Add IPCV edge detection

src/ainotes.cls

@@ -15,7 +15,7 @@
 \usepackage{marginnote}
 \usepackage{scrlayer-scrpage}
 \usepackage{scrhack, algorithm, listings}
-\usepackage{array, makecell, multirow}
+\usepackage{array, makecell, multirow, booktabs}
 \usepackage{acro}
 \usepackage{subcaption}
 \usepackage{eurosym}

src/image-processing-and-computer-vision/module1/ipcv1.tex

@@ -12,6 +12,7 @@
 
     \input{./sections/_image_acquisition.tex}
     \input{./sections/_spatial_filtering.tex}
+    \input{./sections/_edge_detection.tex}
 
     \printbibliography[heading=bibintoc]
  

src/image-processing-and-computer-vision/module1/sections/_edge_detection.tex

@@ -0,0 +1,215 @@
+\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 $A$, $B$ along the direction $\theta$ belonging the the discrete pixel grid,
+it is possible to use linear interpolation to estimate the gradient of these points even if off-grid of 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}

src/image-processing-and-computer-vision/module1/sections/_spatial_filtering.tex

@@ -427,7 +427,36 @@ where $\tilde{I}(p)$ is the real information.
 
         \begin{figure}[H]
             \centering
-            \includegraphics[width=0.95\textwidth]{./img/_bilateral_filter_example.pdf}
+            \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}