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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.

52
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2answers
16k views

What do eigenvalues have to do with pictures?

I am trying to write a program that will perform OCR on a mobile phone, and I recently encountered this article : Can someone explain this to me ?
52
votes
2answers
8k views

Mathematical explanation behind a picture posted (lifted from facebook)

In this image given below, there is an actor's (famous south Indian actor Rajinikanth) image which can be seen only if you shake your head ! I had lifted this from Facebook. I am just curious to know ...
16
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2answers
42k views

What is divergence in image processing?

What is the difference between gradient and divergence? I understood that gradient points in the direction of steepest ascent and divergence measures source strength. I couldn't relate this to the ...
16
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3answers
20k views

What does the Fourier Transform mean in the context of images?

This is clearly a very important equation with tonnes of properties that I see come up a lot in image processing literature, but I don't understand why this equation is important, and what it is ...
10
votes
2answers
15k views

What are the limitations /shortcomings of Fourier Transform and Fourier Series?

I am fond of Fourier series & Fourier transform. But every approach has some outcomes and some shortcomings. It's limitations lead to innovation of new approach. So, can anybody explain about ...
9
votes
1answer
475 views

A mathematical way to represent an image kernel?

How to represent the calculation in this image mathematically? For example: With the discrete convolution and Fourier Transform. It tries to do a calculation on the original image (image $A$/input) ...
9
votes
1answer
299 views

How to measure the irregularity of a hexagon?

I need to evaluate the quality of a list of machine parts, which roughly has one center point surrounded by 6 exterior points. If the quality is good, then the 6 exterior points will form a regular ...
8
votes
2answers
7k views

Why is $8 \times 8$ matrix chosen for Discrete Cosine Transform?

In JPEG and MPEG, why is $8 \times 8$ matrix chosen for Discrete Cosine Transform? Why not any other, say $64 \times 64$?
8
votes
4answers
6k views

Laplacian 2D kernel - is it separable?

I'm wondering if the 2D laplacian kernel 0 1 0 1 -4 1 0 1 0 is also a separable kernel. How can I find that out?
8
votes
1answer
268 views

Homotopy and watershed

homotopy is a new word to me. Upon trying to understand this property, I immediately think of another well-known segmentation algorithm: watersheds. I see that watershed should exhibit some ...
7
votes
1answer
10k views

How can I combine affine transformations into one matrix?

So from what I understand from this picture, the box is stretched to twice its width. And it is then flipped from the x-axis. And then it is rotated 30 degrees anticlockwise. So these three ...
6
votes
1answer
24k views

What does it mean to divide by the standard deviation?

I'm trying to "variance-normalize" an image. In order to do so, I subtract the mean from the pixel value (to have a $0$ mean), and divide by the standard deviation (to have a unit variance) right? But ...
6
votes
2answers
3k views

Two Quadrilaterals Intersection Area (Special Case)

I have two intersecting quadrilaterals (the area of intersection is the grey polygon with thick boundary): These properties holds: One quadrilateral is always a rectangle There is always some ...
6
votes
1answer
1k views

any idea what fractal algorithm might generate this shape?

I Found this image around, and i'm curious what algorithm generates this kind of shape In particular, i'm curious how the flow lines are generated, since usually the Mandelbrot iteration just ...
6
votes
2answers
2k views

Image Reconstruction:Phase vs. Magnitude

Figure 1.(c) shows the Test image reconstructed from MAGNITUDE spectrum only. We can say that the intensity values of LOW frequency pixels are comparatively more than HIGH frequency pixels. $$ f(x,y)=...
5
votes
1answer
981 views

How to mark rational points on a sphere

I found this picture on mathoverflow, which I find very intriguing and so I like to know how to draw such an image with a simple computer program. To calculate the rational point, I can draw a line ...
5
votes
1answer
2k views

How to resize an image?

I am not sure about the title of this question, so if someone knows an appropriate one, please rename it. It's a programming related question (but doesn't involve any programming). I posted it on ...
5
votes
4answers
5k views

PCA - Image compression

I have 2 questions related to principal component analysis: The first is, how do you prove that the principal components matrix forms a orthonormal basis? Are the eigenvalues always orthogonal? The ...
5
votes
5answers
2k views

The mathematics behind Fourier Transform for Image Processing

I am following http://homepages.inf.ed.ac.uk/rbf/HIPR2/fourier.htm . I understand the application of Fourier Transform behind Image Processing, but right now, I am curious about the mathematics behind ...
5
votes
1answer
115 views

A property of the x-ray transform

Problem: Let $P_{\theta}f(x) = \int_{\mathbb{R}}f(x + s \theta) ds$ be defined as the x-ray transform, where $\theta \in S^{n-1}$, and $x$ belongs to $\Theta^{\perp}$, the hyperplane that passes ...
5
votes
2answers
356 views

Shortest distance between two digital blobs

We have two digital blobs of arbitrary shape, i.e. two connected components on a digital image. They are described by a run-length-coded representation, i.e. a list of $(x, y, l)$ triples of integers. ...
5
votes
1answer
90 views

How can I quantify how 'evenly distributed' a given set of points are in a 2D plane?

I am working on an estimation problem involving cameras, where the quality of a location of the camera is quantified by multiple factors: such as how many feature points (salient features in a given ...
5
votes
1answer
207 views

Compute the image intensity on a spherical surface under orthographic projection

I got stuck on the folllowing exercise: Consider a spherical surface of radius $r$ centered at the origin with equation:$$z = d - \sqrt{r^2 - x^2 - y^2}, \quad x^2 + y^2 \leq r$$. The surface is ...
5
votes
0answers
276 views

When is a mapping the proximity operator of some convex function?

Sorry for cross-posting from MO. It's been a few days and the question hasn't received any attention there. So, is there a characterization of mappings $p : \mathbb R^n \rightarrow \mathbb R^n$ which ...
4
votes
1answer
416 views

Minimization problem as PDE

In the article "An Image Interpolation Scheme for Repetitive Structures" Luong, Ledda and Philips propose the following approach to denoising digital image. They consider that regularized total ...
4
votes
2answers
103 views

What does oplus symbol ⊕ do for 2 images in Convolution Neural Networks

So I'm reading this paper on optical flow prediction from two image frames, and I'm having a difficult time finding what this operator does. This paper, and some other ones uses it on the outputs of ...
4
votes
1answer
86 views

How does Fourier transform convert from time domain to frequency domain

I understand the formula, but what does it even mean for a Fourier transform to convert from time domain to frequency domain? And how is it doing so? Even, in the field of image processing, why do ...
4
votes
1answer
457 views

What are Basis images?

I have read that using Fourier transformation we can decompose any arbitrary image into orthogonal basis images and reconstruct it back. But what are basis images actually?
4
votes
2answers
3k views

Scale Space - Scales and Octaves

So I'm desperately trying to understand scale space for signals, specifically for 2D images... I'm having trouble with algorithms that discuss creating a pyramid. Specifically, I don't understand how ...
4
votes
1answer
1k views

Unscramble images without trying all permutations

I try to write an algorithm that unscrambles images that were before scrambled by mixing up small blocks: My idea is that in the bottom image there are more "sharp" corners compared to the image ...
4
votes
3answers
254 views

Robust line segment fitting to a digital path

Robust line fitting to a set of 2D points is a well studied problem for which several approaches are known. They usually consider the point cloud as unstructured. I call a digital path a sequence of ...
4
votes
2answers
9k views

About Sum of Squared differences

I found a paragraph in the book about $SSD$, can't get one thing: Most commonly, the distance measure is the sum of squared differences. For two images $f(x, y)$ and $g (x, y)$ it is defined as $...
4
votes
0answers
73 views

Shearlets - Understanding Anisotropic Features

For $\psi \in L^2(\mathbb R^2)$, the continuous shearlet system $SH(\psi)$ is defined by $SH(\psi) = \{\psi_{a,s,t} = T_t D_{A_a}D_{S_s}\psi:a>0,s\in \mathbb R, t \in \mathbb R^2$}, where $D_{A_a}...
4
votes
0answers
435 views

I am looking for a mathematical equation to warp an image [closed]

Theoretically, I know that to warp an image, each pixel $(x,y)$ in the source image is transformed to $(x', y')$ using a function f (i.e. $x'=f(x,y)$ & $y'=f(x,y)$ ). But what mathematical ...
4
votes
1answer
39 views

What is the simplest way to extract a rough orientation statistics from images

Which is the fastest method to extract a rough orientation statistics from images. I think the most precise way is the scanning with local Gabor filters, but its very time consuming. Is it possible to ...
4
votes
1answer
640 views

Texture mapping from a camera image (knowing the camera pose)

I'm not sure if I should ask this question here or on stackoverflow, so forgive me if I'm wrong. I want to apply a texture (taken from a camera) on a 3D surface, let me explain my problem: I have ...
4
votes
1answer
114 views

How to solve cross-products including matrices?

I'm a programmer and I'm doing a whitebalance-transformation in RGB colorspace. This should work with this transformation matrix that I've found in literature: $$ \begin{pmatrix} R \\ G \\ B ...
4
votes
0answers
664 views

Reprojecting/converting an orthographic image/grid into a cartesian grid

I'm trying to dewarp a fisheye image into a simple rectilinear image of a subset of the fisheye. As part of this, I'm trying to map the azimuth/altitude values into a point on the image. The points ...
3
votes
5answers
914 views

How do I find/predict the center of a circle while only seeing the outer edge?

Question What formula would allow me to predict the center of this circle? In addition, what attributes of this image must be detected in order to predict the center? I figured understanding the ...
3
votes
2answers
243 views

The old and modern definitions of total variation are actually equivalent?

According to wikipedia, the total variation of the real-valued function $f$, defined on an interval $[a,b]\subset \mathbb{R}$, is the quantity $$V_b^a=\sup_{P\in\mathcal{P}}\sum_{i=0}^{n_P-1}\left | f(...
3
votes
2answers
87 views

Heat Equation derivative in terms of Laplace

If the heat equation is $ \frac{\partial u}{\partial t} - \alpha \nabla^2 u=0$ Is the second derivative of u w.r.t t is the laplacian of the lapacian?
3
votes
3answers
5k views

Why is central difference preferred over backward and forward difference in convolution?

It is mentioned in some literature that we should always use central difference when computing the derivatives of an image instead of forward or backward difference. Does anyone knows why is that? ...
3
votes
3answers
12k views

How do I compute the gradient vector of pixels in an image?

I'm trying to find the curvature of the features in an image and I was advised to calculate the gradient vector of pixels. So if the matrix below are the values from a grayscale image, how would I go ...
3
votes
2answers
156 views

How can I measure the properties of a Point Spread Function?

What quantity or property can I use that describes by how much a point spread function distorts/blurs an image?
3
votes
2answers
3k views

How do I apply a Gaussian Blur (low-pass filter) to an image made up from a set of points?

I have an image encoded in the form of a list of points, like so: ...
3
votes
2answers
3k views

Back-projecting Pixel to 3D Rays in World Coordinates using PseudoInverse Method

For perspective projection with given camera matrices and rotation and translation we can compute the 2D pixel coordinate of a 3D point. using the projection matrix, $$ P = K [R | t] $$ where $K$ ...
3
votes
2answers
112 views

Largest four line segments of polygon

I have some polygon (see darkblue contour): It consists of very small segments, pixel by pixel, so angles differ although they seem to be the same. Visually we see 4 large line segments. How can I ...
3
votes
1answer
1k views

Homography with line correspondences

When calculating a homography with line instead of point correspondences, what is the derivation of the formula: $$ l_i = H^T\cdot l^{'}_i $$ I know that: $$ l^T\cdot x = 0 \quad\text{and}\quad l^{'...
3
votes
2answers
213 views

How can I convert camera motion into zoom?

I try to reconstruct a camera of a video sequence via match moving techniques. After the reconstruction process all seemed to work as expected, but then I've realized my camera is moving forward ...
3
votes
1answer
162 views

Geometric Transformations of Images

Related to image processing, I'm familiar with different types of geometric transformations. Translation, scaling, similarity, Affine, Perspectivity, and Projective Transformation. Is there a ...