Questions tagged [computer-vision]

Mathematical methods used in widely understood computer vision.

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Finding Rotation and Translation from 2 known sets of data

As illustrated in the Figure above, I have 2 point clouds $X$ and $X'$ which initially lie on the same plane $\mathcal{P}$. The orientation of $\mathcal{P}$ - i.e. both Rotation $\mathbf{R}_0$ and ...
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Why the epipoles are the Fundamental matrix right and left null space in Computer Vision?

I have known that the Fundamental matrix is $\operatorname{rank}(2)$,and all the epipolar lines meet at the epipoles. Given this identity $(e')^T\cdot (Fx)= \mathbf 0$, in order to make this equation ...
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Deriving the essential matrix using the scalar triple product

I'm trying to test my understanding by deriving famous results from computer vision. This one: the essential matrix. I'm using the scalar triple product to encode the co-planarity constraint. I'm ...
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Object detection uncertainty due to camera limitations

I have a 2-D image of an object, and am using this to detect an object and estimate the object's position relative to the camera's pose (say, as a vector between the origin of the camera's coordinate ...
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Why are these matrices oddly sized?

With regards to the above image which was delivered as a slide in a Computer Vision course, I can't understand why this matrix multiplication is represented as a $4\times4$ * $4\times1$ where I feel ...
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How can I turn a mirrored image of an object into a natural perspective of itself?

I am trying to get the 3D mapping of a fast moving object and only have 1 high fps camera on-hand. Thus I am opting for a setup using a mirror to reflect the 2nd perspective to my camera. The software ...
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29 views

Constants in the mathematics of sRGB [closed]

I am learning about color spaces in digital images, and while researching the precise definition of the sRGB space on wikipedia, found what seemed to me as an unnecessarily complex, full of (to me) ...
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Notation for probability distribution

I am very confused by the notation used here: https://hci.iwr.uni-heidelberg.de/vislearn/HTML/publications/papers/2016/Brachmann_Uncertainty-Driven_6D_Pose_CVPR_2016_paper.pdf The distribution $P_i^D(...
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How to compute the RMS(root-mean-squared) distance between 2D points and the origin?

In Hartley and Zisserman's book Multiple View Geometry in computer vision, when it comes to data normalization, it states Namely the points should be translated so that their centroid is at the ...
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DIY 360 Cam: Fast Fisheye to Equirectangular Image Transform

I'm working on a DIY 360 camera project in which I stitch together the images from 180 fisheye cameras. The images from the 180 fisheye lenses are 2D circles. I wrote a python program that ...
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Graduate School in Computer Vision

I intend to apply to graduate school in the near future (I am a few years out of undergrad), and I want to apply to math departments with researchers that study image processing and computer vision. ...
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Understanding a computer vision gravity vector detection algorithm/ Solving for an eigenvector

Supposing we have a depth picture where objects from the picture have estimates $v \in \Bbb R^3$ of their surface normals, vectors on walls that point horizontally, and vectors on flat surfaces that ...
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Do I apply K to [R|T] transformation from solvePnP for a basic pose matrix

Below is an extrinsic transformation matrix [R|t] from the OpenCV PnP Solve function. To convert this output of extrinsic pose to a projective transform requires multiplication with intrinsic ...
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Compute the camera center of the camera and the principal axis

The subject is Pinhole Camera. I've given a camera matrix $P = \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix}$ and ...
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Geometric interpretation of projection of 3D point w/ camera matrix

The subject is Pinhole Camera. I've given a camera matrix $P = \begin{pmatrix} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix}$ and a ...
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102 views

What is the projection matrix of reverse (Byzantine) perspective?

I would like to construct a projection matrix for reverse perspective. I'm using OpenGL and tried to modify concepts from this excelent tutorial. I came up with: $$ \begin{bmatrix} 2\frac{(\text{near}...
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Converting image plane coordinates to world coordinates such that they lie in a given plane

Let's say we have an RGBD image and we are interested in a polygon region of this image. The region of interest is defined by 4 points in the image plane. The depth information within this region is ...
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How the Maximally Stable Extremal Region (MSER) formula works in detail?

I'm so confused right now, because I still don't understand how to find maxVariation, minArea, maxArea, and thresholdDelta value in manual calculation in MSER with this formula : $$\Psi(R_i^g)=\left(|...
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Algorithm to project a point onto a geodesic polyhedron?

I have a geodesic polyhedron/icosphere and would like to map an arbitrary point onto its surface through the origin. I devised the following algorithm: For the point, find the three closest vertices ...
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Estimating a cost of a graph smoothness

For a given video scene, I have a task to evaluate a score for its stability. I have managed calculate their deltaX and deltaY ...
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1answer
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Can a homography transform a straight-line into another type of curve?

I am trying to find the homography that maps a certain image to another related image. This can be done by selecting at least $4$ point-to-point correspondences, then we can solve a constrained ...
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camera calibration problem - computer vision

I am new to computer vision and I have a problem which may not require a complex algorithm, however im not sure if I have enough data to ignore this. problem statement: I have a 6-joint robot arm and ...
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Why do homographies transform points contravariantly and lines covariantly?

In the context of projective spaces, a homography is the most general transform. In $\mathbb{P}^2$ (projective space of $\mathbb{R^2}$), a homography can be represented as a matrix $H \in \mathbb{R}^{...
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Calculating whitespace given list of rectangles?

It's part of a software application, but should be irrelevant for this. The main thing is the code generates a set of rectangles (with each having x, y position and width, height). Some rectangles ...
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Extracting vanishing points from the camera projection matrix

I’m struggling with the following paragraph from the book Multiple View Geometry: The columns of the projective camera are 3-vectors which have a geometric meaning as particular image points. ...
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Using trigonometry to find projection of a point at $(x, y, z)$

The following slide is part of a lecture on computer graphics, where the author is explaining the pinhole camera model: I don't understand what "trigonometry" the author is referring to here. It ...
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Understanding the formula of anisotropic filtering (Barcelos et al. (2003))

I am currently trying to understand a paper about shadow removal (DESHADOWING OF HIGH SPATIAL RESOLUTION IMAGERY APPLIED TO URBAN AREA DETECTION, Samara Azevedo et al.), but I am currently stuck at ...
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How to find 3d world coordinates from projection matrix?

I have a 3x4 projection matrix for the camera and a set of 2d coordinates of the desired object in the image. How do I find the 3d coordinates of the object? This is how the matrix looks like: $$ \...
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Normalize homography matrix for use in deep learning

I have a number of homography matrices $H$ which are $3x3$. They map a pixel from a 1280x720 image (width times height, in pixels) to another image. The topleft corner is $(0,0)$. However, for use in ...
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50 views

Is central projection of the imaging process a projective transformation?

I am reading the book Multiple View Geometry in Computer Vision(Second Edition) and encounter some questions. On Page 7, it says In applying projective geometry to the imaging process, it is ...
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Hand Selected Points and Homography precision.

I have two calibrated cameras, so I know the essential an intrinsic matrices and I have computed the homography between a set of hand selected points. They are hand selected because the selection of ...
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How to perform angular warping of images?

I am looking for mesh grid based angular warping, e.g.- using some control points to define target shape and deformed grid is used to perform same manipulation on image..
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How to find an object's 3D coordinates (triangulation) given two images and camera positions/orientations

I am given Camera intrinsics: focal length of the pinhole camera in pixels, resolution of the camera in pixels Camera extrinsics: 3D coordinates (X,Y,Z) of 2 points where pictures of the object were ...
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How to generate Maximum Min-SW gray code?

I'm trying to generate Maximum min-SW gray code for structured light described in this paper. Can someone help to understand how to generate this 10 bit gray code?
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Visualize the interpolated unit quaternion on the surface of the unit sphere

I'm reading the following paper Dam, Erik B., Martin Koch, and Martin Lillholm. Quaternions, interpolation and animation. Vol. 2. Copenhagen: Datalogisk Institut, Københavns Universitet, 1998. On ...
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Epipolar geometry and fundamental matrix: Is the product equal to the distance from the epipolar lines?

Given the fundamental matrix of a stereo camera setup $\boldsymbol{F}$ and two potentially corresponding image points from a left image $(u_l | v_l)$ and a right image $(u_r | v_r)$: The product $$ \...
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201 views

Radius of the paraxial impulse response of a pinhole camera

I have the following pinhole camera: The cone of rays that would enter the pinhole from the object would resemble the following: (Image from https://www.optilayer.com/products-and-services/tools/...
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323 views

Calculating half-angle of cone of rays entering pinhole camera

I have the following pinhole camera: The cone of rays that would enter the pinhole from the object would resemble the following: (Image from https://www.optilayer.com/products-and-services/tools/...
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Why is the null-space of A 2-dimensional if 7 point correspondences are used to estimate the fundamental matrix?

Just have a bit of confusion with understanding the 7-point algorithm for calculating the Fundamental Matrix. When reading notes, I see that the 7-point algorithm uses 7 point correspondences in the ...
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Hi, non-mathematician here, looking for any proof regarding gimbal lock being unavoidable for 3-value rotation representations

So while there are many CG-oriented guides explaining gimbal-lock, they always jump into quaternions without first stating the necessity for doing so in that it's a dead end for deriving rotations ...
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8 degrees of freedom in homography

I am working through the math behind homographies, but my math skills are a bit rusty. A homography can be calculated with 8 corresponding points (4-4) because the homography matrix has 8 degrees of ...
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Finding side length from two scalene triangles with common angle and side and ratio between sides. Trigonometry

I have two scalene triangles with a common angle and side. I would like to find the length of $x_2$. I have all variables in green: the angles $\angle$ B,$\angle$ C,$\angle$ D,$\angle$ E and side "...
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Calculating FPIR/FNIR the way NIST does

My question is about the metrics NIST uses to calculate FPIR (false positive identification rate) and FNIR (false negative identification rate). Their official report on testing biometric vendors (...
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Is it possible to constrain an ellipsoid knowing projected points coordinates on image plane?

Simple Case If we know the 3D position of the points and know they are on an ellipsoid Q, we could directly use the equation: $X^TQX=0$, where Q is the 4x4 matrix. Question I'm wondering is it ...
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Minimizing the Frobenius norm for projecting on the essential space

In order to understand the eight-point algorithm I have some trouble with the proof of theorem 0.3 from https://cs.gmu.edu/~kosecka/it835/lect4.pdf in the first step. The idea is to show that for a ...
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Identifying that two planes touch and “merging” them

I have a series of finite planes defined by 4 corner points. Some of the planes are "touching" (to a certain small tolerance), and I would like to identify these touching planes and "merge" them into ...
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Solving PDE produced for $C^1$ isometric embedding and its complexity analysis

I am a beginner in differential geometry and I am investigating $C^1$ isometric embedding of Riemannian manifolds to Euclidean space for computer vision. I know that the PDE produced for a metric $g$ ...
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What's the meaning of a derivative of a parametric curve?

A parametric curve $C$ can be defined as follows $$ C(p) = \{x(p), y(p) \}, \; p \in [0, 1] $$ where $p$ is the parameter. We can define the unnormalised tangent to the point of the curve ...
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Fast Marching Method - How does the procedure start?

I've read about the Fast Marching Method described by Sethian in his book "Level Set Methods and Fast Marching Methods" and also in the paper "Fast Marching Methods", which can be found for free here: ...
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692 views

What are the use cases of the Dirichlet energy in computer vision?

I am reading a paper, in the context of computer vision, that mentions the "famous" Dirichlet energy. I am not familiar with this Dirichlet energy, but apparently we can minimise it. What are specific ...