From 168f167f3160b03d2ae5e70b9f137c0fc8a1271a Mon Sep 17 00:00:00 2001 From: NotXia <35894453+NotXia@users.noreply.github.com> Date: Sat, 9 Mar 2024 15:26:51 +0100 Subject: [PATCH] Add IPCV convolutions --- .gitignore | 4 + .../module1/img/_bilateral_filter_example.pdf | Bin 0 -> 291468 bytes .../module1/img/_convolution_graphical.pdf | Bin 0 -> 116948 bytes .../module1/img/_gaussian_filter_example.pdf | Bin 0 -> 283590 bytes .../module1/img/_impulse_noise_example.pdf | Bin 0 -> 354051 bytes .../module1/img/_mean_filter_example.pdf | Bin 0 -> 466531 bytes .../module1/img/_median_filter_example.pdf | Bin 0 -> 556396 bytes .../img/continuous_convolution_example.png | Bin 0 -> 19915 bytes .../img/crosscorrelation_graphical.png | Bin 0 -> 151941 bytes .../module1/ipcv1.tex | 6 +- .../module1/references.bib | 42 ++ .../module1/sections/_spatial_filtering.tex | 391 +++++++++++++++++- 12 files changed, 440 insertions(+), 3 deletions(-) create mode 100644 src/image-processing-and-computer-vision/module1/img/_bilateral_filter_example.pdf create mode 100644 src/image-processing-and-computer-vision/module1/img/_convolution_graphical.pdf create mode 100644 src/image-processing-and-computer-vision/module1/img/_gaussian_filter_example.pdf create mode 100644 src/image-processing-and-computer-vision/module1/img/_impulse_noise_example.pdf create mode 100644 src/image-processing-and-computer-vision/module1/img/_mean_filter_example.pdf create mode 100644 src/image-processing-and-computer-vision/module1/img/_median_filter_example.pdf create mode 100644 src/image-processing-and-computer-vision/module1/img/continuous_convolution_example.png create mode 100644 src/image-processing-and-computer-vision/module1/img/crosscorrelation_graphical.png create mode 100644 src/image-processing-and-computer-vision/module1/references.bib diff --git a/.gitignore b/.gitignore index 003bf85..96bba81 100644 --- a/.gitignore +++ b/.gitignore @@ -5,6 +5,10 @@ *.aux *.toc 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Y_Ud)_!;*ghu8U9J%BV<}NSOxzFB%0u$N&HU literal 0 HcmV?d00001 diff --git a/src/image-processing-and-computer-vision/module1/ipcv1.tex b/src/image-processing-and-computer-vision/module1/ipcv1.tex index 2d52773..d6110b4 100644 --- a/src/image-processing-and-computer-vision/module1/ipcv1.tex +++ b/src/image-processing-and-computer-vision/module1/ipcv1.tex @@ -1,4 +1,6 @@ \documentclass[11pt]{ainotes} +\usepackage{biblatex} +\addbibresource{./references.bib} \title{Image Processing and Computer Vision\\(Module 1)} \date{2023 -- 2024} @@ -10,5 +12,7 @@ \input{./sections/_image_acquisition.tex} \input{./sections/_spatial_filtering.tex} - + + \printbibliography[heading=bibintoc] + \end{document} \ No newline at end of file diff --git a/src/image-processing-and-computer-vision/module1/references.bib b/src/image-processing-and-computer-vision/module1/references.bib new file mode 100644 index 0000000..94d21ea --- /dev/null +++ b/src/image-processing-and-computer-vision/module1/references.bib @@ -0,0 +1,42 @@ +@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}" +} \ No newline at end of file diff --git a/src/image-processing-and-computer-vision/module1/sections/_spatial_filtering.tex b/src/image-processing-and-computer-vision/module1/sections/_spatial_filtering.tex index 72d0aa4..0a17b95 100644 --- a/src/image-processing-and-computer-vision/module1/sections/_spatial_filtering.tex +++ b/src/image-processing-and-computer-vision/module1/sections/_spatial_filtering.tex @@ -4,7 +4,7 @@ \section{Noise} The noise added to a pixel $p$ is defined by $n_k(p)$, -where $k$ indicates the time step (i.e. noise is different at each time step). +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: @@ -28,7 +28,7 @@ where $\tilde{I}(p)$ is the real information. \end{remark} \item[Spatial mean denoising] \marginnote{Spatial mean denoising} - Given one image, average across neighboring pixels. + Given an image, average across neighboring pixels. Let $K_p$ be the pixels in a window around $p$ (included): \[ @@ -43,4 +43,391 @@ where $\tilde{I}(p)$ is the real information. \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 + & \\ + % + &= \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 $(3\sigma+1) \times (3\sigma+1)$. + + \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=0.95\textwidth]{./img/_bilateral_filter_example.pdf} + \caption{Example of bilateral filter application} + \end{figure} \end{description} \ No newline at end of file

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