Questions tagged [image-processing]
Mathematics of image processing, variational methods (i.e. methods from calculus of variations), questions about denoising, deblurring, segmentation, image registration, imaging modalities (e.g. computed tomography, ultrasound, magnetic resonance tomography)
39 questions
1 vote
0 answers
101 views
How can we calculate the Euler-lagrange equations?
In this paper https://arxiv.org/pdf/1907.09605.pdf \ let $\Omega \subset \mathbb{R}^n$ with $n \geq 1$ be a bounded Lipschitz domain with boundary $\partial \Omega$, $f: \Omega \rightarrow \mathbb{R}$ ...
- 11
1 vote
0 answers
215 views
Reconstructing an object from its shadow
I'm looking into the section "Reconstructing an object from its shadow" in the book Introduction to the Mathematics of Medical Imaging by Charles L. Epstein. I have two questions The ...
- 11
2 votes
0 answers
242 views
Graph Laplacians, Riemannian manifolds, and object collisions
To preface this question, I am a part-time game developer and full-time optimization fiend. I am working on object collisions at the moment and many resources I have found online are more-or-less just ...
- 21
0 votes
0 answers
151 views
Is this formula for 2D Fourier integral of diffraction kernel correct?
Well I have a function parametrized by $z$ $$g_z(x,y) = \frac{z}{i \lambda r^2} e^{i k r}, \quad r = \sqrt{x^2+y^2+z^2},$$ where $\lambda > 0$ is real constant and $k = \frac{2\pi}{\lambda}$. This ...
- 151
1 vote
0 answers
126 views
1D representation of 2D discrete Fourier transformation [closed]
I'm not too familiar with image processing, so I need a little help: In general, if we transform a discrete function $f$ with $n$-variables from the "spatial domain" using the Fourier ...
- 11
1 vote
0 answers
91 views
elaboration on the equation of directional derivative that lead to steepest gradient descent [closed]
I am reading the book Ian Goodfellow and Yoshua Bengio and Aaron Courville, Deep Learning, MIT Press, 2016. I am reaching to the point about directional derivative. Given the $u$ as the unit vector ...
- 11
1 vote
0 answers
93 views
Apply gaussian blur to get original image [closed]
Suppose I have an image A. Is it possible to construct an image A' from A so I can get the ...
- 21
1 vote
0 answers
101 views
Mandelbrot's lacunarity realized by fractal or stochastic field?
It is my understanding that Mandelbrot came up with the notion of lacunarity to classify the homogeneity of 2D functions that only take two distinct values see here. I wonder, does there exist a ...
- 73
1 vote
0 answers
140 views
Analytically compare two 3D heatmaps of the brain
I have two heatmaps of a 3D model of the brain, with the color of each pixel being an intensity of the response to a stimulus, and I want to get a metric of how "alike" those two heatmaps ...
3 votes
2 answers
326 views
Continuity of Radon transform w.r.t the angle
Let $f \in L^1(\mathbb R^n)$ (or in case it helps, actually a probability density on $\mathbb R^n$). Define the Radon transform $R[f]:S_{n-1} \times \mathbb R \to \mathbb R$ of $f$ by $$ R[f](w,b) := ...
- 7,033
1 vote
1 answer
429 views
Hölder continuity of Radon transform of smooth function
Given an integrable function (e.g a probability density function) $f:\mathbb R^n \to \mathbb R$, let $R[f]$ be its Radon transform defined by $$ R[f](w,b) := \int_{\mathbb R^n} \delta(x^\top w - b)f(x)...
- 7,033
1 vote
0 answers
208 views
Geometrical interpretation of back projection operator or adjoint of Radon transform
If $f \in C_{c}^{\infty}\left(\mathbb{R}^{2}\right)$, the Radon transform of $f$ is the function $$R f(s, \omega):=\int_{-\infty}^{\infty} f\left(s \omega+t \omega^{\perp}\right) d t, \quad s \in \...
- 309
5 votes
2 answers
264 views
Fast algorithms for calculating the distance between measures on finite ultrametric spaces
Let $X$ be a finite ultrametric space and $P(X)$ be the space of probability measures on $X$ endowed with the Wasserstein-Kantorovich-Rubinstein metric (briefly WKR-metric) defined by the formula $$\...
- 44k
4 votes
1 answer
325 views
Advantage of fractional Fourier transform over multiscale wavelet
What is the best argument of fractional Fourier transform over multiscale wavelet in data analysis purpose. Optimization of the good time-frequency domain parameter ? "Good" will be, find the %time-%...
- 41
0 votes
1 answer
2k views
Suggestions for Math Equation OCR tool with API [closed]
I am looking forward to converting equation images to tex/mathml. All the equations are computer printed. So, not so worried about the clarity. We have around 100K images and hence looking for some ...
- 17
-1 votes
1 answer
87 views
Calculating Average Ratios Of Human Faces From Images Of Differing Lengths To The Camera [closed]
For a project I'm creating a program that must analyse a database of images to define average ratios for certain parts of the face (i.e. distance between eyes, distance from nose to chin etc.). I've ...
- 5
52 votes
1 answer
5k views
Mathematics of imaging the black hole
The first ever black hole was "pictured" recently, per an announcement made on 10th April, 2019. See for example: https://www.bbc.com/news/science-environment-47873592 . It has been claimed that ...
- 1,821
3 votes
1 answer
232 views
Why do we consider some weakening frames like K-frames, frame sequences, and upper semi-frames?
I have found some applications of the Frame Theory in engineering sciences like signal processing, image processing, data compression, sampling theory, optics, filter-banks, signal detection. As we ...
- 4,140
3 votes
1 answer
508 views
Updating Geman and Geman (1984) on image restoration
I am reading the seminal paper Stuart Geman and Donald Geman, Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images, IEEE Transactions on Pattern Analysis and Machine ...
- 75
2 votes
0 answers
190 views
Lower bound for the sum of cosines between singular vectors of diagonally dominant matrices
Let $A \in \mathbb{R}^{n \times n}$ be a nonsymmetric diagonally dominant matrix with $a_{ij} < 0$ $\forall i \ne j$ and $a_{ii}>0$. Let the singular value decomposition of $A$ be $A=U \Sigma V^...
- 323
4 votes
2 answers
245 views
Geometrical interpretation of pictures transforms and other "high dimensional everyday objects"
During the preparation of a general audience talk on why mathematicians use dimensions higher than three (or four) even for concrete applications, I came up with the following enjoyable observation : ...
- 2,155
6 votes
2 answers
484 views
Mathematical physics applications in present-day image processing
During the past few years several important areas of image processing and image classification or generation became dominated by convolutional neural networks. I'm interested if there are any methods ...
- 161
13 votes
2 answers
1k views
Why does this Moiré pattern look this way?
I was making some gifs of Mobius transformations in Matlab, and some strange patterns began to appear. I'm not sure if a deeper knowledge of the filetype/algorithm is needed to understand this ...
- 397
1 vote
0 answers
93 views
The jump set of $SBV$ function for different value of parameter in image denoising problem
The classical Mumford-Shah image denoisng problem study the minimizer of the following functional, for each $\alpha>0$ where $\Omega\subset \mathbb R^2$ is open bounded with sommth boundary, $$ u_\...
- 699
9 votes
2 answers
988 views
When is a mapping the proximity operator of some convex function?
Is there a characterization of mappings $p : \mathbb R^n \rightarrow \mathbb R^n$ which are proximity operators (in the sense of Moreau) of l.s.c (extended) real-valued functions ? That is, given $p : ...
- 7,033
3 votes
0 answers
119 views
Can Mumford-Shah functional be adapted to lower $L^1$ space?
The well know Mumford-Shah functional functional $$ F(u)=\int_\Omega|\nabla u|^2+\mathcal H^{N-1}(S_u) \tag 1 $$ where $u\in SBV(\Omega)$ and $\nabla u$ is the absolutely continuous part of ...
- 699
3 votes
1 answer
263 views
Why is it important to know if a frame is a Parseval frame?
I understand that a Parseval frame is one in which both upper and lower frame bounds equal 1. What's the main advantage to having this be the case? Or, more specifically, if I'm constructing a frame ...
- 131
4 votes
0 answers
147 views
The best constant in Poincare-liked inequality in $BV$ and $BD$ space
This question has been posted on Math Stack exchange for a while and received no response. So I decide to move it here to get more attention. Let $\Omega\subset \mathbb R^N$ be open, bounded and with ...
- 699
1 vote
1 answer
153 views
question about $TGV^2$ space
Let us just stay in $\mathbb R^1$. The space $TGV^k$ is defined as the function $u\in L^1(I)$ and $$ TGV^k(u,I):=\sup\left\{\int_I u\,\phi^{(k)}\,d\mu, \,\phi\in C_c^\infty(I),\,\|\phi\|_{L^{\infty}(...
- 699
5 votes
3 answers
1k views
Looking for techniques of How to measure the Similarity/Dissimilarity between two images?
I would like to compute the similarity/dissimilarity between two images L and R. I know one way which is : computing the histogram of blocks of each image, and then using Bhattacharyya measure I ...
user74794
2 votes
0 answers
162 views
sets with positive reach with complementary set with positive reach
I am interested in bibliographical references about a special class of sets, those who have positive reach and which complementary has also positive reach. I recall that the reach $R\geq 0$ of a set ...
- 1,362
9 votes
3 answers
4k views
Spectra of spatial and temporal covariance matrices
Suppose $(x_i(t))$ is a $n$-dimensional time-series, where $t$ is an integer between $1$ and $T$ (time is discrete) and $i$ an integer between $1$ and $n$, and I assume $n<T$. From this time-series,...
- 860
1 vote
0 answers
194 views
Indecomposability of image transformations (pure algebra). Open questions
W-transformations -- definitions We will consider a class called finite window transformations $\ T:C^\mathbb Z\rightarrow C^\mathbb Z\ $ defined a paragraph below; $\ \mathbb Z\ $ is the ring of ...
- 7,600
15 votes
2 answers
4k views
Switching from pure mathematics (e.g. geometry) to more applied areas (e.g imaging) after Ph.D., as postdoc and chance of getting such a postdoc?
Before I start my question, I should probably mention that this question might not be the right question to ask here, but I tried academiabeta, and stackoverflow, but without getting any to-the-point ...
2 votes
0 answers
1k views
Optimal transport warping implementation in Matlab
I am implementing the paper "Optimal Mass Transport for Registration and Warping", my goal being to put it online as I just cannot find any eulerian mass transportation code online and this would be ...
- 491
1 vote
2 answers
400 views
Smooth a matrix
I have a matrix in which each element contains the coordinates of a 3D surface. Sometimes, some points will be "out of line" meaning that they will not conform to the general shape. For example you ...
- 87
0 votes
0 answers
757 views
Decomposing max-convolution of sum of functions ?
Hello. $R, F_1, F_2, F_3$ are random (not-convex, not-concave) 2D matrices of size 100x100. $R$ is a linear combination of $F_1, F_2, F_3$. Explicitly, $R = w_1 F_1 + w_2 F_2 + w_3 F_3$ where $w_1,...
11 votes
1 answer
5k views
How to un-pixelate pixelated regions in films?
Everybody knows the effect of pixelated objects (e.g. faces) on TV. Is there a way - and which mathematical method lies behind it - to un-pixelate the region? Beware: I am not talking about smoothing ...
- 6,073
2 votes
1 answer
443 views
Given a function $f(t): \mathbb{R}\to\mathbb{R}^n$, can 2D, or $n$D discrete Fourier transforms be used on $f(t)$ to perform frequency analysis?
$\DeclareMathOperator{\R}{\mathbb{R}}$Frequency analysis is often performed on wave forms (1D DFT = discrete Fourier transform), and images (2D DFT), where the function in question often takes the ...
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