From 14bddf37349199f1a9a6b0e85103d306e9d277fd Mon Sep 17 00:00:00 2001 From: nrupatunga Date: Thu, 7 May 2020 16:34:33 +0530 Subject: [PATCH] Update documentation --- .gitignore | 4 ++ docs/FFT Based 2D Cyclic Convolution.pdf | Bin 0 -> 433710 bytes docs/L0-Smoothing.html | 77 +++++++++++++++++++++++ docs/images/cyclicconv1.png | Bin 0 -> 50310 bytes docs/images/cyclicconv2.png | Bin 0 -> 54991 bytes docs/images/cyclicshift.png | Bin 0 -> 39255 bytes src/L0Smoothing.py | 2 +- src/psf2otf.py | 8 +++ 8 files changed, 90 insertions(+), 1 deletion(-) create mode 100644 docs/FFT Based 2D Cyclic Convolution.pdf create mode 100644 docs/L0-Smoothing.html create mode 100644 docs/images/cyclicconv1.png create mode 100644 docs/images/cyclicconv2.png create mode 100644 docs/images/cyclicshift.png diff --git a/.gitignore b/.gitignore index b6e4761..046ad09 100644 --- a/.gitignore +++ b/.gitignore @@ -127,3 +127,7 @@ dmypy.json # Pyre type checker .pyre/ + +tags +docs/images/.DS_Store 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+$$S=\mathscr{F}^{-1}\left(\frac{\mathscr{F}(I)+\beta\left(\mathscr{F}\left(\partial_{x}\right)^{*} \mathscr{F}(h)+\mathscr{F}\left(\partial_{y}\right)^{*} \mathscr{F}(v)\right)}{\mathscr{F}(1)+\beta\left(\mathscr{F}\left(\partial_{x}\right)^{*} \mathscr{F}\left(\partial_{x}\right)+\mathscr{F}\left(\partial_{y}\right)^{*} \mathscr{F}\left(\partial_{y}\right)\right)}\right)$$ + +$\mathscr{F}$: Fourier transform, $\partial_{x}$: gradient in x-direction, $\partial_{y}$: gradient in y-direction. $I$: Input +$S$: output image + +$$\left(h_{p}, v_{p}\right)=\left\{\begin{array}{ll} +(0,0) & \left(\partial_{x} S_{p}\right)^{2}+\left(\partial_{y} S_{p}\right)^{2} \leq \lambda / \beta \\ +\left(\partial_{x} S_{p}, \partial_{y} S_{p}\right) & \text { otherwise } +\end{array}\right.$$ + +Brief understanding: +- First from the input image, take the horizontal and vertical gradient of the image. + +- Then we have some edges (some with strong gradients, some with medium gradients, some with weak gradients). + +- Now, we threshold some of these gradients to zero them out using the condition $\left(\partial_{x} S_{p}\right)^{2}+\left(\partial_{y} S_{p}\right)^{2} \leq \lambda / \beta$. $\lambda$ and $\beta$ are the hyperparameters. + +- Intuitively, this means that, we want the output $S$ to have the new set of gradients after applying this condition. + +- This is nothing but we are reducing the number of edges in the output image $S$ that are not strong. $h_{p}$ and $v_{p}$ is what we find. + +Before implementing the algorithm, I went through the theory in Gonzalez Digital Image Processing book, chapter-4. +This is really a great intro to Fourier Transform properties. + +**TODO**: Write some important properties of Fourier Transform, that we need to remember. + +psf2otf +---------------------------------------------------------------------------------------- +This function takes the input filter (kernel) in our case it is [-1, 1] or [-1; +1] and circularly shifts the kernel while making the size of the input +kernel same as that of the image. After that we take the FFT + +While implementing this function, I got stuck wondering why we shift the +kernel circularly, it is basically because FFT does cyclic convolution. Below is what I mean by that. + +The given kernel is padded with zeros first to match the input image dims, +then we circularly shift to make the center of kernel to be at (0, 0) as shown in Figure [CycShift] + +![Figure [CycShift]: Step-1 Cyclic Shift](images/cyclicshift.png width="1280px") + +Figure [Cycconv1] shows the cyclic convolution for non-border pixels + +![Figure [Cycconv1]: Cyclic Convolution Non Border Case](images/cyclicconv1.png width="1280px") + +Figure [Cycconv2] shows the cyclic convolution for border pixels + +![Figure [Cycconv2]: Cyclic Convolution Border Case](images/cyclicconv2.png width="1280px") + +Figure [Cycconv2] and Figure [Cycconv1], is showing the cyclic convolution in spatial domain, and FFT(input) .* FFT(cyclic shifted kernel) +does the same job as (input) convolution with (cyclic shifted kernel) + + +Reference to Cyclic Convolution +---------------------------------------------------------------------------------------- +In order to read more and verify, refer to following document and the code + +Document: [FFT implements cyclic +convolution](https://www.docdroid.net/YSKkZ5Y/fft-based-2d-cyclic-convolution-pdf#page=5) + +Code: [Matlab +implementation of Cyclic Convolution](https://github.com/RoyiAvital/StackExchangeCodes/blob/master/SignalProcessing/Q38542/Q38542.m) + +Python implementation: [psf2otf](https://github.com/nrupatunga/L0-Smoothing/blob/master/src/psf2otf.py) + + + + diff --git a/docs/images/cyclicconv1.png b/docs/images/cyclicconv1.png new file mode 100644 index 0000000000000000000000000000000000000000..145ab671ef8b1d12a8ac77f851e5f48214069e3d GIT binary patch literal 50310 zcma&ObzGER^e#FyN{4`Si6}L6he(Zd2}pNJcQ?`kN|zuZ2uMkXG*ZG)(jC&>aQA%A z`Q3B=x}Q7$D9kYLe)nGStYyf|8lIWE^FE#$Q(heR2QzhlU8byD+ZMwlBngqRe>xE_I>jia*F?_OAHhz(_S0;_ zl#dye1MUhG{S}pOzFKG2IL-5H!{(TtcMWxy6B*L>@ENt!;!8>*>wn~FT;$+p7#eIX zTcn{NMx(&)k)uBKzc)Gm-f}?>8*HwgY+C3R*rW}`m&BxFk=NeCMf@l9yEJhq!GHfj 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