Questions tagged [image-processing]
This tag is for the mathematics involved in the field of image processing. Many such questions are also appropriate for Signal Processing Stack Exchange.
654
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Additive white Gaussian noise (AWGN)
In my research work regarding wireless communication, I came across many research papers wherein AWGN is assumed to be modelled as "complex Gaussian with zero mean and unit variance".
I ...
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Get 3d ray in world coordinates from a 2d pixel location
I am trying to project a 3d ray from a 2d image pixel location. I have the actual 3d coordinates of a pixel location and want the ray from the camera to intersect this point but currently, my line is ...
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Transforming an equation from complex space to quaternion space and how phase is interpreted?
I'm new to quaternions im trying to find an extention of my equation from complex space to it's from in the complex space so i have this so to start i have this equation
\begin{equation}
I_{f}(p,q) = ...
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Discrete Fourier Transform of the Gaussian
I encountered the following question in a Digital Image Processing examination:
Find the 2D DFT of $\frac{1}{2 \pi \sigma^2} e^{-\frac{(x - x_0)^2 + (y - y_0)^2}{2 \sigma^2}}$ where $x_0, y_0$ are ...
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$2$-dimensional Fourier transform of the same function after different sample resolution
I have a function $f :\mathbf{R}^2 \to \mathbf{R}$ under different discretizations, e.g., $64 \times 64$ and $32 \times 32$ (can be viewed as a $64 \times 64$ image downsampled to $32 \times 32$). Now,...
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What are the family of geodesics/curves between point weights/balls on an elastic sheet?
ASSUMPTIONS:
no deformation is separate, all deformations touch with at least another points deformation, i.e., no deformation is by itself.
point force preferred, but small dense balls okay too.
The ...
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histogram equalization and CDF
Im having hard time understading one of the derivation steps in histogram equalization, Wikipedia (added picture).
derivation
In the last sentence it seems to me like they are suggesting that the CDF ...
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Efficient algorithms to detect connected components
I am reading the book "Computational Homology" by Tomasz Kaczynski, Konstantin Mischaikow and Marian Mrozek and in several places it says something to the effect of "of course, from the ...
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Mean Squared Error incompatible definitions
Let $x$ be a vectorial estimator, and we aim to estimate $y$. Then the mean squared error can be defined as $MSE(x) = \mathbb{E}[\|x-y\|^2]$. On the other hand, in image processing the mean squared ...
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Finding an algorithm EF[1,1] and PO division for more than two agents
From this research paper I want to write an algorithm for finding envy-freeness(EF) and Pareto optimality(PO) division for more than two agents.
We consider the problem of fairly and efficiently ...
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Convolving an image with mean filter infinite times
I was taught in my image processing class that when a mean filter is applied infinite times on a given image, the intensity of each pixel reaches the same value. I understood this that time entirely ...
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Extrapolating 2D image pixels using second order derivative
I intend to use image pixel data prediction to improve image compression. This means that I need predict image pixels based on previous rows and columns. I cannot use the next rows and columns for the ...
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Weighted sum of N images, which minimizes their TV norm
I have $K$ images $I_i, i\in{1 \ldots K}$ of the size $M \times N$. I wish to find weights $w_i$, s.t. $w_i \in [0,1]$ and $\sum_1^K
w_i = 1$ so that
$$|\sum_{i=1}^{K} w_i I_i|_{TV}$$
is minimal. I ...
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What determines the coefficients of the laplacian filter?
I am building an approximately isotropic Laplace kernel based on the guidance in enter link description here. What I don't understand is the derivation process from (2) to (3). I don't know why the ...
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Convolution of a $2$-dimensional function with a $1$-dimensional function
Below is the schematic diagram from a very popular paper in this field of retinotopy.
In the diagram above, $g$ is a bivariate Gaussian function of the form
$$ g (x,y) = \exp \left( - \left( \frac{ (...
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What is the distribution of distances between two random points in RGB space?
Suppose we pick pairs of triples from $\{ 0, 1, 2, \dots, 255\}^3$ with a uniform distribution and would like to find a closed form for the distribution of the Euclidean distances
$$ d((x_1, x_2, x_3),...
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How to choose pixel value in case of nearest neighbor image down scaling
I am working on implementing a fast algorithm for image resizing (more generally 2D array of numbers resizing).
It is supposed to be fast, and should not add new values (meaning the result 2D array ...
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Pinhole camera projection of 3D multivariate Gaussian
Consider a 3D Gaussian with $3\times 1$ mean $\boldsymbol \mu$ and $3\times 3$ covariance $\boldsymbol \Sigma$ (which is symmetric positive semidefinite):
$$
p(\mathbf x) = \frac{1}{\sqrt{\...
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Finding the dimensions of a box (cuboid) given a hexagon filled in to look like the box
Suppose I have hexagons that like the ones below but I know the area and the points of each hexagon that represent a cuboid of dimensions g,h,d. How can I find the values for g, h, and d? Any pointers ...
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How is the convolution of images properly defined?
In the Wikipedia article, the convolution is defined as $$(g*f)(x,y):=\sum_{dx=-a}^a\sum_{dy=-b}^b\omega(dx,dy)f(x-dx,y-dy).\tag1$$ The image $f$ is obviously understood as a function on $[-a,a]\times[...
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How to interpret this set and corresponding summation?
Background: I'm working on a 2D Moving-Average (MA) estimator for image processing stuff and I'm trying to follow along from a paper that defines the algorithm for estimation.
The main equation for ...
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152
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Determining camera orientation from a Minecraft screenshot
I am currently working on a project where I need to get the orientation of a camera in a Minecraft world using only a screenshot. The purpose of this project is to make seedcracking (the process of ...
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Recreating an image with a Haar function basis instead of Fourier basis
I'm stuck on recreating an image with a Haar function basis. I already succeed to make it with a Fourier basis and I wanted to use a similar approach for a Haar basis. I found some valuables sources ...
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Inverse of image connected component graph labeling
For a given image I one can compute connected components and define a graph structure G on those components. Each connected ...
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Why do these patterns appear in custom fractals? [closed]
Last year I programmed a fractal renderer and played around a lot with fractals such as the Julia set and Mandelbrot set. Eventually I got curious and inputted my own algorithms for the fractal, and ...
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Trying to understand an application of expectation-Maximization algorithm
In this image downscaling paper, the authors apply the Expectation-Maximization algorithm to solve an image downscaling problem. I've never done Bayesian statistics, so I find some parts a bit hard to ...
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Fast Fourier Transform in computer vision
Can someone explain me how does FFT works in computer vision, please. I know something about FFT as an algorithm of competitive programming but I can't understand how it perform an image in computer ...
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Calculate point coordinates after image rotation
So I have two rectangle pictures with known Resolution
On the first image we have a point on $(900, 300)$ $x,y$ pixel coordinate (guessed number, not exactly).
The second picture is rotated by $15^\...
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Linear algebra: is there a matrix representation of cropping an image? [closed]
Can I write a matrix description of cropping an image to a given size? It doesn't even need to be an image, simply some rectangle $\square ABCD$, and a crop boundary $\square EFGH$. Is there a ...
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alpha in Cauchy weight function for edge weights
I'm reading an article about graph construction for image compression, and it says that cauchy and gaussian are commonly used weighting functions to determine edge weights. The cauchy is:
$$\omega_{ij}...
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Measuring the consistency of aspect ratio
Given the following rectangle:
To measure the consistency of the aspect ratio between the two rectangles, the following formula is applied:
$$
v = \frac{4}{\pi}(arctan\frac{w^{gt}}{h^{gt}}-arctan\...
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How can one construct polynomial bases for positional estimation?
In the field of signal processing a popular problem is to try and estimate how something has moved as compared to a previous point in time.
In attacking this problem a long standing popular approach ...
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Solving a ratio of Lambert W branches: $y=\frac{W_{-1}(x)}{W_0(x)},$ where y is positive and real.
How might I go about solving a ratio of Lambert W branches? Namely,
$y=\frac{W_{-1}(x)}{W_0(x)},$ where y is positive and real.
Motivation:
I want to define a Cauchy filter for edge detection given ...
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Relationship between the discrete cosine transform and the Karhunen–Loève transform
The Wikipedia article on Discrete cosine transform states:
For strongly correlated Markov processes, the DCT can approach the compaction efficiency of the Karhunen-Loève transform (which is optimal ...
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Connection between image space and Hough space? [closed]
The other day I got very interested in learning about Hough Transform that is used to detect edges in images.
After going through OpenCV documentation, I still couldn't piece together an understanding ...
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Map coords of a 3d rendered image of a textured grid to the coords of the texture using a height map
I'm not sure if this is the right place to ask questions like mine, so forgive me if it isn't.
Here's what I'm trying to do:
I have a rendered image of a grid with a texture on it, that is deformed: ...
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Calculate magnitude of the gradient using higher order statistics
I am making a model for detecting blurred part of an image. I'm using features described in the paper Blurred Image Region Detection And Segmentation by Hyukzae Lee and Changick Kim, and I have a ...
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How can a 2D convolution retain the size of the first matrix
I am supposed to create a 2D convolution function for image filtering that works for both padded convolution and "convolution of the same size". I know that if we have a matrix A of size N x ...
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How to generate the order-n Fourier basis transform given a 4-d state in Python?
Let's say I have the following 4-d vector x = $[x_1, x_2, x_3, x_4]$.
I want to get the n-basis Fourier transform for the above x how would I do that? I can't seem to find anything on Python that ...
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What is the result of the filtering of a 8x8 picture that goes by a SQRT VAR(N(n,m)) filter?
A picture 8x8 is defined as follows:
$I[m,n] = {200 if (round(n/2)+m)mod8) = 0 or n=m 0 otherwise}
This picture goes through a 3x3 filter defined as:
$T(I[m,n]) = \sqrt{ VAR(N(n,m)) }
around the ...
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Prerequisits on eulerian video magnification
From this article http://people.csail.mit.edu/mrub/papers/vidmag.pdf I became extremely interested in the subject, but I couldn't find too much content that starts from the "basics". This ...
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How to normalize signal from 3 images
My question is mathematical related but it concerns a bit biology since they are images of cells. Image we have 3 images of negative control. Since I have to only use 1 I need to normalize these 3 ...
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Name / type of image filter
Occasionally I stumble across a very pretty image filter and I always wonder what the underlying algorithm is. The term wavelet-transformation pops up in my head but my searches let to nothing.
It ...
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Unit and step of coordinates of spatial frequency Discret Fourier Transform
I'm trying to understand spacial frequency Fourier Transform.
If I perform the 2D Fourier Transform of an image of $n_x*n_y$ pixels and I know that the measurement step is $\Delta L$ (in $\mu m$) in ...
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Viterbi algorithm for object-tracking
I have a sequence of images, and I need to find and track the creation of the objects, then their movement and then their disappearance. There can be up to $3$ objects overall, and sometimes there are ...
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Summing the discretized values of the density function of $\mathcal{N}(0, \sigma)$ with a step of $1$ is equal to $1$?
Given the density function of the Gaussian distribution $\mathcal{N}(0, \sigma)$:
$$ \frac{1}{\sigma\sqrt{2\pi}} \exp\left(-0.5\,\frac{x^2}{\sigma^2}\right)$$
I have noted that emprically, summing ...
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Trying to implement a loss function read from a journal-article in python
Computer science undergrad here. I am trying to understand Eqn 12 from this paper so that I can implement it in python code. In this paper, the NN model takes a blurred image as input and outputs a ...
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Can we use mathematics and logic to estimate probability of extremely absurd events?
I'd like to detail my question over the example below.
Let's say I have a random pixel generator which has $1024 \times 768$ screen resolution. It also has $24$ bit color which means $2^{24}= 16,777,...
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Combining two fundamental matrices
Let $\mathcal{F_{ab}}$ be the fundamental matrix obtained from images $A$ and $B$
$$ \mathcal{F_{ab}} = \begin{bmatrix}
ab_{11} & ab_{12} & ab_{13} \\
ab_{21} & ab_{22} & ab_{23} \\
...
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Frequency peak always appearing at half of Nyquist frequency in Fourier transform
When taking the FT of a signal I always get a sharp peak at exactly half the Nyquist frequency. My signal is shown here:
and its FT here:
The Nyquist frequency is 36.7 KHz. As can been seen in the ...
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