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Our goal is to obtain proximal operator of the following function $$g \left( x \right) = {\left\| x \right\|}_{1} + \operatorname{TV}(x).$$ The involved optimization problem for any $z \in \mathbb{R}^d$ is the following $$\text{argmin}_{x}\left\{g(x) + \frac{1}{2}\|x-z\|^2_2\right\}$$ Denote the following $$g_1(x) := {\left\| x \right\|}_{1} + \frac{1}... 3 Indeed the model for the Proximal Gradient Method (Also see Proximal Gradient Methods for Learning) is in the form of:$$ F \left( x \right) = f \left( x \right) + g \left( x \right) $$Where usually  f \left( x \right)  is convex smooth function and  g \left( x \right)  is convex non smooth function. Yet the model is quite flexible and you may define ... 3 For the sake of putting some numbers to this question, I implemented a basic histogram of gradient from scikit-image (skimage.feature.hog). Here is the timing data for HOG with default parameters applied to the skimage.data.astronaut image in b&w and rescaled to have the given dimensions: Image dimensions......(102, 102) 3.49 ms ± 26.3 µs per loop (mean ... 2 I'd try a very "tinkery" approach here: Erode the image, so that the black area is shrunk by a fixed radius of pixels from its border (say, 5px). Dilate the resulting image by the same amount measure the amount of difference between original and processed image. The idea is that something that is a locally convex border doesn't suffer through erosion (it's ... 2 ...is the quality measure SSIM between two images (one base-line, one distorted) directly correlated to the mutual information between the two [?] The short answer to this is "yes" but it tells you nothing. Because the question is a little bit ill posed. You have two signals, let them be signals or images, it doesn't matter, you have two observations. And ... 2 There might be several versions, yet the most probable is as follows. An SNR is an energy-dependent measure. In the time or spatial domain, it is agnostic to location: all samples have the same "weight" (uniform weighting). As an energetic measure, it can be computed in the Fourier domain as well (or any orthogonal transformation). Then, from the sensory ... 2 In our days the Deep Neural Network methods certainly are generating best results. Due to the intense research going on in this field the best method is a moving target hence one can not pin point to one. One generation before them the best methods were based on Dictionary Learning. For example you can use the K-SVD for Single Image Super Resolution. Those ... 1 The important message is: "it can indeed be reconstructed", meaning under certain conditions, and not "always". An image pyramid is hierarchical representation of an image with a collection of derived images at different resolutions (thus, sizes). In a Gaussian pyramid, derived images are smoothed at level l by an operator S_l (eg by a Gaussian filter) ... 1 This depends on the order of upsampling and downsampling. If the order is correct, then you won't throw away anything and thus you should in principle be able to reconstruct the image. In general:$$ \left(\uparrow_n\downarrow_n f\right) \neq \left(\downarrow_n\uparrow_n f\right)  Similar things are used when using the Wavelet decomposition on a signal, ...

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I agree with Peter K. It looks like the authors of the Multidirectional Scratch Detection and Restoration in Digitized Old Images (E. Ardizzone, H. Dindo, and G. Mazzola, EURASIP Journal on Image and Video Processing, 2010) paper tried to reformulate the equation (shown by Peter K.) from the Analyzing Oriented Patterns paper by Kass and Witkin (1987; ...

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Usually one could divide the methods according to: Global Method The function optimized is all over the image with no local data. Namely it finds best solution involving all pixels. Diffusion Based is usually iterative method to solve such a Global model. Local Method The optimization function is local for a group of patches and holes. Most "Patch" based ...

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