(i+1, j-1) \Vert \\
|
||||
\end{split}
|
||||
\]
|
||||
\[
|
||||
\begin{split}
|
||||
G_A &\approx G1 + (G2 - G1)\Delta x \\
|
||||
G_B &\approx G3 - (G3 - G4)\Delta x \\
|
||||
\end{split}
|
||||
\]
|
||||
\end{minipage}
|
||||
\begin{minipage}{0.35\linewidth}
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{./img/_nms_interpolation.pdf}
|
||||
\end{minipage}
|
||||
|
||||
|
||||
|
||||
\section{Canny's edge detector}
|
||||
|
||||
Method based on three criteria:
|
||||
\begin{descriptionlist}
|
||||
\item[Good detection] Correctly detect edges in noisy images.
|
||||
\item[Good localization] Minimize the distance between the found edges and the true edges.
|
||||
\item[One response to one edge] Detect only one pixel at each true edge.
|
||||
\end{descriptionlist}
|
||||
|
||||
In the 1D case, the optimal operator consists in finding the local extrema of the
|
||||
signal obtained by convolving the image and the Gaussian first-order derivative.
|
||||
|
||||
Generalized for the 2D case, Canny's edge detector does the following: \marginnote{Canny's edge detector}
|
||||
\begin{enumerate}
|
||||
\item Gaussian smoothing.
|
||||
\item Gradient computation.
|
||||
\item NMS and thresholding.
|
||||
\end{enumerate}
|
||||
|
||||
It is possible to exploit the convolutions property of being commutative w.r.t. differentiation
|
||||
and simplify the smoothing and gradient computation:
|
||||
\[
|
||||
\begin{split}
|
||||
\partial_x I(x, y) &= \frac{\partial}{\partial x} ( I(x, y) * G(x, y) ) = I(x, y) * \frac{\partial G(x, y)}{\partial x} \\
|
||||
\partial_y I(x, y) &= \frac{\partial}{\partial y} ( I(x, y) * G(x, y) ) = I(x, y) * \frac{\partial G(x, y)}{\partial y}
|
||||
\end{split}
|
||||
\]
|
||||
By leveraging the separability of a 2D Gaussian ($G(x, y) = G_1(x)G_2(y)$),
|
||||
the computation can be reduced to 1D convolutions:
|
||||
\[
|
||||
\begin{split}
|
||||
\partial_x I(x, y) &= I(x, y) * (G_1'(x)G_2(y)) = (I(x, y) * G_1'(x)) * G_2(y) \\
|
||||
\partial_y I(x, y) &= I(x, y) * (G_1(x)G_2'(y)) = (I(x, y) * G_1(x)) * G_2'(y)
|
||||
\end{split}
|
||||
\]
|
||||
|
||||
\begin{remark}
|
||||
When magnitude varies along the object contour, thresholding might remove true edges (edge streaking).
|
||||
\end{remark}
|
||||
|
||||
\begin{description}
|
||||
\item[Hysteresis thresholding] \marginnote{Hysteresis thresholding}
|
||||
Given a high threshold $T_\text{h}$ and a low threshold $T_\text{l}$,
|
||||
depending on its magnitude, a pixel $(i, j)$ can be considered a:
|
||||
\begin{descriptionlist}
|
||||
\item[Strong edge] $\nabla I(i, j) > T_h$.
|
||||
\item[Weak edge] $\nabla I(i, j) > T_l$ and the pixel $(i, j)$ is a neighbor of a strong/weak edge.
|
||||
\end{descriptionlist}
|
||||
In practice, the algorithm starts from strong edges and "propagates" them.
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.5\linewidth]{./img/hysteresis thresholding.png}
|
||||
\caption{Application of hysteresis thresholding}
|
||||
\end{figure}
|
||||
\end{description}
|
||||
|
||||
\begin{remark}
|
||||
The output of Canny's edge detector is not an image with edges but a list of edges.
|
||||
|
||||
Note that if the edges of two objects intersect, this will be recognized as a single edge.
|
||||
\end{remark}
|
||||
|
||||
|
||||
|
||||
\section{Zero-crossing edge detector}
|
||||
|
||||
\begin{description}
|
||||
\item[Zero-crossing edge detector] \marginnote{Zero-crossing}
|
||||
Detect edges by finding zero-crossing of the second derivative of the signal.
|
||||
|
||||
\begin{remark}
|
||||
A zero-crossing is a point at 0 where the function changes sign.
|
||||
\end{remark}
|
||||
|
||||
\begin{remark}
|
||||
This approach does not require a threshold anymore but is computationally more expensive.
|
||||
\end{remark}
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\begin{subfigure}{0.3\linewidth}
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{./img/zero_crossing_example1.png}
|
||||
\caption{Input signal}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{0.3\linewidth}
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{./img/zero_crossing_example2.png}
|
||||
\caption{First-order derivative}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{0.3\linewidth}
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{./img/zero_crossing_example3.png}
|
||||
\caption{Second-order derivative}
|
||||
\end{subfigure}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\item[Laplacian] \marginnote{Laplacian}
|
||||
Approximation of the second-order derivative:
|
||||
\[ \nabla^2 I(x, y) \approx \frac{\partial^2 I(x, y)}{\partial x^2} + \frac{\partial^2 I(x, y)}{\partial y^2} = \partial_{x,x} I + \partial_{y, y} I \]
|
||||
|
||||
|
||||
\item[Discrete Laplacian] \marginnote{Discrete Laplacian}
|
||||
Use forward difference to compute first-order derivatives,
|
||||
followed by backward difference to compute second-order derivatives.
|
||||
\[
|
||||
\begin{split}
|
||||
\partial_{x,x} I(i, j) &\approx \partial_x I(i, j) - \partial_x I(i, j-1) \\
|
||||
&= \big ( I(i, j+1) - I(i, j) \big) - \big ( I(i, j) - I(i, j-1) \big) \\
|
||||
&= I(i, j+1) - 2I(i, j) + I(i, j-1)
|
||||
\end{split}
|
||||
\]
|
||||
\[
|
||||
\begin{split}
|
||||
\partial_{y,y} I(i, j) &\approx \partial_y I(i, j) - \partial_y I(i-1, j) \\
|
||||
&= \big ( I(i+1, j) - I(i, j) \big) - \big ( I(i, j) - I(i-1, j) \big) \\
|
||||
&= I(i+1, j) - 2I(i, j) + I(i-1, j)
|
||||
\end{split}
|
||||
\]
|
||||
|
||||
This is equivalent to applying the cross-correlation kernel:
|
||||
\[
|
||||
\nabla^2 = \begin{pmatrix}
|
||||
0 & 1 & 0 \\
|
||||
1 & -4 & 1 \\
|
||||
0 & 1 & 0 \\
|
||||
\end{pmatrix}
|
||||
\]
|
||||
|
||||
\begin{remark}
|
||||
It can be shown that zero-crossings of the Laplacian typically lay close to those of the second-order derivative.
|
||||
\end{remark}
|
||||
\end{description}
|
||||
|
||||
|
||||
\subsection{Laplacian of Gaussian (LoG)}
|
||||
|
||||
Laplacian of Gaussian (LoG) does the following: \marginnote{Laplacian of Gaussian (LoG)}
|
||||
\begin{enumerate}
|
||||
\item Gaussian smoothing.
|
||||
\item Second-order differentiation using the Laplacian filter.
|
||||
\item Zero-crossings extraction.
|
||||
\end{enumerate}
|
||||
|
||||
\begin{description}
|
||||
\item[Discrete zero-crossing]
|
||||
As the image consists of a discrete grid of pixels, finding a zero-crossing as per definition is not always possible.
|
||||
Instead, an edge is detected when there is a change of sign in the magnitude.
|
||||
The edge pixel can be identified as:
|
||||
\begin{itemize}
|
||||
\item The pixel where the magnitude is positive.
|
||||
\item The pixel where the magnitude is negative.
|
||||
\item The pixel where the absolute value of the magnitude is smaller.
|
||||
This is usually the best choice as it is closer to the true zero-crossing.
|
||||
\end{itemize}
|
||||
\end{description}
|
||||
|
||||
\begin{remark}
|
||||
A final thresholding might still be useful to remove uninteresting edges.
|
||||
\end{remark}
|
||||
|
||||
\begin{remark}
|
||||
Smaller values of $\sigma$ of the smoothing operator detect more detailed edges,
|
||||
while higher $\sigma$ captures more general edges.
|
||||
\end{remark}
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.8\linewidth]{./img/_example_laplacian_gaussian.pdf}
|
||||
\end{figure}
|
||||
@ -0,0 +1,532 @@
|
||||
\chapter{Image acquisition and formation}
|
||||
|
||||
|
||||
\section{Pinhole camera}
|
||||
|
||||
\begin{description}
|
||||
\item[Imaging device] \marginnote{Imaging device}
|
||||
Gathers the light reflected by 3D objects in a scene and creates a 2D representation of them.
|
||||
|
||||
\item[Computer vision] \marginnote{Computer vision}
|
||||
Infer knowledge of the 3D scene from 2D digital images.
|
||||
\end{description}
|
||||
|
||||
\begin{description}
|
||||
\item[Pinhole camera] \marginnote{Pinhole camera}
|
||||
Imaging device where the light passes through a small pinhole and hits the image plane.
|
||||
Geometrically, the image is obtained by drawing straight rays from the scene to the image plane passing through the pinhole.
|
||||
|
||||
\begin{remark}
|
||||
Larger aperture size of the pinhole results in blurry images (circle of confusion),
|
||||
while smaller aperture results in sharper images but requires longer exposure time (as less light passes through).
|
||||
\end{remark}
|
||||
|
||||
\begin{remark}
|
||||
The pinhole camera is a good approximation of the geometry of the image formation mechanism of modern imaging devices.
|
||||
\end{remark}
|
||||
|
||||
\begin{figure}[h]
|
||||
\begin{subfigure}{.4\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=0.8\linewidth]{./img/pinhole.png}
|
||||
\caption{Pinhole camera model}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{.45\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=0.7\linewidth]{./img/pinhole_hole_size.png}
|
||||
\caption{Images with varying pinhole aperture size}
|
||||
\end{subfigure}
|
||||
\end{figure}
|
||||
\end{description}
|
||||
|
||||
|
||||
|
||||
\section{Perspective projection}
|
||||
\marginnote{Perspective projection}
|
||||
|
||||
Geometric model of a pinhole camera.\\
|
||||
|
||||
\begin{minipage}{0.65\textwidth}
|
||||
\begin{description}
|
||||
\setlength\itemsep{0em}
|
||||
\item[Scene point] $M$ (the object in the real world).
|
||||
\item[Image point] $m$ (the object in the image).
|
||||
\item[Image plane] $I$.
|
||||
\item[Optical center] $C$ (the pinhole).
|
||||
\item[Image center/piercing point] $c$ (intersection between the optical axis -- the line orthogonal to $I$ passing through $C$ -- and $I$).
|
||||
\item[Focal length] $f$.
|
||||
\item[Focal plane] $F$.
|
||||
\end{description}
|
||||
\end{minipage}
|
||||
\begin{minipage}{0.3\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{./img/perspective_projection1.png}
|
||||
\end{minipage}\\
|
||||
|
||||
\begin{minipage}{0.55\textwidth}
|
||||
\begin{itemize}[leftmargin=*]
|
||||
\item $u$ and $v$ are the horizontal and vertical axis of the image plane, respectively.
|
||||
\item $x$ and $y$ are the horizontal and vertical axis of the 3D reference system, respectively,
|
||||
and form the \textbf{camera reference system}. \marginnote{Camera reference system}
|
||||
\end{itemize}
|
||||
|
||||
\begin{remark}
|
||||
For the perspective model, the coordinate systems $(U, V)$ and $(X, Y)$ must be parallel.
|
||||
\end{remark}
|
||||
\end{minipage}
|
||||
\begin{minipage}{0.35\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{./img/perspective_projection2.png}
|
||||
\end{minipage}
|
||||
|
||||
\begin{description}
|
||||
\item[Scene--image mapping] \marginnote{Scene--image mapping}
|
||||
The equations to map scene points into image points are the following:
|
||||
\[ u = x \frac{f}{z} \hspace*{3em} v = y \frac{f}{z} \]
|
||||
|
||||
\begin{proof}
|
||||
This is the consequence of the triangle similarity theorems.
|
||||
|
||||
\begin{minipage}{0.45\textwidth}
|
||||
\[
|
||||
\begin{split}
|
||||
\frac{u}{x} = -\frac{f}{z} &\iff u = -x \frac{f}{z} \\
|
||||
\frac{v}{y} = -\frac{f}{z} &\iff v = -y \frac{f}{z}
|
||||
\end{split}
|
||||
\]
|
||||
The minus is needed as the axes are inverted
|
||||
\end{minipage}
|
||||
\begin{minipage}{0.50\textwidth}
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.7\textwidth]{./img/_perspective_projection_eq_proof.pdf}
|
||||
\caption{\small Visualization of the horizontal axis. The same holds on the vertical axis.}
|
||||
\end{figure}
|
||||
\end{minipage}
|
||||
|
||||
By inverting the axis horizontally and vertically (i.e. inverting the sign),
|
||||
the image plane can be adjusted to have the same orientation of the scene:
|
||||
\[ u = x \frac{f}{z} \hspace*{3em} v = y \frac{f}{z} \]
|
||||
\end{proof}
|
||||
|
||||
\begin{remark}
|
||||
The image coordinates are a scaled version of the scene coordinates.
|
||||
The scaling is inversely proportioned with respect to the depth.
|
||||
\begin{itemize}
|
||||
\item The farther the point, the smaller the coordinates.
|
||||
\item The larger the focal length, the bigger the object is in the image.
|
||||
\end{itemize}
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.4\textwidth]{./img/perspective_projection_proportion.png}
|
||||
\caption{Coordinate space by varying focal length}
|
||||
\end{figure}
|
||||
\end{remark}
|
||||
|
||||
\begin{remark}
|
||||
The perspective projection mapping is not a bijection:
|
||||
\begin{itemize}
|
||||
\item A scene point is mapped into a unique image point.
|
||||
\item An image point is mapped onto a 3D line.
|
||||
\end{itemize}
|
||||
Therefore, reconstructing the 3D structure of a single image is an ill-posed problem (i.e. it has multiple solutions).
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.3\textwidth]{./img/perspective_projection_loss.png}
|
||||
\caption{Projection from scene and image points}
|
||||
\end{figure}
|
||||
\end{remark}
|
||||
\end{description}
|
||||
|
||||
|
||||
\subsection{Stereo geometry}
|
||||
|
||||
\begin{description}
|
||||
\item[Stereo vision] \marginnote{Stereo vision}
|
||||
Use multiple images to triangulate the 3D position of an object.
|
||||
|
||||
\item[Stereo correspondence] \marginnote{Stereo correspondence}
|
||||
Given a point $L$ in an image, find the corresponding point $R$ in another image.
|
||||
|
||||
Without any assumptions, an oracle is needed to determine the correspondences.
|
||||
\end{description}
|
||||
|
||||
\begin{description}
|
||||
\item[Standard stereo geometry] \marginnote{Standard stereo geometry}
|
||||
Given two reference images, the following assumptions must hold:
|
||||
\begin{itemize}
|
||||
\item The $X$, $Y$, $Z$ axes are parallel.
|
||||
\item The cameras that took the two images have the same focal length $f$ (coplanar image planes) and
|
||||
the images have been taken at the same time.
|
||||
\item There is a horizontal translation $b$ between the two cameras (baseline).
|
||||
\item The disparity $d$ is the difference of the $U$ coordinates of the object in the left and right image.
|
||||
\end{itemize}
|
||||
|
||||
\begin{theorem}[Fundamental relationship in stereo vision] \marginnote{Fundamental relationship in stereo vision}
|
||||
If the assumptions above hold, the following equation holds:
|
||||
\[ z = b\frac{f}{d} \]
|
||||
|
||||
\begin{proof}
|
||||
Let $P_L = \begin{pmatrix}x_L & y & z\end{pmatrix}$ and $P_R = \begin{pmatrix}x_R & y & z\end{pmatrix}$ be the
|
||||
coordinates of the object $P$ with respect to the left and right camera reference system, respectively.
|
||||
Let $p_L = \begin{pmatrix}u_L & v\end{pmatrix}$ and $p_R = \begin{pmatrix}u_R & v\end{pmatrix}$
|
||||
be the coordinates of the object $P$ in the left and right image plane, respectively.
|
||||
|
||||
By assumption, we have that $P_L - P_R = \begin{pmatrix} b & 0 & 0 \end{pmatrix}$, where $b$ is the baseline.
|
||||
|
||||
\begin{minipage}{0.6\textwidth}
|
||||
|
||||
By the perspective projection equation, we have that:
|
||||
\[ u_L = x_L\frac{f}{z} \hspace{3em} u_R = x_R\frac{f}{z} \]
|
||||
Disparity is computed as follows:
|
||||
\[ d = u_L - u_R = x_L\frac{f}{z} - x_R\frac{f}{z} = b\frac{f}{z} \]
|
||||
We can therefore obtain the $Z$ coordinate of $P$ as:
|
||||
\[ z = b\frac{f}{d} \]
|
||||
\end{minipage}
|
||||
\begin{minipage}{0.3\textwidth}
|
||||
\begin{center}
|
||||
\includegraphics[width=\textwidth]{./img/_standard_stereo_geometry.pdf}
|
||||
\end{center}
|
||||
Note: the $Y$/$V$ axes are not in figure.
|
||||
\end{minipage}\\
|
||||
\end{proof}
|
||||
|
||||
\begin{remark}
|
||||
Disparity and depth are inversely proportional:
|
||||
the disparity of two points decreases if the points are farther in depth.
|
||||
\end{remark}
|
||||
\end{theorem}
|
||||
|
||||
\begin{description}
|
||||
\item[Stereo matching] \marginnote{Stereo matching}
|
||||
If the assumptions for standard stereo geometry hold,
|
||||
to find the object corresponding to $p_L$ in another image,
|
||||
it is sufficient to search along the horizontal axis of $p_L$ looking for the same colors or patterns.
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.5\textwidth]{./img/stereo_matching.png}
|
||||
\caption{Example of stereo matching}
|
||||
\end{figure}
|
||||
\end{description}
|
||||
|
||||
\item[Epipolar geometry] \marginnote{Epipolar geometry}
|
||||
Approach applied when the two cameras are no longer aligned according to the standard stereo geometry assumption.
|
||||
Still, the focal lengths and the roto-translation between the two cameras must be known.
|
||||
|
||||
Given two images, we can project the epipolar line related to the point $p_L$ in the left plane onto the right plane
|
||||
to reduce the problem of correspondence search to a single dimension.
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.3\textwidth]{./img/_epipolar_geometry.pdf}
|
||||
\caption{Example of epipolar geometry}
|
||||
\end{figure}
|
||||
|
||||
\begin{remark}
|
||||
It is nearly impossible to project horizontal epipolar lines and
|
||||
searching through oblique lines is awkward and computationally less efficient than straight lines.
|
||||
\end{remark}
|
||||
|
||||
\begin{description}
|
||||
\item[Rectification] \marginnote{Rectification}
|
||||
Transformation applied to convert epipolar geometry to a standard stereo geometry.
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\begin{subfigure}{0.35\linewidth}
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{./img/rectification_no.png}
|
||||
\caption{Images before rectification}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{0.35\linewidth}
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{./img/rectification_yes.png}
|
||||
\caption{Images after rectification}
|
||||
\end{subfigure}
|
||||
\end{figure}
|
||||
\end{description}
|
||||
\end{description}
|
||||
|
||||
|
||||
\subsection{Ratios and parallelism}
|
||||
|
||||
Given a 3D line of length $L$ lying in a plane parallel to the image plane at distance $z$,
|
||||
then its length $l$ in the image plane is:
|
||||
\[ l = L\frac{f}{z} \]
|
||||
|
||||
In all the other cases (i.e. when the line is not parallel to the image plane),
|
||||
the ratios of lengths and the parallelism of lines are not preserved.
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.25\textwidth]{./img/_perspective_projection_ratio.pdf}
|
||||
\caption{Example of not preserved ratios. It holds that $\frac{\overline{AB}}{\overline{BC}} \neq \frac{\overline{ab}}{\overline{bc}}$.}
|
||||
\end{figure}
|
||||
|
||||
\begin{description}
|
||||
\item[Vanishing point] \marginnote{Vanishing point}
|
||||
Intersection point of lines that are parallel in the scene but not in the image plane.
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.6\textwidth]{./img/_vanishing_point.pdf}
|
||||
\caption{Example of vanishing point}
|
||||
\end{figure}
|
||||
\end{description}
|
||||
|
||||
|
||||
|
||||
\section{Lens}
|
||||
|
||||
\begin{description}
|
||||
\item[Depth of field (DOF)] \marginnote{Depth of field (DOF)}
|
||||
Distance at which a scene point is in focus (i.e. when all its light rays gathered by the imaging device hit the image plane at the same point).
|
||||
|
||||
\begin{remark}
|
||||
Because of the small size of the aperture, a pinhole camera has infinite depth of field
|
||||
but requires a long exposure time making it only suitable for static scenes.
|
||||
\end{remark}
|
||||
|
||||
|
||||
\item[Lens] \marginnote{Lens}
|
||||
A lens gathers more light from the scene point and focuses it on a single image point.
|
||||
|
||||
This allows for a smaller exposure time but limits the depth of field (i.e. only a limited range of distances in the image can be in focus at the same time).
|
||||
|
||||
|
||||
\item[Thin lens] \marginnote{Thin lens}
|
||||
Approximate model for lenses.
|
||||
|
||||
\begin{minipage}{0.65\textwidth}
|
||||
\begin{description}
|
||||
\setlength\itemsep{0em}
|
||||
\item[Scene point] $P$ (the object in the real world).
|
||||
\item[Image point] $p$ (the object in the image).
|
||||
\item[Object--lens distance] $o$.
|
||||
\item[Image--lens distance] $i$ (i.e. focal length of the camera).
|
||||
\item[Center of the lens] $C$.
|
||||
\item[Focal length of the lens] $f$.
|
||||
\item[Focal plane/focus of the lens] $F$.
|
||||
\end{description}
|
||||
\end{minipage}
|
||||
\begin{minipage}{0.4\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{./img/_thin_lens.pdf}
|
||||
\end{minipage}
|
||||
|
||||
A thin lens has the following properties:
|
||||
\begin{itemize}
|
||||
\item Rays hitting the lens parallel to the optical axis are deflected to pass through the focal plane of the lens $F$.
|
||||
\item Rays passing through the center of the lens $C$ are undeflected.
|
||||
\item The following equation holds: \marginnote{Thin lens equation}
|
||||
\[ \frac{1}{o} + \frac{1}{i} = \frac{1}{f} \]
|
||||
\end{itemize}
|
||||
|
||||
\item[Image formation]
|
||||
When the image is in focus, the image formation process follows the normal rules of the perspective projection model where:
|
||||
\begin{itemize}
|
||||
\item $C$ is the optical center.
|
||||
\item $i$ is the focal length of the camera.
|
||||
\end{itemize}
|
||||
|
||||
By fixing the focal length of the lens ($f$),
|
||||
we can determine the distance of the scene point ($o$) or the image point ($i$) required to have the object in focus.
|
||||
\[ \frac{1}{o}+\frac{1}{i} = \frac{1}{f} \iff o = \frac{if}{i - f} \hspace{3em} \frac{1}{o}+\frac{1}{i} = \frac{1}{f} \iff i = \frac{of}{o - f} \]
|
||||
|
||||
\begin{remark}
|
||||
Points projected in front or behind the image plane will create a circle of confusion (blur).
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\begin{subfigure}{0.3\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{./img/thin_lens_formation1.png}
|
||||
\caption{Image in focus}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{0.3\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=0.8\textwidth]{./img/thin_lens_formation2.png}
|
||||
\caption{Projection behind the image plane}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{0.3\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=\textwidth]{./img/thin_lens_formation3.png}
|
||||
\caption{Projection in front of the image plane}
|
||||
\end{subfigure}
|
||||
\end{figure}
|
||||
\end{remark}
|
||||
|
||||
\item[Adjustable diaphragm] \marginnote{Adjustable diaphragm}
|
||||
Device to control the light gathered by the effective aperture of the lens.
|
||||
|
||||
Reducing the aperture will result in less light and an increased depth of field.
|
||||
|
||||
\begin{remark}
|
||||
On a theoretical level, images that are not in focus appear blurred (circles of confusion).
|
||||
Despite that, if the circle is smaller than the photo-sensing elements (i.e. pixels), it will appear in focus.
|
||||
\end{remark}
|
||||
|
||||
\item[Focusing mechanism] \marginnote{Focusing mechanism}
|
||||
Allows the lens to translate along the optical axis to increase its distance to the image plane.
|
||||
|
||||
At the minimum extension (\Cref{fig:focus_mechanism_min}), we have that:
|
||||
\[ i = f \text{ and } o = \infty \text{ as the thin lens equation states that } \frac{1}{o} + \frac{1}{i} = \frac{1}{f} \]
|
||||
By increasing the extension (i.e. increase $i$), we have that the distance to the scene point $o$ decreases.
|
||||
The maximum extension determines the minimum focusing distance.
|
||||
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\begin{subfigure}{0.4\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=0.45\textwidth]{./img/_focus_mechanism1.pdf}
|
||||
\caption{Minimum extension} \label{fig:focus_mechanism_min}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{0.4\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=0.45\textwidth]{./img/_focus_mechanism2.pdf}
|
||||
\caption{Maximum extension} \label{fig:focus_mechanism_max}
|
||||
\end{subfigure}
|
||||
\caption{Extension of a focusing mechanism}
|
||||
\end{figure}
|
||||
\end{description}
|
||||
|
||||
|
||||
|
||||
\section{Image digitalization}
|
||||
|
||||
|
||||
\subsection{Sampling and quantization}
|
||||
|
||||
The image plane of a camera converts the received irradiance into electrical signals.
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.6\textwidth]{./img/_digitalization.pdf}
|
||||
\caption{Image digitalization steps}
|
||||
\end{figure}
|
||||
|
||||
\begin{description}
|
||||
\item[Sampling] \marginnote{Sampling}
|
||||
The continuous electrical signal is sampled to produce a $N \times M$ matrix of pixels:
|
||||
\[
|
||||
I(x, y) = \begin{pmatrix}
|
||||
I(0, 0) & \hdots & I(0, M-1) \\
|
||||
\vdots & \ddots & \vdots \\
|
||||
I(N-1, 0) & \hdots & I(N-1, M-1) \\
|
||||
\end{pmatrix}
|
||||
\]
|
||||
|
||||
\item[Quantization] \marginnote{Quantization}
|
||||
Let $m$ be the number of bits used to encode a pixel.
|
||||
The value of each pixel is quantized into $2^m$ discrete gray levels.
|
||||
|
||||
\begin{remark}
|
||||
A grayscale image usually uses $8$ bits
|
||||
|
||||
An RGB image usually uses $3 \cdot 8$ bits.
|
||||
\end{remark}
|
||||
\end{description}
|
||||
|
||||
\begin{remark}
|
||||
The more bits are used for the representation, the higher the quality of the image will be.
|
||||
\begin{itemize}
|
||||
\item Sampling with fewer bits will result in a lower resolution (aliasing).
|
||||
\item Quantization with fewer bits will result in less representable colors.
|
||||
\end{itemize}
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.7\textwidth]{./img/_digitalization_quality.pdf}
|
||||
\caption{Sampling and quantization using fewer bits}
|
||||
\end{figure}
|
||||
\end{remark}
|
||||
|
||||
|
||||
\subsection{Camera sensors}
|
||||
|
||||
\begin{description}
|
||||
\item[Photodetector] \marginnote{Photodetector}
|
||||
Sensor that, during the exposure time, converts the light into a proportional electrical charge that
|
||||
will be processed by a circuit and converted into a digital or analog signal.
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.55\textwidth]{./img/_camera_sensors.pdf}
|
||||
\caption{Components of a camera}
|
||||
\end{figure}
|
||||
|
||||
The two main sensor technologies are:
|
||||
\begin{descriptionlist}
|
||||
\item[Charge Coupled Device (CCD)] \marginnote{Charge Coupled Device (CCD)}
|
||||
Typically produces higher quality images but are more expensive.
|
||||
|
||||
\item[Complementary Metal Oxide Semiconductor (CMOS)] \marginnote{Complementary Metal Oxide Semiconductor (CMOS)}
|
||||
Generally produces lower quality images but is more compact and less expensive.
|
||||
Each sensor has integrated its own circuitry that allows to read an arbitrary window of the sensors.
|
||||
\end{descriptionlist}
|
||||
|
||||
\item[Color sensors] \marginnote{Color sensors}
|
||||
CCD and CMOS sensors are sensitive to a wide spectrum of light frequencies (both visible and invisible) but
|
||||
are unable to sense colors as they produce a single value per pixel.
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.35\textwidth]{./img/sensors_sensitivity.png}
|
||||
\caption{CCD and CMOS spectral sensitivity}
|
||||
\end{figure}
|
||||
|
||||
\begin{description}
|
||||
\item[Color Filter Array (CFA)] \marginnote{Color Filter Array (CFA)}
|
||||
Filter placed in front of a photodetector to allow it to detect colors.
|
||||
|
||||
Possible approaches are:
|
||||
\begin{descriptionlist}
|
||||
\item[Bayer CFA]
|
||||
A grid of green, blue, and red filters with the greens being twice as much as the others (the human eye is more sensible to the green range).
|
||||
To determine the RGB value of each pixel, missing color channels are sampled from neighboring pixels (demosaicking).
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.15\textwidth]{./img/_bayer_cfa.pdf}
|
||||
\caption{Example of Bayer filter}
|
||||
\end{figure}
|
||||
|
||||
\item[Optical prism]
|
||||
A prism splits the incoming light into 3 RGB beams, each directed to a different sensor.
|
||||
It is more expensive than Bayer CFA.
|
||||
\end{descriptionlist}
|
||||
\end{description}
|
||||
\end{description}
|
||||
|
||||
|
||||
\subsection{Metrics}
|
||||
|
||||
\begin{description}
|
||||
\item[Signal to Noise Ratio (SNR)] \marginnote{Signal to Noise Ratio (SNR)}
|
||||
Quantifies the strength of the actual signal with respect to unwanted noise.
|
||||
|
||||
Sources of noise are:
|
||||
\begin{descriptionlist}
|
||||
\item[Photon shot noise] Number of photons captured during exposure time.
|
||||
\item[Elecronic circuitry noise] Generated by the electronics that read the sensors.
|
||||
\item[Quantization noise] Caused by the digitalization of the image (ADC conversion).
|
||||
\item[Dark current noise] Random charge caused by thermal excitement.
|
||||
\end{descriptionlist}
|
||||
|
||||
SNR is usually expressed in decibels or bits:
|
||||
\[ \texttt{SNR}_\text{db} = 20 \cdot \log_{10}(\texttt{SNR}) \hspace{3em} \texttt{SNR}_\text{bit} = \log_{2}(\texttt{SNR}) \]
|
||||
|
||||
\item[Dynamic Range (DR)] \marginnote{Dynamic Range (DR)}
|
||||
Measures the ability of a sensor to capture both the dark and bright structure of the scene.
|
||||
|
||||
Let:
|
||||
\begin{itemize}
|
||||
\item $E_\text{min}$ be the minimum detectable irradiation. This value depends on the noise.
|
||||
\item $E_\text{max}$ be the saturation irradiation (i.e. the maximum amount of light that fills the capacity of the photodetector).
|
||||
\end{itemize}
|
||||
DR is defined as:
|
||||
\[ \texttt{DR} = \frac{E_\text{max}}{E_\text{mix}} \]
|
||||
As with SNR, DR can be expressed in decibels or bits.
|
||||
\end{description}
|
||||
@ -0,0 +1,350 @@
|
||||
\chapter{Instance-level object detection}
|
||||
|
||||
|
||||
\begin{description}
|
||||
\item[Instance-level object detection] \marginnote{Instance-level object detection}
|
||||
Given a reference/model image of a specific object,
|
||||
determine if the object is present in the target image and estimate its pose.
|
||||
|
||||
\begin{remark}
|
||||
This is different from category-level object detection which deals with detecting classes of objects
|
||||
independent of appearance and pose.
|
||||
\end{remark}
|
||||
\end{description}
|
||||
|
||||
|
||||
|
||||
\section{Template matching}
|
||||
\marginnote{Template matching}
|
||||
|
||||
Slide the model image (template) across the target image and
|
||||
use a similarity/dissimilarity function with a threshold to determine if a match has been found.
|
||||
|
||||
|
||||
\subsection{Similarity/dissimilarity functions}
|
||||
|
||||
Let $I$ be the target image and $T$ the template image of shape $M \times N$.
|
||||
Possible similarity/dissimilarity functions are:
|
||||
\begin{descriptionlist}
|
||||
\item[Pixel-wise intensity differences] \marginnote{Pixel-wise intensity differences}
|
||||
Squared difference between the intensities of the template image and the patch in the target image:
|
||||
\[ \texttt{SSD}(i, j) = \sum_{m=0}^{M-1} \sum_{n=0}^{N-1} \big( I(i+m, j+n) - T(m, n) \big)^2 \]
|
||||
|
||||
\begin{remark}
|
||||
\texttt{SSD} is fast but not intensity invariant.
|
||||
\end{remark}
|
||||
|
||||
\item[Sum of absolute differences] \marginnote{Sum of absolute differences}
|
||||
Difference in absolute value between the intensities of the template image and the patch in the target image:
|
||||
\[ \texttt{SAD}(i, j) = \sum_{m=0}^{M-1} \sum_{n=0}^{N-1} \big\vert I(i+m, j+n) - T(m, n) \big\vert \]
|
||||
|
||||
\begin{remark}
|
||||
\texttt{SAD} is fast but not intensity invariant.
|
||||
\end{remark}
|
||||
|
||||
\item[Normalized cross-correlation] \marginnote{Normalized cross-correlation}
|
||||
Cosine similarity between the template image and the target image patch (which are seen as flattened vectors):
|
||||
\[
|
||||
\texttt{NCC}(i, j) =
|
||||
\frac{ \sum\limits_{m=0}\limits^{M-1} \sum\limits_{n=0}\limits^{N-1} \big( I(i+m, j+n) \cdot T(m, n) \big) }
|
||||
{ \sqrt{\sum\limits_{m=0}\limits^{M-1} \sum\limits_{n=0}\limits^{N-1} I(i+m, j+n)^2} \cdot \sqrt{\sum\limits_{m=0}\limits^{M-1} \sum\limits_{n=0}\limits^{N-1} T(m, n)^2} }
|
||||
\]
|
||||
|
||||
In other words, let $\tilde{I}_{i,j}$ be a $M \times N$ patch of the target image $I$ starting at the coordinates $(i, j)$.
|
||||
In vector form, $\texttt{NCC}(i, j)$ computes the cosine of the angle $\theta$ between $\tilde{I}_{i,j}$ and $T$.
|
||||
\[ \texttt{NCC}(i, j) = \frac{\tilde{I}_{i,j} \cdot T}{\Vert \tilde{I}_{i,j} \Vert \cdot \Vert T \Vert} = \cos \theta \]
|
||||
|
||||
\begin{remark}
|
||||
\texttt{NCC} is invariant to a linear intensity change but not to an additive bias.
|
||||
\end{remark}
|
||||
|
||||
\item[Zero-mean normalized cross-correlation] \marginnote{Zero-mean normalized cross-correlation}
|
||||
Let $\tilde{I}_{i,j}$ be a $M \times N$ patch of $I$ starting at the coordinates $(i, j)$.
|
||||
Zero-mean normalized cross-correlation subtracts the mean of the template image and the patch of the target image
|
||||
before computing \texttt{NCC}:
|
||||
\[ \mu(\tilde{I}_{i,j}) = \frac{1}{MN} \sum_{m=0}^{M-1} \sum_{n=0}^{N-1} I(i+m, j+n) \hspace{3em} \mu(T) = \frac{1}{MN} \sum_{m=0}^{M-1} \sum_{n=0}^{N-1} T(m, n) \]
|
||||
\[
|
||||
\texttt{NCC}(i, j) =
|
||||
\frac{ \sum\limits_{m=0}\limits^{M-1} \sum\limits_{n=0}\limits^{N-1} \Big( \big(I(i+m, j+n) - \mu(\tilde{I}_{i,j})\big) \cdot \big(T(m, n) - \mu(T)\big) \Big) }
|
||||
{ \sqrt{\sum\limits_{m=0}\limits^{M-1} \sum\limits_{n=0}\limits^{N-1} \big(I(i+m, j+n) - \mu(\tilde{I}_{i,j})\big)^2} \cdot \sqrt{\sum\limits_{m=0}\limits^{M-1} \sum\limits_{n=0}\limits^{N-1} \big(T(m, n) - \mu(T)\big)^2} }
|
||||
\]
|
||||
|
||||
\begin{remark}
|
||||
\texttt{ZNCC} is invariant to an affine intensity change.
|
||||
\end{remark}
|
||||
\end{descriptionlist}
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.95\linewidth]{./img/template_matching_example.png}
|
||||
\caption{Examples of template matching}
|
||||
\end{figure}
|
||||
|
||||
|
||||
|
||||
\section{Shape-based matching}
|
||||
\marginnote{Shape-based matching}
|
||||
|
||||
Edge-based template matching that works as follows:
|
||||
\begin{enumerate}
|
||||
\item Use an edge detector to extract a set of control points $\{ P_1, \dots, P_n \}$ in the template image $T$.
|
||||
\item Compute the gradient normalized to unit vector at each point $P_k$:
|
||||
\[ \nabla T(P_k) = \begin{pmatrix} \partial_x T(P_k) \\ \partial_y T(P_k) \end{pmatrix} \hspace{2em} \vec{u}_k(P_k) = \frac{\nabla T(P_k)}{\Vert \nabla T(P_k) \Vert} \]
|
||||
\item Given a patch $\tilde{I}_{i,j}$ of the target image,
|
||||
compute the gradient normalized to unit vector at the points $\{ \tilde{P}_1(i, y), \dots, \tilde{P}_n(i, y) \}$
|
||||
corresponding to the control points of the template image:
|
||||
\[
|
||||
\nabla \tilde{I}_{i,j}(\tilde{P}_k) = \begin{pmatrix} \partial_x \tilde{I}_{i,j}(\tilde{P}_k) \\ \partial_y \tilde{I}_{i,j}(\tilde{P}_k) \end{pmatrix} \hspace{2em}
|
||||
\tilde{\vec{u}}_k(\tilde{P}_k) = \frac{\nabla \tilde{I}_{i,j}(\tilde{P}_k)}{\Vert \nabla \tilde{I}_{i,j}(\tilde{P}_k) \Vert}
|
||||
\]
|
||||
\item Compute the similarity as the sum of the cosine similarities of each pair of gradients:
|
||||
\[ S(i, j) = \frac{1}{n} \sum_{k=1}^{n} \vec{u}_k(P_k) \cdot \tilde{\vec{u}}_k(\tilde{P}_k) = \frac{1}{n} \sum_{k=1}^{n} \cos \theta_k \in [-1, 1] \]
|
||||
$S(i, j) = 1$ when the gradients perfectly match. A minimum threshold $S_\text{min}$ is used to determine if there is a match.
|
||||
\end{enumerate}
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.5\linewidth]{./img/shape_based_matching.png}
|
||||
\caption{Example of control points matching}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\subsection{Invariance to global inversion of contrast polarity}
|
||||
|
||||
As an object might appear on a darker or brighter background, more robust similarity functions can be employed:
|
||||
\begin{description}
|
||||
\item[Global polarity inversion contrast]
|
||||
\[ S(i, j) = \frac{1}{n} \left\vert \sum_{k=1}^{n} \vec{u}_k(P_k) \cdot \tilde{\vec{u}}_k(\tilde{P}_k) \right\vert =
|
||||
\frac{1}{n} \left\vert \sum_{k=1}^{n} \cos \theta_k \right\vert \]
|
||||
|
||||
\item[Local polarity inversion contrast]
|
||||
\[ S(i, j) = \frac{1}{n} \sum_{k=1}^{n} \left\vert \vec{u}_k(P_k) \cdot \tilde{\vec{u}}_k(\tilde{P}_k) \right\vert =
|
||||
\frac{1}{n} \sum_{k=1}^{n} \left\vert \cos \theta_k \right\vert \]
|
||||
|
||||
\begin{remark}
|
||||
This is the most robust one.
|
||||
\end{remark}
|
||||
\end{description}
|
||||
|
||||
|
||||
|
||||
\section{Hough transform}
|
||||
|
||||
Detect objects of a known shape that can be expressed through an analytic equation
|
||||
by means of a projection from the image space to a parameter space.
|
||||
|
||||
\begin{description}
|
||||
\item[Parameter space] \marginnote{Parameter space}
|
||||
Euclidean space parametrized on the parameters $\phi$ of an analytic shape $\mathcal{S}_\phi$ (e.g. line, sphere, \dots).
|
||||
A point $(x, y)$ in the image space is projected into the curve (which also intends lines) that contains the set of parameters $\hat{\phi}$
|
||||
such that the shape $\mathcal{S}_{\hat{\phi}}$ defined on those parameters passes through $(x, y)$ in the image space.
|
||||
|
||||
\begin{remark}
|
||||
If many curves intersect at $\hat{\phi}$ in the parameter space of a shape $S_\phi$, then, there is high evidence of the fact that
|
||||
the image points that were projected into those curves are part of the shape $S_{\hat{\phi}}$.
|
||||
\end{remark}
|
||||
|
||||
\begin{example}[Space of straight lines]
|
||||
Consider the line equation:
|
||||
\[ \hat{y} - m\hat{x} - c = 0 \]
|
||||
where $(\hat{x}, \hat{y})$ are fixed while $(m, c)$ vary (in the usual equation, it is the opposite).
|
||||
|
||||
This can be seen as a mapping of points $(\hat{x}, \hat{y})$ into the space
|
||||
of points $(m, c)$ such that the straight line parametrized on $m$ and $c$ passes through $(\hat{x}, \hat{y})$
|
||||
(i.e. a point in the image space is mapped into a line in the parameter space as infinite lines pass through a point).
|
||||
|
||||
For instance, consider two points $p_1$, $p_2$ in the image space and
|
||||
their projection in the parameter space.
|
||||
If the two lines intersect at the point $(\tilde{m}, \tilde{c})$,
|
||||
then the line parametrized on $\tilde{m}$ and $\tilde{c}$ passes through $p_1$ and $p_2$ in the image space.
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.4\linewidth]{./img/hough_line_parameter_space.png}
|
||||
\end{figure}
|
||||
|
||||
\begin{remark}
|
||||
By projecting $n$ points of the image space, there are at most $\frac{n(n-1)}{2}$ intersections in the parameter space.
|
||||
\end{remark}
|
||||
\end{example}
|
||||
|
||||
\item[Algorithm] \marginnote{Object detection using Hough transform}
|
||||
Given an analytic shape that we want to detect,
|
||||
object detection using the Hough transform works as follows:
|
||||
\begin{enumerate}
|
||||
\item Map image points into curves of the parameter space.
|
||||
\item The parameter space is quantized into $M \times N$ cells and an accumulator array $A$ of the same shape is initialized to $0$.
|
||||
\item For each cell $(i, j)$ of the discretized grid,
|
||||
the corresponding cell $A(i, j)$ in the accumulator array counts how many curves lie in that cell (i.e. voting process).
|
||||
\item Find the local maxima of $A$ and apply a threshold if needed.
|
||||
The points that were projected into the curves passing through a maximum cell are points belonging to an object of the sought shape.
|
||||
\end{enumerate}
|
||||
|
||||
\begin{remark}
|
||||
The Hough transform reduces a global detection problem into a local detection in the parameter space.
|
||||
\end{remark}
|
||||
|
||||
\begin{remark}
|
||||
The Hough transform is usually preceded by an edge detection phase so that the input consists of the edge pixels of the image.
|
||||
\end{remark}
|
||||
|
||||
\begin{remark}
|
||||
The Hough transform is robust to noise and can detect partially occluded images (with a suitable threshold).
|
||||
\end{remark}
|
||||
\end{description}
|
||||
|
||||
|
||||
\subsection{Hough transform for line detection}
|
||||
\marginnote{Hough transform for line detection}
|
||||
|
||||
For line detection, points in the image space are projected into lines of the parameter space.
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.85\linewidth]{./img/hough_line_detection.png}
|
||||
\caption{Example of line detection}
|
||||
\end{figure}
|
||||
|
||||
In practice, the parametrization on $(m, c)$ of the equation $y-mx-c = 0$ is impractical as $m$ and $c$ have an infinite range.
|
||||
An alternative approach is to describe straight lines as linear trigonometric equations parametrized on $(\theta, \rho)$:
|
||||
\[ x \cos \theta + y \sin \theta - \rho = 0 \]
|
||||
In this form, $\theta \in [ -\frac{\pi}{2}, \frac{\pi}{2} ]$ while $\rho \in [-\rho_\text{max}, \rho_\text{max}]$
|
||||
where $\rho_\text{max}$ is usually taken of the same size as the diagonal of the image (e.g. for square $N \times N$ images, $\rho_\text{max} = N\sqrt{2}$).
|
||||
|
||||
|
||||
|
||||
\section{Generalized Hough transform}
|
||||
|
||||
Hough transform extended to detect an arbitrary shape.
|
||||
|
||||
|
||||
\subsection{Naive approach}
|
||||
|
||||
\begin{description}
|
||||
\item[Off-line phase] \marginnote{Generalized Hough transform}
|
||||
Phase in which the model of the template object is built.
|
||||
Given a template shape, the algorithm works as follows:
|
||||
\begin{enumerate}
|
||||
\item Fix a reference point $\vec{y}$ (barycenter). $\vec{y}$ is typically within the shape of the object.
|
||||
\item For each point $\vec{x}$ at the border $B$ of the object:
|
||||
\begin{enumerate}
|
||||
\item Compute its gradient direction $\varphi(\vec{x})$ and discretize it according to a chosen step $\Delta \varphi$.
|
||||
\item Compute the vector $\vec{r} = \vec{y} - \vec{x}$ as the distance of $\vec{x}$ to the barycenter.
|
||||
\item Store $\vec{r}$ as a function of $\Delta \varphi$ in a table (R-table).
|
||||
Note that more than one vector might be associated with the same $\Delta \varphi$.
|
||||
\end{enumerate}
|
||||
\end{enumerate}
|
||||
|
||||
\begin{example}
|
||||
\phantom{}\\[0.5em]
|
||||
\begin{minipage}{0.7\linewidth}
|
||||
\small
|
||||
\begin{tabular}{lll}
|
||||
\toprule
|
||||
$i$ & $\varphi_i$ & $R_{\varphi_i}$ \\
|
||||
\midrule
|
||||
$0$ & $0$ & $\{ \vec{r} \mid \vec{r}=\vec{y}-\vec{x}.\, \forall \vec{x} \in B: \varphi(\vec{x}) = 0 \}$ \\
|
||||
$1$ & $\Delta\varphi$ & $\{ \vec{r} \mid \vec{r}=\vec{y}-\vec{x}.\, \forall \vec{x} \in B: \varphi(\vec{x}) = \Delta\varphi \}$ \\
|
||||
$2$ & $2\Delta\varphi$ & $\{ \vec{r} \mid \vec{r}=\vec{y}-\vec{x}.\, \forall \vec{x} \in B: \varphi(\vec{x}) = 2\Delta\varphi \}$ \\
|
||||
\multicolumn{1}{c}{\vdots} & \multicolumn{1}{c}{\vdots} & \multicolumn{1}{c}{\vdots} \\
|
||||
\bottomrule
|
||||
\end{tabular}
|
||||
\end{minipage}
|
||||
\begin{minipage}{0.2\linewidth}
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{./img/generalized_hough_offline.png}
|
||||
\end{minipage}
|
||||
\end{example}
|
||||
|
||||
\item[On-line phase]
|
||||
Phase in which the object is detected.
|
||||
Given a $M \times N$ image, the algorithm works as follows:
|
||||
\begin{enumerate}
|
||||
\item Find the edges $E$ of the image.
|
||||
\item Initialize an accumulator array $A$ of the same shape of the image.
|
||||
\item For each edge pixel $\vec{x} \in E$:
|
||||
\begin{enumerate}
|
||||
\item Compute its gradient direction $\varphi(\vec{x})$ discretized to match the step $\Delta \varphi$ of the R-table.
|
||||
\item For each $\vec{r}_i$ in the corresponding row of the R-table:
|
||||
\begin{enumerate}
|
||||
\item Compute an estimate of the barycenter as $\vec{y} = \vec{x} - \vec{r}_i$.
|
||||
\item Cast a vote in the accumulator array $A[\vec{y}] \texttt{+=} 1$
|
||||
\end{enumerate}
|
||||
\end{enumerate}
|
||||
\item Find the local maxima of the accumulator vector to estimate the barycenters.
|
||||
The shape can then be visually found by overlaying the template barycenter to the found barycenters.
|
||||
\end{enumerate}
|
||||
\end{description}
|
||||
|
||||
\begin{remark}
|
||||
This approach is not rotation and scale invariant.
|
||||
|
||||
Therefore, if rotation or scale is changed, the method should try different rotations or scales.
|
||||
\end{remark}
|
||||
|
||||
|
||||
\subsection{Star model}
|
||||
\marginnote{Star model}
|
||||
|
||||
Generalized Hough transform based on local invariant features.
|
||||
|
||||
% \begin{remark}
|
||||
% Local invariant features usually prune features found along edges.
|
||||
% \end{remark}
|
||||
|
||||
\begin{description}
|
||||
\item[Off-line phase]
|
||||
Given a template, its model is obtained as follows:
|
||||
\begin{enumerate}
|
||||
\item Detect local invariant features $F = \{ F_1, \dots, F_N \}$ and compute their descriptors.
|
||||
Each feature $F_i$ is described by the tuple:
|
||||
\[ F_i = (\vec{P}_i, \varphi_i, S_i, \vec{D}_i) = \text{(position, canonical orientation, scale, descriptor)} \]
|
||||
\item Compute the position of the barycenter $\vec{P}_C$ as:
|
||||
\[ \vec{P}_C = \frac{1}{N} \sum_{i=1}^{N} \vec{P}_i \]
|
||||
\item For each feature $F_i$ compute the joining vector $\vec{V}_i = \vec{P}_C - \vec{P}_i$ as the distance to the barycenter
|
||||
and add it to its associated tuple to obtain:
|
||||
\[ F_i = (\vec{P}_i, \varphi_i, S_i, \vec{D}_i, \vec{V}_i) \]
|
||||
\end{enumerate}
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.2\linewidth]{./img/star_model_offline.png}
|
||||
\end{figure}
|
||||
|
||||
\begin{remark}
|
||||
The R-table is not needed anymore.
|
||||
\end{remark}
|
||||
|
||||
\item[On-line phase]
|
||||
Given a target image, detection works as follows:
|
||||
\begin{enumerate}
|
||||
\item Extract the local invariant features $\tilde{F} = \{ \tilde{F}_1, \dots, \tilde{F}_M \}$ of the target image.
|
||||
Each feature $\tilde{F}_i$ is describe by the tuple:
|
||||
\[ \tilde{F}_i = (\tilde{\vec{P}}_i, \tilde{\varphi}_i, \tilde{S}_i, \tilde{\vec{D}}_i) \]
|
||||
\item Match the features $\tilde{F}$ of the target image to the features $F$ of the template image through the descriptors.
|
||||
\item Initialize a 4-dimensional accumulator array. Two dimensions match the image shape.
|
||||
The other two represent different rotations and scales.
|
||||
\item For each target feature $\tilde{F}_i = (\tilde{\vec{P}}_i, \tilde{\varphi}_i, \tilde{S}_i, \tilde{\vec{D}}_i)$
|
||||
with matching template feature $F_j = (\vec{P}_j, \varphi_j, S_j, \vec{D}_j, \vec{V}_j)$:
|
||||
\begin{enumerate}
|
||||
\item Align the joining vector $\vec{V}_j$ to the scale and rotation of $\tilde{F}_i$.
|
||||
|
||||
Let $\matr{R}(\phi)$ be a $\phi$ degree rotation matrix. The aligned joining vector $\tilde{\vec{V}}_i$ is obtained as:
|
||||
\[ \Delta \varphi_i = \tilde{\varphi}_i - \varphi_j \hspace{3em} \Delta S_i = \frac{\tilde{S}_i}{S_j} \]
|
||||
\[ \tilde{\vec{V}}_i = \Delta S_i \cdot \matr{R}(\Delta \varphi_i) \vec{V}_i \]
|
||||
\item Estimate the barycenter $\tilde{\vec{P}}_{C_i}$ associated to the feature $\tilde{F}_i$ as:
|
||||
\[ \tilde{\vec{P}}_{C_i} = \tilde{\vec{P}}_i + \tilde{\vec{V}}_i \]
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.55\linewidth]{./img/star_model_online.png}
|
||||
\end{figure}
|
||||
\item Cast a vote in the accumulator array $A[\tilde{\vec{P}}_{C_i}, \Delta S_i, \Delta \varphi_i] \texttt{+=} 1$.
|
||||
\end{enumerate}
|
||||
\item Find the local maxima of the accumulator vector to estimate the barycenters.
|
||||
The shape can then be visually found by overlaying the template barycenter to the found barycenters with the proper scaling and rotation.
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.6\linewidth]{./img/star_model_voting.png}
|
||||
\end{figure}
|
||||
\end{enumerate}
|
||||
\end{description}
|
||||
@ -0,0 +1,539 @@
|
||||
\chapter{Local features}
|
||||
|
||||
\begin{description}
|
||||
\item[Correspondence points] \marginnote{Correspondence points}
|
||||
Image points projected from the same 3D point from different views of the scene.
|
||||
|
||||
\begin{example}[Homography]
|
||||
Align two images of the same scene to create a larger image.
|
||||
Homography requires at least 4 correspondences.
|
||||
To find them, it does the following:
|
||||
\begin{itemize}
|
||||
\item Independently find salient points in the two images.
|
||||
\item Compute a local description of the salient points.
|
||||
\item Compare descriptions to find matching points.
|
||||
\end{itemize}
|
||||
\end{example}
|
||||
|
||||
|
||||
\item[Local invariant features] \marginnote{Local invariant features}
|
||||
Find correspondences in three steps:
|
||||
\begin{descriptionlist}
|
||||
\item[Detection] \marginnote{Detection}
|
||||
Find salient points (keypoints).
|
||||
|
||||
The detector should have the following properties:
|
||||
\begin{descriptionlist}
|
||||
\item[Repeatability] Find the same keypoints across different images.
|
||||
\item[Saliency] Find keypoints surrounded by informative patterns.
|
||||
\item[Fast] As it must scan the entire image.
|
||||
\end{descriptionlist}
|
||||
|
||||
|
||||
\item[Description] \marginnote{Description}
|
||||
Compute a descriptor for each salient point based on its neighborhood.
|
||||
|
||||
A descriptor should have the following properties:
|
||||
\begin{descriptionlist}
|
||||
\item[Invariant] Robust to as many transformations as possible (i.e. illumination, weather, scaling, viewpoint, \dots).
|
||||
\item[Distinctiveness/robustness trade-off] The description should only capture important information around a keypoint and
|
||||
ignore irrelevant features or noise.
|
||||
\item[Compactness] The description should be concise.
|
||||
\end{descriptionlist}
|
||||
|
||||
|
||||
\item[Matching] \marginnote{Matching}
|
||||
Identify the same descriptor across different images.
|
||||
\end{descriptionlist}
|
||||
\end{description}
|
||||
|
||||
\begin{remark}
|
||||
Edges are not good interest points as they are locally ambiguous (i.e. pixels are very similar along the direction of the gradient).
|
||||
|
||||
Corners on the other hand are more suited as they have a larger variation along all directions.
|
||||
\end{remark}
|
||||
|
||||
|
||||
\section{Moravec's corner detector}
|
||||
\marginnote{Moravec's corner detector}
|
||||
|
||||
Given a window $W$ of size $n \times n$,
|
||||
the cornerness of a pixel $p$ is given by the minimum squared difference between
|
||||
the intensity of $W$ centered on $p$ and the intensity of $W$ centered on each of its neighbors:
|
||||
\[ C(p) = \min_{q \in \mathcal{N}(p)} \Vert W(p) - W(q) \Vert^2 \]
|
||||
|
||||
After computing the cornerness of each pixel, one can apply thresholding and then NMS to obtain a matrix where $1$ indicates a corner.
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\begin{subfigure}{0.3\linewidth}
|
||||
\includegraphics[width=0.9\linewidth]{./img/_corner_detector_example_flat.pdf}
|
||||
\caption{Flat region: $C(p)$ is low.}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{0.3\linewidth}
|
||||
\includegraphics[width=0.8\linewidth]{./img/_corner_detector_example_edge.pdf}
|
||||
\caption{Edge: $C(p)$ is low.}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{0.3\linewidth}
|
||||
\includegraphics[width=0.8\linewidth]{./img/_corner_detector_example_corner.pdf}
|
||||
\caption{Corner: $C(p)$ is high.}
|
||||
\end{subfigure}
|
||||
\end{figure}
|
||||
|
||||
\begin{remark}
|
||||
Moravec corner detector is isotropic (i.e. independent of the direction).
|
||||
\end{remark}
|
||||
|
||||
|
||||
|
||||
\section{Harris' corner detector}
|
||||
|
||||
\subsection{Structure matrix}
|
||||
|
||||
Harris' corner detector uses an error function formulated as the continuous version of Moravec's detector and
|
||||
assumes an infinitesimal shift $(\Delta x, \Delta y)$ of the image:
|
||||
\[ E(\Delta x, \Delta y) = \sum_{x, y} w(x, y) \big( I(x+\Delta x, y+\Delta y) - I(x, y) \big)^2 \]
|
||||
where $w(x, y)$ in a window centered on $(x, y)$ and can be seen as a mask with $1$ when the pixel belongs to the window and $0$ otherwise.
|
||||
|
||||
By employing the Taylor's series $f(x + \Delta x) = f(x) + f'(x) \Delta x$,
|
||||
we can expand the intensity difference as:
|
||||
\[
|
||||
\begin{split}
|
||||
I(x+\Delta x, y+\Delta y) - I(x, y) &\approx \big( I(x, y) + \partial_x I(x, y)\Delta x + \partial_y I(x, y)\Delta y \big) - I(x, y) \\
|
||||
&= \partial_x I(x, y)\Delta x + \partial_y I(x, y)\Delta y
|
||||
\end{split}
|
||||
\]
|
||||
|
||||
By developing the error function into matrix form, we obtain the following:
|
||||
\[
|
||||
\begin{split}
|
||||
E(\Delta x, \Delta y) &= \sum_{x, y} w(x, y) \big( I(x+\Delta x, y+\Delta y) - I(x, y) \big)^2 \\
|
||||
&= \sum_{x, y} w(x, y) \big( \partial_x I(x, y)\Delta x + \partial_y I(x, y)\Delta y \big)^2 \\
|
||||
&= \sum_{x, y} w(x, y) \big( \partial_x^2 I(x, y)\Delta x^2 + 2 \partial_x I(x, y) \partial_y I(x, y) \Delta x \Delta y + \partial_y^2 I(x, y)\Delta y^2 \big) \\
|
||||
&= \sum_{x, y} w(x, y) \left(
|
||||
\begin{bmatrix} \Delta x & \Delta y \end{bmatrix}
|
||||
\begin{bmatrix}
|
||||
\partial_x^2 I(x, y) & \partial_x I(x, y) \partial_y I(x, y) \\
|
||||
\partial_x I(x, y) \partial_y I(x, y) & \partial_y^2 I(x, y)
|
||||
\end{bmatrix}
|
||||
\begin{bmatrix} \Delta x \\ \Delta y \end{bmatrix}
|
||||
\right) \\
|
||||
&= \begin{bmatrix} \Delta x & \Delta y \end{bmatrix}
|
||||
\begin{bmatrix}
|
||||
\sum_{x, y} w(x, y) \partial_x^2 I(x, y) & \sum_{x, y} w(x, y) (\partial_x I(x, y) \partial_y I(x, y)) \\
|
||||
\sum_{x, y} w(x, y) (\partial_x I(x, y) \partial_y I(x, y)) & \sum_{x, y} w(x, y) \partial_y^2 I(x, y)
|
||||
\end{bmatrix}
|
||||
\begin{bmatrix} \Delta x \\ \Delta y \end{bmatrix} \\
|
||||
&= \begin{bmatrix} \Delta x & \Delta y \end{bmatrix}
|
||||
\matr{M}
|
||||
\begin{bmatrix} \Delta x \\ \Delta y \end{bmatrix} \\
|
||||
\end{split}
|
||||
\]
|
||||
|
||||
\begin{description}
|
||||
\item[Structure matrix] \marginnote{Structure matrix}
|
||||
Matrix $\matr{M}_w$ that encodes the local structure of the image at the pixels within a window $w$.
|
||||
\[ \matr{M}_w = \begin{pmatrix}
|
||||
\sum_{x, y} w(x, y) \partial_x^2 I(x, y) & \sum_{x, y} w(x, y) (\partial_x I(x, y) \partial_y I(x, y)) \\
|
||||
\sum_{x, y} w(x, y) (\partial_x I(x, y) \partial_y I(x, y)) & \sum_{x, y} w(x, y) \partial_y^2 I(x, y)
|
||||
\end{pmatrix} \]
|
||||
|
||||
$\matr{M}_w$ is real and symmetric, thus it is diagonalizable through an orthogonal matrix $\matr{R}$:
|
||||
\[ \matr{M}_w = \matr{R} \begin{pmatrix} \lambda_1^{(w)} & 0 \\ 0 & \lambda_2^{(w)} \end{pmatrix} \matr{R}^T \]
|
||||
$\matr{R}^T$ is the rotation matrix that aligns the image to the eigenvectors of $\matr{M}_w$,
|
||||
while the eigenvalues remain the same for any rotation of the same patch.
|
||||
|
||||
Therefore, the eigenvalues $\lambda_1^{(w)}, \lambda_2^{(w)}$ of $\matr{M}_w$ allow to detect intensity changes along the shift directions:
|
||||
\[
|
||||
\begin{split}
|
||||
E(\Delta x, \Delta y) &= \begin{pmatrix} \Delta x & \Delta y \end{pmatrix}
|
||||
\begin{pmatrix} \lambda_1^{(w)} & 0 \\ 0 & \lambda_2^{(w)} \end{pmatrix}
|
||||
\begin{pmatrix} \Delta x \\ \Delta y \end{pmatrix} \\
|
||||
&= \lambda_1^{(w)} \Delta x^2 + \lambda_2^{(w)} \Delta y^2
|
||||
\end{split}
|
||||
\]
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.6\linewidth]{./img/_harris_rotation.pdf}
|
||||
\caption{Eigenvalues relationship at different regions of an image.}
|
||||
\end{figure}
|
||||
\end{description}
|
||||
|
||||
|
||||
\subsection{Algorithm}
|
||||
\marginnote{Harris' corner detector}
|
||||
|
||||
As computing the eigenvalues of $\matr{M}_w$ at each pixel is expensive, a more efficient cornerness function is the following:
|
||||
\[
|
||||
\begin{split}
|
||||
C(x, y) &= \lambda_1^{(w)}\lambda_2^{(w)} - k(\lambda_1^{(w)} + \lambda_2^{(w)})^2 \\
|
||||
&= \det(\matr{M}_{w(x,y)}) - k \cdot \text{trace}(\matr{M}_{w(x,y)})^2
|
||||
\end{split}
|
||||
\]
|
||||
where $k$ is a hyperparameter (empirically in $[0.04, 0.06]$).
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.5\linewidth]{./img/_harris_efficient.pdf}
|
||||
\caption{Cornerness at different regions.}
|
||||
\end{figure}
|
||||
|
||||
After computing the cornerness of each pixel, one can apply thresholding and then NMS.
|
||||
|
||||
\begin{remark}
|
||||
The window function $w(x, y)$ in the original work follows a Gaussian distribution.
|
||||
\end{remark}
|
||||
|
||||
|
||||
\subsection{Properties}
|
||||
|
||||
Harris' corner detector enjoys the following properties:
|
||||
\begin{descriptionlist}
|
||||
\item[Rotation invariance]
|
||||
The eigenvalues are invariant to a rotation of the image.
|
||||
|
||||
\item[No affine intensity change invariance]
|
||||
An affine intensity change of a signal consists of a gain factor and the addition of a bias (i.e. $I' = \alpha I + \beta$).
|
||||
\begin{description}
|
||||
\item[Invariance to bias]
|
||||
Harris' detector is invariant to an additive bias ($I' = I + \beta$) as a consequence of the approximate derivative computation:
|
||||
\[
|
||||
\partial_x I'(i, j) = I'(i, j+1) - I'(i, j) = (I(i, j+1) + \cancel{\beta}) - (I(i, j) + \cancel{\beta})
|
||||
\]
|
||||
\item[No invariance to gain factor]
|
||||
Harris' detector is not invariant to a gain factor ($I' = \alpha I$) as the multiplicative factor is carried in the derivatives.
|
||||
\end{description}
|
||||
\begin{remark}
|
||||
In other words, Harris' detector is not illumination invariant.
|
||||
\end{remark}
|
||||
|
||||
\item[No scale invariance]
|
||||
Harris' detector is not scale invariant as the use of a fixed window size makes it impossible to recognize the same features when the image is scaled.
|
||||
\end{descriptionlist}
|
||||
|
||||
|
||||
|
||||
\section{Multi-scale feature detector}
|
||||
|
||||
% \subsection{Scale invariance}
|
||||
|
||||
Depending on the scale, an image may exhibit more or less details.
|
||||
A naive approach consists of using a smaller window size for images with a smaller scale,
|
||||
but this is not always able to capture the same features due to the details difference.
|
||||
|
||||
\begin{description}
|
||||
\item[Scale-space] \marginnote{Scale-space}
|
||||
One-parameter family of images obtained by increasingly smoothing the input image.
|
||||
|
||||
\begin{remark}
|
||||
When smoothing, small details should disappear and no new structures should be introduced.
|
||||
\end{remark}
|
||||
|
||||
\begin{remark}
|
||||
It is possible to use the same window size when working with scale-space images.
|
||||
\end{remark}
|
||||
|
||||
\begin{description}
|
||||
\item[Gaussian scale-space] \marginnote{Gaussian scale-space}
|
||||
Scale-space obtained using Gaussian smoothing:
|
||||
\[ L(x, y, \sigma) = I(x, y) * G(x, y, \sigma) \]
|
||||
where $\sigma$ is the standard deviation but also the level of scaling.
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.8\linewidth]{./img/_scale_space_example.pdf}
|
||||
\caption{Gaussian scale-space example}
|
||||
\end{figure}
|
||||
\end{description}
|
||||
\end{description}
|
||||
|
||||
|
||||
\subsection{Scale-normalized LoG blob detection}
|
||||
|
||||
\begin{description}
|
||||
\item[Scale-normalized Laplacian of Gaussian] \marginnote{Scale-normalized Laplacian of Gaussian}
|
||||
LoG scaled by a factor of $\sigma^2$:
|
||||
\[ F(x, y, \sigma) = \sigma^2 \nabla^{(2)} L(x, y, \sigma) = \sigma^2 (I(x, y) * \nabla^{(2)} G(x, y, \sigma)) \]
|
||||
$\sigma^2$ avoids small derivatives when the scaling ($\sigma$) is large.
|
||||
|
||||
\item[Characteristic scale] Scale $\sigma$ that produces a peak in the Laplacian response at a given pixel \cite{slides:scale_normalized_log}.
|
||||
|
||||
\item[Algorithm] \marginnote{Scale-normalized LoG blob detection}
|
||||
Blob (circle) detection using scale-normalized LoG works as follows \cite{slides:scale_normalized_log}:
|
||||
\begin{enumerate}
|
||||
\item Create a Gaussian scale-space by applying the scale-normalized Laplacian of Gaussian with different values of $\sigma$.
|
||||
\item For each pixel, find the characteristic scale and its corresponding Laplacian response across the scale-space (automatic scale selection).
|
||||
\item Filter out the pixels whose response is lower than a threshold and apply NMS.
|
||||
\item The remaining pixels are the centers of the blobs.
|
||||
It can be shown that the radius is given by $r = \sigma\sqrt{2}$.
|
||||
\end{enumerate}
|
||||
\end{description}
|
||||
|
||||
|
||||
When detecting a peak, there are two cases:
|
||||
\begin{descriptionlist}
|
||||
\item[Maximum] Dark blobs on a light background.
|
||||
\item[Minimum] Light blobs on a dark background.
|
||||
\end{descriptionlist}
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.6\linewidth]{./img/LOG_blob_detection_example.png}
|
||||
\caption{Example of application of the algorithm}
|
||||
\end{figure}
|
||||
|
||||
\begin{remark}
|
||||
The intuitive idea of the detection algorithm is the following:
|
||||
\begin{itemize}
|
||||
\item $F(x, y, \sigma)$ keeps growing for $\sigma$s that capture areas within the blob (i.e. with similar intensity).
|
||||
\item The Laplacian reaches its peak when its weights capture the entire blob (virtually it detects an edge).
|
||||
\item After the peak, the LoG filter will also capture intensities outside the blob and therefore decrease.
|
||||
\end{itemize}
|
||||
\end{remark}
|
||||
|
||||
\begin{remark}
|
||||
Using different scales creates the effect of searching in a 3D space.
|
||||
\end{remark}
|
||||
|
||||
\begin{remark}
|
||||
Scale-normalized LoG blob detection is scale and rotation invariant.
|
||||
\end{remark}
|
||||
|
||||
\begin{remark}
|
||||
It empirically holds that, given two points representing the centers of two blobs,
|
||||
the ratio between the two characteristic scales is approximately the ratio between the diameters of the two blobs.
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.4\linewidth]{./img/_scaled_log_blob_diameter.pdf}
|
||||
\caption{
|
||||
\begin{varwidth}[t]{0.55\linewidth}
|
||||
Scale-normalized LoG computed on varying $\sigma$.\\
|
||||
Note that, in the second image, the characteristic scale is
|
||||
higher as the scale is larger.
|
||||
\end{varwidth}
|
||||
}
|
||||
\end{figure}
|
||||
\end{remark}
|
||||
|
||||
|
||||
|
||||
\subsection{Difference of Gaussians blob detection}
|
||||
|
||||
\begin{description}
|
||||
\item[Difference of Gaussians (DoG)] \marginnote{Difference of Gaussians (DoG)}
|
||||
Approximation of the scale-normalized LoG computed as:
|
||||
\[
|
||||
\begin{split}
|
||||
\texttt{DoG}(x, y, \sigma) &= \big( G(x, y, k\sigma) - G(x, y, \sigma) \big) * I(x, y) \\
|
||||
&= L(x, y, k\sigma) - L(x, y, \sigma)
|
||||
\end{split}
|
||||
\]
|
||||
|
||||
\begin{theorem}
|
||||
It can be proven that the DoG kernel is a scaled version of the LoG kernel:
|
||||
\[ G(x, y, k\sigma) - G(x, y, \sigma) \approx (k-1)\sigma^2 \nabla^{(2)}G(x, y, \sigma) \]
|
||||
|
||||
\begin{remark}
|
||||
As we are interested in extrema, the scaling factor is irrelevant.
|
||||
\end{remark}
|
||||
\end{theorem}
|
||||
|
||||
\item[Extrema detection] \marginnote{DoG extrema}
|
||||
Given three DoG images with scales $\sigma_i$, $\sigma_{i-1}$ and $\sigma_{i+1}$,
|
||||
a pixel $(x, y, \sigma_i)$ is an extrema (i.e. keypoint) iff:
|
||||
\begin{itemize}
|
||||
\item It is an extrema in a $3 \times 3$ patch centered on it (8 pixels as $(x, y, \sigma_i)$ is excluded).
|
||||
\item It is an extrema in a $3 \times 3$ patch centered on the pixels at $(x, y, \sigma_{i-1})$ and at $(x, y, \sigma_{i+1})$ ($9+9$ pixels).
|
||||
\end{itemize}
|
||||
|
||||
\begin{center}
|
||||
\includegraphics[width=0.35\linewidth]{./img/_DoG_extrema.pdf}
|
||||
\end{center}
|
||||
|
||||
\item[Algorithm] \marginnote{DoG blob detection}
|
||||
To detect blob centers (i.e. a DoG extrema), an octave of $s$ DoG images is computed as follows:
|
||||
\begin{enumerate}
|
||||
\item Compute a scale-space $L$ of $s+1$ Gaussian smoothed images with $\sigma$ varying by a factor $k = 2^{1/s}$.
|
||||
As the extrema detection method requires checking the DoG images above and below,
|
||||
two additional Gaussians (with scales $k^{-1}\sigma$ and $k^{s+1}\sigma$) are computed.
|
||||
\item The DoG image $\texttt{DoG}(\cdot, \cdot, k^i\sigma)$ is obtained as the difference between the
|
||||
images $L(\cdot, \cdot, k^{i+1}\sigma)$ and $L(\cdot, \cdot, k^i\sigma)$ of the Gaussian scale-space.
|
||||
\begin{figure}[H]
|
||||
\small
|
||||
\centering
|
||||
\includegraphics[width=0.6\linewidth]{./img/_DoG_octave.pdf}
|
||||
\caption{Example of octave computation with $s=4$}
|
||||
\end{figure}
|
||||
\item Extrema detection is done as described above across the $s$ DoG images.
|
||||
\item Points with a weak DoG response can be pruned through thresholding.
|
||||
Furthermore, it has been observed that strong DoG points along edges are unstable and can be also pruned.
|
||||
\end{enumerate}
|
||||
|
||||
Octaves should be computed using different starting $\sigma$s.
|
||||
Instead of recomputing the Gaussian scale-space,
|
||||
it is possible to simply down-sample the already computed Gaussians and compute the DoG images starting from shrunk smoothed images.
|
||||
|
||||
\begin{remark}
|
||||
In the original work, the input image is first enlarged by a factor of 2.
|
||||
Then, four octaves are computed starting from the enlarged image
|
||||
(i.e. images of size factor $\times 2$, $\times 1$, $\times \frac{1}{2}$ and $\times \frac{1}{4}$ are considered).
|
||||
\end{remark}
|
||||
|
||||
\begin{remark}
|
||||
The original work found out that the best hyperparameters are $s=3$ and $\sigma=1.6$.
|
||||
\end{remark}
|
||||
\end{description}
|
||||
|
||||
\begin{remark}
|
||||
DoG blob detection is scale and rotation invariant.
|
||||
\end{remark}
|
||||
|
||||
|
||||
|
||||
\section{Descriptor}
|
||||
|
||||
After finding the keypoints, a descriptor of a keypoint is computed from the pixels within a patch centered on it.
|
||||
|
||||
|
||||
\subsection{DoG descriptor}
|
||||
|
||||
\begin{description}
|
||||
\item[Canonical / Characteristic orientation] \marginnote{Canonical / Characteristic orientation}
|
||||
Direction along which the magnitudes of the gradients of the neighboring pixels of a keypoint are the highest.
|
||||
|
||||
Given a pixel $(x, y)$, its gradient magnitude and direction is computed from the Gaussian smoothed image $L$:
|
||||
\[
|
||||
\begin{split}
|
||||
\vert \nabla L(x, y) \vert &= \sqrt{ \big( L(x+1, y) - L(x-1, y) \big)^2 + \big( L(x, y+1) - L(x, y-1) \big)^2 } \\
|
||||
\theta_L(x, y) &= \tan^{-1}\left( \frac{L(x, y+1) - L(x, y-1)}{L(x+1, y) - L(x-1, y)} \right)
|
||||
\end{split}
|
||||
\]
|
||||
|
||||
\begin{description}
|
||||
\item[Orientation histogram] \marginnote{Orientation histogram}
|
||||
By dividing the directions into bins (e.g. bins of size $10^\circ$),
|
||||
it is possible to define for each keypoint a histogram by considering its neighboring pixels within a patch.
|
||||
For each pixel $(x, y)$ neighboring a keypoint $(x_k, y_k)$, its contribution to the histogram along the direction $\theta_L(x, y)$ is given by:
|
||||
\[ G_{(x_k, y_k)}(x, y, \frac{3}{2} \sigma_s(x_k, y_k)) \cdot \vert \nabla L(x, y) \vert \]
|
||||
where $G_{(x_k, y_k)}$ is a Gaussian centered on the keypoint and $\sigma_s(x_k, y_k)$ is the scale of the keypoint.
|
||||
|
||||
The characteristic orientation of a keypoint is given by the highest peak of the orientation histogram.
|
||||
Other peaks that are higher than at least $80\%$ of the main one are also considered characteristic orientations
|
||||
(i.e. a keypoint might have multiple canonical orientations and, therefore, multiple descriptors).
|
||||
|
||||
For a more accurate estimation, a parabola is interpolated on the neighborhood of each peak and
|
||||
the two bins adjacent to the peak of the parabola are considered.
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.45\linewidth]{./img/_canonical_histogram.pdf}
|
||||
\caption{Orientation histogram and parabola interpolation}
|
||||
\end{figure}
|
||||
\end{description}
|
||||
\end{description}
|
||||
|
||||
\begin{description}
|
||||
\item[DoG descriptor] \marginnote{DoG descriptor}
|
||||
Keypoints are found using the DoG detector and the descriptors are computed through patches along the canonical orientations.
|
||||
|
||||
\begin{remark}
|
||||
DoG descriptor is scale and rotation invariant.
|
||||
\end{remark}
|
||||
\end{description}
|
||||
|
||||
|
||||
\subsection{Scale invariant feature transform (SIFT) descriptor}
|
||||
\marginnote{SIFT descriptor}
|
||||
|
||||
Given a keypoint, SIFT detector works as follows:
|
||||
\begin{enumerate}
|
||||
\item Center on the keypoint a $16 \times 16$ grid divided into $4 \times 4$ regions.
|
||||
\item Compute for each region its orientation histogram with eight bins (i.e. bins of size $45^\circ$).
|
||||
The Gaussian weighting function is centered on the keypoint and has $\sigma$ equal to half the grid size.
|
||||
\item The descriptor is obtained by concatenating the histograms of each region.
|
||||
This results in a feature vector with $128$ elements ($(4 \cdot 4) \cdot 8$).
|
||||
\item Normalize the descriptor to unit length. Pixels larger than $0.2$ are saturated and normalized again (for illumination invariance).
|
||||
\end{enumerate}
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.6\linewidth]{./img/sift.png}
|
||||
\caption{SIFT detector example}
|
||||
\end{figure}
|
||||
|
||||
\begin{description}
|
||||
\item[Trilinear interpolation]
|
||||
Bins are assigned in a soft manner to avoid boundary effects.
|
||||
The contribution of a pixel is spread between its two adjacent bins weighted by the distance to the bin centers:\\
|
||||
\begin{minipage}{0.55\linewidth}
|
||||
\centering
|
||||
\includegraphics[width=0.6\linewidth]{./img/_sift_interpolation.pdf}
|
||||
\end{minipage}
|
||||
\begin{minipage}{0.3\linewidth}
|
||||
\[
|
||||
\begin{cases}
|
||||
\text{weight}_k = 1 - d_k \\
|
||||
\text{weight}_{k+1} = 1 - d_{k+1} \\
|
||||
\end{cases}
|
||||
\]
|
||||
\end{minipage}
|
||||
|
||||
This is done both on the histogram within a region and on the histograms between the four neighboring regions.
|
||||
\end{description}
|
||||
|
||||
\begin{remark}
|
||||
SIFT descriptor is scale, rotation and affine intensity change invariant.
|
||||
\end{remark}
|
||||
|
||||
|
||||
|
||||
\section{Matching}
|
||||
|
||||
Matching the keypoints across different views is a nearest-neighbor search problem. \marginnote{Nearest-neighbor search problem}
|
||||
|
||||
Given a target image $T$ and a reference image $R$,
|
||||
we want to match each keypoint in $T$ to the most similar one in $R$ (usually using the Euclidean distance).
|
||||
|
||||
\begin{description}
|
||||
\item[Matching criteria] \marginnote{Matching criteria}
|
||||
Given the distance $d_\text{NN}$ to the nearest-neighbor of a keypoint of $T$ in $R$,
|
||||
the match is accepted by respecting one of the following criteria:
|
||||
\begin{descriptionlist}
|
||||
\item[Threshold]
|
||||
Given a threshold $T$, the match is accepted iff:
|
||||
\[ d_\text{NN} \leq T \]
|
||||
|
||||
\item[Ratio of distances]
|
||||
Given a threshold $T$ and the distance to the second nearest-neighbor $d_\text{NN2}$, the match is accepted iff:
|
||||
\[ \frac{d_\text{NN}}{d_\text{NN2}} \leq T \]
|
||||
This method allows to avoid ambiguity if two neighbors are too close.
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\begin{subfigure}{0.4\linewidth}
|
||||
\centering
|
||||
\includegraphics[width=0.2\linewidth]{./img/_nn_matching_example1.pdf}
|
||||
\caption{Non ambiguous match}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{0.4\linewidth}
|
||||
\centering
|
||||
\includegraphics[width=0.2\linewidth]{./img/_nn_matching_example2.pdf}
|
||||
\caption{Ambiguous match}
|
||||
\end{subfigure}
|
||||
\end{figure}
|
||||
\end{descriptionlist}
|
||||
|
||||
\begin{remark}
|
||||
It has been empirically shown that a threshold $T=0.8$ allows to reject $90\%$ of wrong matches while missing $5\%$ of correct matches.
|
||||
\end{remark}
|
||||
|
||||
\item[Efficient NN search] \marginnote{Efficient NN search}
|
||||
As an exhaustive search is inefficient, indexing techniques may be employed.
|
||||
|
||||
The main indexing technique applied for feature matching is the k-d tree in the best bin first (BBF) variant.
|
||||
|
||||
\begin{remark}
|
||||
Best bin first is not optimal.
|
||||
\end{remark}
|
||||
\end{description}
|
||||
@ -0,0 +1,462 @@
|
||||
\chapter{Spatial filtering}
|
||||
|
||||
|
||||
\section{Noise}
|
||||
|
||||
The noise added to a pixel $p$ is defined by $n_k(p)$,
|
||||
where $k$ indicates the time step (i.e. noise changes depending on the moment the image is taken).
|
||||
It is assumed that $n_k(p)$ is i.i.d and $n_k(p) \sim \mathcal{N}(0, \sigma)$.
|
||||
|
||||
The information of a pixel $p$ is therefore defined as:
|
||||
\[ I_k(p) = \tilde{I}(p) + n_k(p) \]
|
||||
where $\tilde{I}(p)$ is the real information.
|
||||
|
||||
\begin{description}
|
||||
\item[Temporal mean denoising] \marginnote{Temporal mean denoising}
|
||||
Averaging $N$ images taken at different time steps.
|
||||
\[
|
||||
\begin{split}
|
||||
O(p) &= \frac{1}{N} \sum_{k=1}^{N} I_k(p) \\
|
||||
&= \frac{1}{N} \sum_{k=1}^{N} \Big( \tilde{I}(p) + n_k(p) \Big) \\
|
||||
&= \frac{1}{N} \sum_{k=1}^{N} \tilde{I}(p) + \overbrace{\frac{1}{N} \sum_{k=1}^{N} n_k(p)}^{\mathclap{\text{$\mu = 0$}}} \\
|
||||
&\approx \tilde{I}(p)
|
||||
\end{split}
|
||||
\]
|
||||
|
||||
\begin{remark}
|
||||
As multiple images of the same object are required, this method is only suited for static images.
|
||||
\end{remark}
|
||||
|
||||
\item[Spatial mean denoising] \marginnote{Spatial mean denoising}
|
||||
Given an image, average across neighboring pixels.
|
||||
|
||||
Let $K_p$ be the pixels in a window around $p$ (included):
|
||||
\[
|
||||
\begin{split}
|
||||
O(p) &= \frac{1}{\vert K_p \vert} \sum_{q \in K_p} I(p) \\
|
||||
&= \frac{1}{\vert K_p \vert} \sum_{q \in K_p} \Big( \tilde{I}(q) + n(q) \Big) \\
|
||||
&= \frac{1}{\vert K_p \vert} \sum_{q \in K_p} \tilde{I}(q) + \frac{1}{\vert K_p \vert} \sum_{q \in K_p} n(q) \\
|
||||
&\approx \frac{1}{\vert K_p \vert} \sum_{q \in K_p} \tilde{I}(q)
|
||||
\end{split}
|
||||
\]
|
||||
|
||||
\begin{remark}
|
||||
As the average of neighboring pixels is considered, this method is only suited for uniform regions.
|
||||
\end{remark}
|
||||
\end{description}
|
||||
|
||||
|
||||
|
||||
\section{Convolutions}
|
||||
|
||||
|
||||
\subsection{Preliminaries}
|
||||
|
||||
\begin{description}
|
||||
\item[Convolution] \marginnote{Continuous convolution}
|
||||
Given two functions $f$ and $g$, their 1D convolution is defined as \cite{wiki:1d_convolution}:
|
||||
\[ (f * g)(t) = \int_{-\infty}^{+\infty} f(\tau)g(t - \tau) \,\text{d}\tau \]
|
||||
|
||||
In other words, at each $t$, a convolution can be interpreted as the area under $f(\tau)$
|
||||
weighted by $g(t - \tau)$ (i.e. $g(\tau)$ flipped w.r.t. the y-axis and with the argument shifted by $t$).
|
||||
|
||||
Alternatively, it can be seen as the amount of overlap between $f(\tau)$ and $g(t - \tau)$.
|
||||
|
||||
\begin{figure}[h]
|
||||
\centering
|
||||
\includegraphics[width=0.4\textwidth]{./img/continuous_convolution_example.png}
|
||||
\caption{Example of convolution}
|
||||
\end{figure}
|
||||
|
||||
Extended to the 2-dimensional case, the definition becomes:
|
||||
\[ (f * g)(x, y) = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} f(\alpha, \beta)g(x-\alpha, y-\beta) \,\text{d}\alpha\,\text{d}\beta \]
|
||||
|
||||
A convolution enjoys the following properties: \marginnote{Convolution properties}
|
||||
\begin{descriptionlist}
|
||||
\item[Associative] $f * (g * h) = (f * g) * h$.
|
||||
\item[Commutative] $f * g = g * f$.
|
||||
\item[Distributive w.r.t. sum] $f * (g + h) = f*g + f*h$.
|
||||
\item[Commutative with differentiation] $(f*g)' = f' * g = f * g'$
|
||||
\end{descriptionlist}
|
||||
|
||||
\item[Dirac delta] \marginnote{Dirac delta}
|
||||
The Dirac delta "function" $\delta$ is defined as follows \cite{wiki:dirac,book:sonka}:
|
||||
\[ \forall x \neq 0: \delta(x) = 0 \text{, constrained to } \int_{-\infty}^{+\infty} \delta(x) \,\text{d}x = 1 \]
|
||||
|
||||
Extended to the 2-dimensional case, the definition is the following:
|
||||
\[ \forall (x, y) \neq (0, 0): \delta(x, y) = 0 \text{, constrained to } \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} \delta(x, y) \,\text{d}x\,\text{d}y = 1 \]
|
||||
|
||||
\begin{description}
|
||||
\item[Sifting property] \marginnote{Sifting property}
|
||||
The following property holds:
|
||||
\[ \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} f(x, y) \delta(\alpha-x, \beta-y) \,\text{d}x\,\text{d}y = f(\alpha, \beta) \]
|
||||
|
||||
\begin{remark}
|
||||
Exploiting the sifting property, the signal of an image can be expressed through an integral of Dirac deltas
|
||||
(i.e. a linear combination) \cite{slides:filters,book:sonka}:
|
||||
\[ i(x, y) = \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} i(\alpha, \beta) \delta(x-\alpha, y-\beta) \,\text{d}\alpha\,\text{d}\beta \]
|
||||
\end{remark}
|
||||
\end{description}
|
||||
|
||||
\item[Kronecker delta] \marginnote{Kronecker delta}
|
||||
Discrete version of the Dirac delta \cite{wiki:kronecker}:
|
||||
\[ \delta(x) = \begin{cases}
|
||||
0 & \text{if $x \neq 0$} \\
|
||||
1 & \text{if $x = 0$} \\
|
||||
\end{cases} \]
|
||||
Extended to the 2-dimensional case, the definition is the following:
|
||||
\[ \delta(x, y) = \begin{cases}
|
||||
0 & \text{if $(x, y) \neq (0, 0)$} \\
|
||||
1 & \text{if $(x, y) = (0, 0)$} \\
|
||||
\end{cases} \]
|
||||
\begin{description}
|
||||
\item[Sifting property] \marginnote{Sifting property}
|
||||
The following property holds:
|
||||
\[ i(x, y) = \sum_{\alpha=-\infty}^{+\infty} \sum_{\beta=-\infty}^{+\infty} i(\alpha, \beta) \delta(x-\alpha, y-\beta) \]
|
||||
\end{description}
|
||||
\end{description}
|
||||
|
||||
|
||||
\subsection{Continuous convolutions}
|
||||
|
||||
\begin{description}
|
||||
\item[Image filter] \marginnote{Image filter}
|
||||
Operator that computes the new intensity of a pixel $p$ based on the intensities of a neighborhood of $p$.
|
||||
|
||||
\begin{remark}
|
||||
Image filters are useful for denoising and sharpening operations.
|
||||
\end{remark}
|
||||
|
||||
\item[Linear translation-equivariant (LTE) operator] \marginnote{LTE operator}
|
||||
A 2D operator $T\{ \cdot \}$ is denoted as:
|
||||
\[ T\{ i(x, y) \} = o(x, y) \]
|
||||
$T\{ i(x, y) \}$ is LTE iff it is:
|
||||
\begin{descriptionlist}
|
||||
\item[Linear]
|
||||
Given two input 2D signals $i(x, y)$, $j(x, y)$ and two constants $\alpha$, $\beta$, it holds that:
|
||||
\[ T\{ \alpha \cdot i(x, y) + \beta \cdot j(x, y) \} = \alpha T\{ i(x, y) \} + \beta T\{ j(x, y) \} \]
|
||||
|
||||
\item[Translation-equivariant]
|
||||
Given an input 2D signal $i(x, y)$ and two offsets $x_o$, $y_o$, it holds that:
|
||||
\[ \text{if } T\{ i(x, y) \} = o(x, y) \text{ then } T\{ i(x-x_o, y-y_o) \} = o(x-x_o, y-y_o) \]
|
||||
\end{descriptionlist}
|
||||
|
||||
\begin{description}
|
||||
\item[Impulse response/Point spread function/Kernel]
|
||||
Given a 2D operator $T\{ \cdot \}$,
|
||||
its impulse response, denoted with $h$, is the output of the operator when the input signal is a Dirac delta \cite{slides:filters}:
|
||||
\[ h(x, y) \triangleq T\{ \delta(x, y) \} \]
|
||||
\end{description}
|
||||
\end{description}
|
||||
|
||||
\begin{theorem}[LTE operators as convolutions] \marginnote{LTE operators as convolutions}
|
||||
Applying an LTE operator on an image is equivalent to computing the convolution between the image and the impulse response $h$ of the operator.
|
||||
\[
|
||||
\begin{split}
|
||||
T\{ i(x, y) \} &= \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} i(\alpha, \beta) h(x-\alpha, y-\beta) \,\text{d}\alpha\,\text{d}\beta \\
|
||||
&= i(x, y) * h(x, y)
|
||||
\end{split}
|
||||
\]
|
||||
In other words, the impulse response allows to compute the output of any input signal through a convolution.
|
||||
\begin{proof}
|
||||
Let $i(x, y)$ be an input signal and $T\{ \cdot \}$ be a 2D operator.
|
||||
We have that:
|
||||
\begin{align*}
|
||||
T\{ i(x, y) \}
|
||||
&= T\left\{ \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} i(\alpha, \beta) \delta(x-\alpha, y-\beta) \,\text{d}\alpha\,\text{d}\beta \right\}
|
||||
& \text{sifting property} \\
|
||||
%
|
||||
&= \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} T\left\{ i(\alpha, \beta) \delta(x-\alpha, y-\beta) \right\} \,\text{d}\alpha\,\text{d}\beta
|
||||
& \text{linearity of $T\{ \cdot \}$} \\
|
||||
%
|
||||
&= \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} i(\alpha, \beta) T\left\{ \delta(x-\alpha, y-\beta) \right\} \,\text{d}\alpha\,\text{d}\beta
|
||||
& \text{linearity of $T\{ \cdot \}$} \\
|
||||
%
|
||||
&= \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} i(\alpha, \beta) h(x-\alpha, y-\beta) \,\text{d}\alpha\,\text{d}\beta
|
||||
& \text{\small translation-equivariance of $T\{ \cdot \}$} \\
|
||||
%
|
||||
&= i(x, y) * h(x, y)
|
||||
& \text{definition of convolution} \\
|
||||
\end{align*}
|
||||
\end{proof}
|
||||
\end{theorem}
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.4\textwidth]{./img/_convolution_graphical.pdf}
|
||||
\caption{Visualization of a convolution}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\begin{description}
|
||||
\item[Cross-correlation] \marginnote{Cross-correlation}
|
||||
Given two signals $i(x, y)$ and $h(x, y)$,
|
||||
their cross-correlation computes their similarity and is defined as follows \cite{wiki:crosscorrelation}:
|
||||
\[
|
||||
i(x, y) \circ h(x, y) =
|
||||
\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} i(\alpha, \beta) h(x+\alpha, y+\beta) \,\text{d}\alpha\,\text{d}\beta =
|
||||
\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} h(\alpha, \beta) i(\alpha-x, \beta-y) \,\text{d}\alpha\,\text{d}\beta
|
||||
\]
|
||||
\[
|
||||
h(x, y) \circ i(x, y) =
|
||||
\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} h(\alpha, \beta) i(x+\alpha, y+\beta) \,\text{d}\alpha\,\text{d}\beta =
|
||||
\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} i(\alpha, \beta) h(\alpha-x, \beta-y) \,\text{d}\alpha\,\text{d}\beta
|
||||
\]
|
||||
|
||||
\begin{remark}
|
||||
Cross-correlation is not commutative.
|
||||
\end{remark}
|
||||
|
||||
\begin{remark}
|
||||
The cross-correlation $h \circ i$ is similar to a convolution without flipping the kernel.
|
||||
|
||||
If $h$ is an even function (i.e. $h(x, y) = h(-x, -y)$), we have that $h \circ i$ has the same result of a convolution:
|
||||
\begin{align*}
|
||||
h(x, y) * i(x, y) &= \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} i(\alpha, \beta)h(x-\alpha, y-\beta) \,\text{d}\alpha\,\text{d}\beta \\
|
||||
&= \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} i(\alpha, \beta)h(\alpha-x, \beta-y) \,\text{d}\alpha\,\text{d}\beta
|
||||
& \parbox[b]{0.25\textwidth}{\raggedleft signs in $h$ swappable for Dirac delta} \\
|
||||
&= h(x, y) \circ i(x, y)
|
||||
\end{align*}
|
||||
\end{remark}
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.6\textwidth]{./img/crosscorrelation_graphical.png}
|
||||
\caption{Visualization of cross-correlation}
|
||||
\end{figure}
|
||||
\end{description}
|
||||
|
||||
|
||||
\subsection{Discrete convolutions}
|
||||
|
||||
\begin{description}
|
||||
\item[Discrete convolution] \marginnote{Discrete convolution}
|
||||
Given an input 2D signal $I(i, j)$ and the kernel $H(i, j) = T\{ \delta(i, j) \}$ of a discrete LTE operator (where $\delta(i, j)$ is the Kronecker delta),
|
||||
a discrete convolution is defined as:
|
||||
\[ T\{ I(i, j) \} = \sum_{m=-\infty}^{+\infty} \sum_{n=-\infty}^{+\infty} I(m, n)H(i-m, j-n) = O(i, j) \]
|
||||
|
||||
In practice, the kernel is finitely defined and is applied to each pixel of the image:
|
||||
\[ T\{ I(i, j) \} = \sum_{m=-k}^{k} \sum_{n=-k}^{k} K(m, n)I(i-m, j-n) = O(i, j) \]
|
||||
|
||||
\begin{example}
|
||||
For simplicity, a kernel of size 3 is considered.
|
||||
Given an image $I$ and a kernel $K$, the output $O(1, 1)$ of the pixel $(1, 1)$ is computed as:
|
||||
\[
|
||||
\begin{split}
|
||||
O(1, 1) &= \begin{pmatrix}
|
||||
I(0, 0) & I(0, 1) & I(0, 2) \\
|
||||
I(1, 0) & I(1, 1) & I(1, 2) \\
|
||||
I(2, 0) & I(2, 1) & I(2, 2) \\
|
||||
\end{pmatrix}
|
||||
*
|
||||
\begin{pmatrix}
|
||||
K(0, 0) & K(0, 1) & K(0, 2) \\
|
||||
K(1, 0) & K(1, 1) & K(1, 2) \\
|
||||
K(2, 0) & K(2, 1) & K(2, 2) \\
|
||||
\end{pmatrix} \\
|
||||
&= I(0, 0)K(2, 2) + I(0, 1)K(2, 1) + I(0, 2)K(2, 0) + \\
|
||||
&\,\,\,\,\,+ I(1, 0)K(1, 2) + I(1, 1)K(1, 1) + I(1, 2)K(1, 0) + \\
|
||||
&\,\,\,\,\,+ I(2, 0)K(0, 2) + I(2, 1)K(0, 1) + I(2, 2)K(0, 0)
|
||||
\end{split}
|
||||
\]
|
||||
Note that by definition, $K$ has to be flipped.
|
||||
\end{example}
|
||||
|
||||
\begin{remark}
|
||||
In convolutional neural networks, the flip of the learned kernels can be considered implicit.
|
||||
\end{remark}
|
||||
|
||||
\begin{description}
|
||||
\item[Border handling] \marginnote{Border handling}
|
||||
Computing the convolution of the pixels at the borders of the image might be an issue as it goes out-of-bounds,
|
||||
possible solutions are:
|
||||
\begin{descriptionlist}
|
||||
\item[Crop] Ignore border pixels on which the convolution overflows.
|
||||
\item[Pad] Add a padding to the image:
|
||||
\begin{descriptionlist}
|
||||
\item[Zero-padding] Add zeros (e.g. $\texttt{000}\vert a \dots d \vert\texttt{000}$).
|
||||
\item[Replicate] Repeat the bordering pixel (e.g. $\texttt{aaa}\vert a \dots d \vert\texttt{ddd}$).
|
||||
\item[Reflect] Use the $n$ pixels closest to the border (e.g. $\texttt{cba}\vert abc \dots dfg \vert\texttt{gfd}$).
|
||||
\item[Reflect\_101] Use the $n$ pixels closest to the border, skipping the first/last one (e.g. $\texttt{dcb}\vert abcd \dots efgh \vert\texttt{gfe}$).
|
||||
\end{descriptionlist}
|
||||
\end{descriptionlist}
|
||||
\end{description}
|
||||
\end{description}
|
||||
|
||||
|
||||
\subsection{Common linear kernels}
|
||||
|
||||
\begin{description}
|
||||
\item[Mean filter] \marginnote{Mean filter}
|
||||
LTE operator that computes the intensity of a pixel as the average intensity of the pixels in its neighborhood.
|
||||
|
||||
The kernel has the form (example with a $3 \times 3$ kernel):
|
||||
\[
|
||||
\begin{pmatrix}
|
||||
\frac{1}{9} & \frac{1}{9} & \frac{1}{9} \\
|
||||
\frac{1}{9} & \frac{1}{9} & \frac{1}{9} \\
|
||||
\frac{1}{9} & \frac{1}{9} & \frac{1}{9} \\
|
||||
\end{pmatrix}
|
||||
=
|
||||
\frac{1}{9} \begin{pmatrix}
|
||||
1 & 1 & 1 \\
|
||||
1 & 1 & 1 \\
|
||||
1 & 1 & 1 \\
|
||||
\end{pmatrix}
|
||||
\]
|
||||
|
||||
\begin{remark}
|
||||
The mean filter has a low-pass effect which allows the removal of details from the signal.
|
||||
This allows for image smoothing and, to some extent, denoising (but adds blur).
|
||||
\end{remark}
|
||||
\begin{remark}
|
||||
As the intensity of a pixel is computed by averaging its neighborhood,
|
||||
the results for pixels located between low-intensity and high-intensity areas might not be ideal.
|
||||
\end{remark}
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.85\textwidth]{./img/_mean_filter_example.pdf}
|
||||
\caption{Example of mean filter application}
|
||||
\end{figure}
|
||||
|
||||
\item[Gaussian filter] \marginnote{Gaussian filter}
|
||||
LTE operator whose kernel follows a 2D Gaussian distribution with $\mu=0$ and given $\sigma$.
|
||||
|
||||
\begin{remark}
|
||||
The smoothing strength of the filter grows with $\sigma$.
|
||||
\end{remark}
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.85\textwidth]{./img/_gaussian_filter_example.pdf}
|
||||
\caption{Example of Gaussian filter application}
|
||||
\end{figure}
|
||||
|
||||
\begin{description}
|
||||
\item[Sampling]
|
||||
In practice, the kernel is created by sampling from the wanted Gaussian distribution.
|
||||
One can notice that a higher $\sigma$ results in a more spread distribution and therefore a larger kernel is more suited,
|
||||
on the other hand, a smaller $\sigma$ can be represented using a smaller kernel as it is more concentrated around the origin.
|
||||
|
||||
As a rule-of-thumb, given $\sigma$, an ideal kernel is of size $(2\lceil 3\sigma \rceil + 1) \times (2\lceil 3\sigma \rceil + 1)$.
|
||||
|
||||
\item[Separability]
|
||||
As a 2D Gaussian $G(x, y)$ can be decomposed into a product of two 1D Gaussians $G(x, y) = G_1(x)G_2(y)$,
|
||||
it is possible to split the convolution into two 1D convolutions.
|
||||
\[
|
||||
\begin{split}
|
||||
I(x, y) * G(x, y) &= \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} I(\alpha, \beta) G(x-\alpha, y-\beta) \,\text{d}\alpha\,\text{d}\beta \\
|
||||
&= \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} I(\alpha, \beta) G_1(x-\alpha)G_2(y-\beta) \,\text{d}\alpha\,\text{d}\beta \\
|
||||
&= \int_{-\infty}^{+\infty} G_2(y-\beta) \left( \int_{-\infty}^{+\infty} I(\alpha, \beta) G_1(x-\alpha) \,\text{d}\alpha \right) \,\text{d}\beta \\
|
||||
&= (I(x, y) * G_1(x)) * G_2(y)
|
||||
\end{split}
|
||||
\]
|
||||
|
||||
\begin{remark}
|
||||
The speed-up in number-of-operations is linear.
|
||||
\end{remark}
|
||||
\end{description}
|
||||
\end{description}
|
||||
|
||||
|
||||
\subsection{Common non-linear kernels}
|
||||
|
||||
\begin{remark}
|
||||
Linear filters are ineffective when dealing with impulse noise and
|
||||
have the side effect of blurring the image.
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.85\textwidth]{./img/_impulse_noise_example.pdf}
|
||||
\caption{Example of impulse noise and denoising with mean filter}
|
||||
\end{figure}
|
||||
\end{remark}
|
||||
|
||||
\begin{remark}
|
||||
As they lose linearity, non-linear filters are technically not convolutions anymore.
|
||||
\end{remark}
|
||||
|
||||
\begin{description}
|
||||
\item[Median filter] \marginnote{Median filter}
|
||||
The intensity of a pixel is obtained as the median intensity of its neighborhood.
|
||||
|
||||
\begin{remark}
|
||||
Median filters are effective in removing impulse noise (as outliers are excluded) without introducing significant blur.
|
||||
It also tends to result in sharper edges.
|
||||
\end{remark}
|
||||
|
||||
\begin{remark}
|
||||
Median filters are not suited for Gaussian noise.
|
||||
It might be useful to apply a linear filter after a median filter.
|
||||
\end{remark}
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.85\textwidth]{./img/_median_filter_example.pdf}
|
||||
\caption{Example of median filter application}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\item[Bilateral filter] \marginnote{Bilateral filter}
|
||||
Given two pixels $p$ and $q$, the following can be computed:
|
||||
\begin{descriptionlist}
|
||||
\item[Spatial distance] $d_s(p, q) = \Vert p - q \Vert_2$
|
||||
\item[Range/intensity distance] $d_r(p, q) = \vert \texttt{intensity}(p) - \texttt{intensity}(q) \vert$
|
||||
\end{descriptionlist}
|
||||
|
||||
Given a pixel $p$, its neighborhood $\mathcal{N}(p)$ and the variances $\sigma_s$, $\sigma_r$ of two Gaussians,
|
||||
the bilateral filter applied on $p$ is computes as follows:
|
||||
\[
|
||||
\begin{split}
|
||||
O(p) &= \sum_{q \in \mathcal{N}(p)} H(p, q) \cdot \texttt{intensity}(q) \\
|
||||
&\text{where } H(p, q) = \frac{G_{\sigma_s}(d_s(p, q)) G_{\sigma_r}(d_r(p, q))}{\sum_{z \in \mathcal{N}(p)} G_{\sigma_s}(d_s(p, z)) G_{\sigma_r}(d_r(p, z))}
|
||||
\end{split}
|
||||
\]
|
||||
where the denominator of $H$ is a normalization factor.
|
||||
|
||||
\begin{remark}
|
||||
Bilateral filters allow to deal with Gaussian noise without the introduction of blur.
|
||||
\end{remark}
|
||||
|
||||
\begin{remark}
|
||||
Neighboring pixels with similar intensities result in larger weights in the filter,
|
||||
while pixels with different intensities (i.e. near an edge) result in smaller weights.
|
||||
This allows to effectively ignore pixels that belong to a different object from being considered when computing the intensity of a pixel.
|
||||
\end{remark}
|
||||
|
||||
\begin{figure}[H]
|
||||
\centering
|
||||
\includegraphics[width=0.9\textwidth]{./img/_bilateral_filter_example.pdf}
|
||||
\caption{Example of bilateral filter application}
|
||||
\end{figure}
|
||||
|
||||
|
||||
\item[Non-local means filter] \marginnote{Non-local means filter}
|
||||
Exploits patches with similar pixels to denoise the image.
|
||||
|
||||
\begin{minipage}{0.75\textwidth}
|
||||
Let:
|
||||
\begin{itemize}
|
||||
\item $I$ be the image plane.
|
||||
\item $\mathcal{N}_i$ be the intensity matrix of a patch centered on the pixel $i$.
|
||||
\item $S_i$ be a neighborhood of the pixel $i$.
|
||||
\item $h$ be the bandwidth (a hyperparameter).
|
||||
\end{itemize}
|
||||
The intensity of a pixel $p$ is computed as follows:
|
||||
\[
|
||||
\begin{split}
|
||||
O(p) &= \sum_{q \in S_p} w(p, q) \cdot \texttt{intensity}(q) \\
|
||||
\text{where }& w(p, q) = \frac{1}{Z(p)} e^{-\frac{\Vert \mathcal{N}_p - \mathcal{N}_q \Vert_2^2}{h^2}} \\
|
||||
& Z(p) = \sum_{q \in I} e^{\frac{\Vert \mathcal{N}_p - \mathcal{N}_q \Vert_2^2}{h^2}}
|
||||
\end{split}
|
||||
\]
|
||||
\end{minipage}
|
||||
\begin{minipage}{0.25\textwidth}
|
||||
\centering
|
||||
\includegraphics[width=\linewidth]{./img/non_local_means_filter.png}
|
||||
\end{minipage}
|
||||
|
||||
Instead of using the full image plane $I$, a neighborhood $S_p$ is used for computational purposes.
|
||||
$Z(p)$ is a normalization factor.
|
||||
\end{description}
|
||||
Reference in New Issue
Block a user