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Laplace of Gaussian The Laplace of Gaussian (LoG) of image $f$ can be written as $$\nabla^2 (f * g) = f * \nabla^2 g$$ with $g$ the Gaussian kernel and $*$ the convolution. That is, the Laplace of the image smoothed by a Gaussian kernel is identical to the image convolved with the Laplace of the Gaussian kernel. This convolution can be further expanded, ...

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What's SURF? In order to correctly understand what is going on, you also need to be familiar with SIFT: SURF is basically an approximation of SIFT. Now, the real question becomes: what's SIFT?. SIFT is both a keypoint detector and a keypoint descriptor. In the detector part, SIFT is essentially a multi-scale variant of classical corner detectors such as ...

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In order to identify the potential interest points, difference-of-Gaussian function (DOG) is often used to process the image, thus making it invariant to scale and orientation. In SIFT, image pyramids are established by filtering each layer with DOG of increasing sigma values and taking the difference. On the other hand, SURF applies a much faster ...

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The easiest way doing so would be Laplacian Pyramid. Yet, it can be done just by using simple Addition operator. Just add the High Frequency of one image to the Low Frequency of the other. Keep in mind few things: Dimensions must be the same. Otherwise, interpolate to the same dimensions. It is better to use HPF which is built from the same LPF used. To ...

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What the authors meant is to create a matrix of Weights, ${U}^{\left( i, j \right)}$. It is a matrix of the size of the image. The given calculation is by the exponent of two terms (Each of them is a matrix, the calculation is element wise). The final step is to blur it using a Gaussian Blue (2D Blur, like on an image). In equation 10 you can see they ...

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The Ricker wavelet, the (isotropic) Marr wavelet, the Mexican hat or the Laplacian of Gaussians belong to be the same concept: continuous admissible wavelets (satisfying certain conditions). Traditionally, the Ricker wavelet is the 1D version. The Marr wavelet or the Mexican hat are names given in the context of 2D image decompositions, you can consider ...

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Actually the down sampling has no role here. It is all based on a real simple equation: $$I = A + B$$ It is always enough to keep 2 terms of the 3 to restore completely and perfectly the information. So let's look on this: $${I}_{0} = \left( \left( {I}_{0} \downarrow \right) \uparrow \right) + {R}_{0}$$ So if we keep ${R}_{0}$ and we have $\left( ... 4 Questioner's answer... sigma have the same units as x and y i.e. number of pixels. In multi-scale filtering, the size of the filter must change when the sigma changes. Obtain the number of pixels per one millimeter or the vice-versa. (I did this using the property of pixel spacing included in the DICOM metadata in Matlab you can do this as info=dicominfo('... 3 For posterity, I'm going to add that you can build the pyramid in this way. In other words, if you choose the correct standard deviation for the gaussians, you can do all the low-pass filtering to the original image first, and then downsample later to make identical results to if you had used the normal blur-downsample-blur-downsample method. Here is ... 2 You could weigh together the images Discrete Wavelet Transforms (DWT), where one image has declining weights with increasing scale and vice versa. The DWT filters are designed to have the perfect reconstruction property so you don't need to worry about cut-off frequencies. This algorithm could have complexity$\mathcal{O}(n)$where$n$is the number of ... 2 The reconstruction will remove aliasing, no matter which filter kernel is used. The lower layer of the 2-layer Laplacian pyramid on page 8 of the Toronto lecture notes, adapted here: Figure 1. A 2-layer Laplacian pyramid. can be redrawn as: Figure 2. An alternative presentation of the lower layer of the Laplacian pyramid of fig. 1. This layer is an ... 2 When we have a discrete signal it is usually sampled on a grid of indices. Both sub sampling and down scaling changes the grid. The classic definition is that Sub Sampling is a step in Down Scaling. Sub Sampling Given a signal which is sampled on a grid of indices Sub Sampling means to keep only the samples which on a sub set of the indices grid. Down ... 2 A very simple example on a$2\times 2$image $$I_0=\begin{bmatrix}a&b\\c&d\end{bmatrix}$$ with (very crude Gaussian) low-pass: $$g=1/4\begin{bmatrix}1&1\\1&1\end{bmatrix}$$ yields a downsampled$I_1$after filtering, with only one pixel (out of 4): $$I_1=\begin{bmatrix}(a+b+c+d/4)\end{bmatrix}$$ It can be upsampled as: $$U(I_1) = I_1^\... 1 This expression is more a discretization of a continuous wavelet transform than an actual DWT (discrete wavelet transform), provided \psi is a genuine wavelet. It only computes the wavelet coefficient c_{j,k} associated to a specific shifted and dilated continuous wavelet \psi_{j,k}(t). This yields a frame-like wavelet decomposition, if you picture all ... 1 The important message is: "it can indeed be reconstructed", meaning under certain conditions, and not "always". An image pyramid is hierarchical representation of an image with a collection of derived images at different resolutions (thus, sizes). In a Gaussian pyramid, derived images are smoothed at level l by an operator S_l (eg by a Gaussian filter) ... 1 This depends on the order of upsampling and downsampling. If the order is correct, then you won't throw away anything and thus you should in principle be able to reconstruct the image. In general:$$ \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, ... 1 We give a precise description of the pyramid algorithm in this paper: http://www.ipol.im/pub/art/2014/79/ It's a wavelet transform. If you like to use the exact filters you can use Fourier transform but the filters are usually well approximated in spatial domain on a small support (8 x 8 px^2). Note that the direct convolution is faster than the Fourier ... 1 On A panorama on multiscale geometric representations, you find a tutorial paper on those directional 2D wavelets, starting with an historical perspective (eg Gaussian & Laplacian pyramids): and highlighting details between the main fixed and adaptive decompositions (below: a contourlet atom). It was published in 2011 in Signal processing, special ... 1 It's due to the weighting function used in the Gaussian pyramid downsampling. Lets take a 1D patch(1D patch because it makes our calculation easy and also the kernel is separable) with p10,p11,p12,p13,p14,p15 in the Level 1. Assume maximum gradient possible is Max - Min between any two pixels. The pixel p20 in level 2 is formed by influence of pixels p10,... 1 Apparently, you are not doing wrong. You are doing a$k\$-level transform on the rows, then on the columns. Giving you rectangular subband. This is fine, and is one of the modes of DWT in 2D. It is called non-separable, standard, anisotropic, and there is a bibliography in A panorama on Multiscale Geometric Representations (2D wavelets). The classic square ...

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Yes, most of the known multiscale or multirate decompositions, as long as they combine at one stage a non-ideal filter and a subsampling operator, induce some kind of aliasing at the analysis stage. And the Laplacian pyramid does so. But proper designs allow perfect reconstruction, so the aliasing can be reversed or cancelled, and kept reasonable in the case ...

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Have you looked to the pyUp and pyDown functions in opencv? They upscale and downscale images using a gaussian pyramids. opencv image filtering page

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Your description sounds very similar to the "Red Black" Wavelet Transform. It is a useful transform, and is used for instance in "Edge Avoiding Wavelets".

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