From 7cff114d6ec7f6c2028439df3f68cabad6553f3c Mon Sep 17 00:00:00 2001 From: NotXia <35894453+NotXia@users.noreply.github.com> Date: Fri, 15 Mar 2024 20:29:40 +0100 Subject: [PATCH] Add IPCV Canny, zero-crossing, and local features --- .../img/_example_laplacian_gaussian.pdf | Bin 0 -> 347567 bytes .../module1/img/hysteresis thresholding.png | Bin 0 -> 170669 bytes .../module1/img/zero_crossing_example1.png | Bin 0 -> 10447 bytes .../module1/img/zero_crossing_example2.png | Bin 0 -> 7697 bytes .../module1/img/zero_crossing_example3.png | Bin 0 -> 6919 bytes .../module1/ipcv1.tex | 1 + .../module1/sections/_edge_detection.tex | 177 +++++++++++++++++- .../module1/sections/_local_features.tex | 49 +++++ 8 files changed, 226 insertions(+), 1 deletion(-) create mode 100644 src/image-processing-and-computer-vision/module1/img/_example_laplacian_gaussian.pdf create mode 100644 src/image-processing-and-computer-vision/module1/img/hysteresis thresholding.png create mode 100644 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+ \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} \ No newline at end of file diff --git a/src/image-processing-and-computer-vision/module1/sections/_local_features.tex b/src/image-processing-and-computer-vision/module1/sections/_local_features.tex new file mode 100644 index 0000000..355fcec --- /dev/null +++ b/src/image-processing-and-computer-vision/module1/sections/_local_features.tex @@ -0,0 +1,49 @@ +\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} +