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Questions tagged [computer-vision]

Mathematical methods used in widely understood computer vision.

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how to find an angle between the person's body and camera?

I am working on project which requires me to find the angle between the person's body and the camera. I already have the pose coordinates of the person but I am not sure how to find the angle. Do you ...
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44 views

Unknown mathematical symbol in context of computer vision

I'm currently reading the paper "A Template Matching Method for Multi-Scale and Rotated Images Using Ring Projection Vector Conversion" and in one formula there is a mathematical symbol I don't know. ...
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Obtaining depth to an object of known dimensions from a monocular calibrated camera

I am using notation defined in this page: Camera Calibration In summary: The standard frame is the frame that has the origin at the projection center and z axis pointed "out" from the lens. The ...
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1answer
102 views

Solving the matrix equation $X^tA+A^tX=0$ for $X$ in terms of $A$

Suppose that I know $A$. And all matrices in the equation are square matrices. I want to solve for $X$ given that $$X^tA + A^tX = 0$$ I'm not really good at matrix calculus. Is it possible to solve ...
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Partial derivatives of surface curvature relative to a tangent plane

If we take the tangent plane (say, $\mathsf{T}$) to a convex surface $\mathcal{S}$ associated with a given surface normal vector $\mathtt{n}$, I assume the surface will have a fixed rate of change ...
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100 views

Identifying the Plane at Infinity in the World Necessitates Determining the Affine Geometry of the World?

Page 18 of my computer vision textbook, Multiple View Geometry in Computer Vision (Second Edition), by Hartley and Zisserman, states the following: 1.8 Auto Calibration $\vdots$ Generally ...
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32 views

Graph-cut and pairwise MRF

I try to use graph-cut method but I got some trouble with it. I want to segment an image into foreground and background. For each pixel $x$, I got the probability that it belongs to the background $p(...
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28 views

Parameterization of uncommon curves

Preface: Out of all the Stack Exchange networks with which I am familiar, this seems the most appropriate for the following question. Please let me know if there's a more suitable network for similar ...
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2answers
256 views

If All the Points Lie On a Plane, Then Why Does the Linear Mapping Reduce to …?

I previously asked a question with regards to what the matrix $\mathrm{H}_{3 \times 3}$ is/represents in the following textbook excerpt: In applying projective geometry to the imaging process, it ...
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1answer
43 views

Projective Transformations: “If all the points lie on a plane, then the linear mapping reduces to …”

Page 7 of my computer vision textbook, Multiple View Geometry in Computer Vision, says the following: In applying projective geometry to the imaging process, it is customary to model the world as a ...
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Textbook Error? “, … and singling out the line at infinity in *the image* or the plane at infinity in *space* when that becomes necessary.”

Page 3 of my computer vision textbook, Multiple View Geometry in Computer Vision, says the following: In computer vision problems, projective space is used as a convenient way of representing the ...
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1answer
26 views

Create Wall 3D math oriented away from camera

I have 2 Points which has x,y,z let's say from and to I am drawing wall between them using ARkit ios To draw wall I use static ...
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2answers
34 views

Prove that two linear map are same — from “A invitation to 3-d vision"

I found a question from the book to prove the following equation: $A^T\hat{\omega}A=\widehat{A^{-1}\omega}$. where $\hat{}$ means turn a vector $(x_1, x_2, x_3)$ into a skew-symmetric matrix $ \...
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Displacement and rotation of the template pattern relative to the camera in 3D space

I have the template pattern (for example, rectangle), so I know it's size in mm. Also I have several frames with this pattern. So I know the intrinsic matrix $K$ of camera with which these frames were ...
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1answer
36 views

Calculation of angle of rear wing of race car above x axis from an image

I am planning an application measuring the angle of rear wing of race car from image or realtime video (from lateral perspective target angle from this kind of image). This is the equation I am ...
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1answer
36 views

Boundary conditions in minimizing Dirichlet energy for an image processing problem.

Suppose $$\mathcal{L} =\mathcal{L}(x,y,u,u_x,u_y) = \frac{1}{2} \lVert \nabla u \rVert^2$$ and I want to find $u$ such that the functional $$ E(u)=\int_{\Omega} \mathcal{L}dxdy $$ is minimized, ...
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1answer
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What is the meaning of “linear projections onto rows of some matrix”?

I was reading through a research paper on compressive particle filtering for target tracking, and I came across the following: Let $z \in d_{z}$ denote a vectorized image with $d_{z}$ pixels. Assume ...
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Substitute a Fully-Connected layer with 1x1 Conv Layer

I am following Andrew Ng's class on ConvNets and I don't get the part where we try to replace a FC layer with a Conv layer and say that both are mathematically equivalent. Here is a picture from the ...
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Comparison of 3 Computer Vision Metrics

right now i do a comparison of 2 Algorithms, with the help of 3 metrics. The problem is that in one paper the metrics are differently described than in the other so i have to "proof" that they are the ...
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0answers
30 views

Calculating depth image from surface normal images, confused about integration/summation algorithm

(Originally posted in Stack Overflow, re-posting here since my question's probably a better fit here) I'm going through Forsyth/Ponce and working through their reading on recreating a depth map from ...
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Is it reasonable to use a log-concave density to model the joint density of natural images?

That is, something like $$p(x) \propto \exp(-V(x) - \phi(x)), $$ where $V: \mathbb R^p \rightarrow \mathbb R$ is a convex function on the image manifold (here assumed to be $\mathbb R^p$, where $p$ ...
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1answer
63 views

How to get interpolated normal of triangle?

I have this 3D Triangle of which I want to get the interpolated normal of V3. I know how to get the normal per triangle face, but I don't know how to get to interpolated normal. I would calculate ...
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3answers
39 views

How to mathematically write: last matrix position that equals one?

Consider the following (4 x 5) 2D matrix: 0 0 0 0 0 0 1 1 0 0 0 0 1 1 0 0 0 0 0 0 I need to write in a mathematical language "The last column at row 3 that ...
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47 views

Euler lagrange equations for a curve integral

I'm reading through this paper, where a variational framework to some computer vision feature detection techniques is given. I'm familiar with variational techniques, though I'm not an expert. There's ...
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Cannot get laplacian operator from leads given by a computer vision book

This a follow up question to this one. I've tried to follow the explanation given that should allow me to derive a laplacian operator, But what I got isn't correct. The quote is from "Computer Vision ...
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43 views

Parabola Equation from its trajectory points in a Video

I have a video of an object describing a ballistic trajectory (i.e. A ball being thrown) The video is filmed with an angle and not orthogonally to the moving object, the camera is calibrated. I am ...
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1answer
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I don't know how to find coordinate systems for a question. [closed]

MST is $3 \times 3$ homogeneous matrix that transform points from coordinate system $S$ to coordinate system $T$. 1- What are the coordinates of $P$ in coordinate system $O$? 2- What are the ...
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3answers
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How to get the transform the coordinates of a 3D point from the standard basis to another basis?

Suppose I have the coordinates of a point in the standard $\mathbb{R}^3$ basis $B_0 = (\vec{e_0}, \vec{e_1}, \vec{e_2})$, with $\vec{e_0} = (1,0,0)$, $\vec{e_1} = (0,1,0)$ and $\vec{e_2} = (0,0,1)$. ...
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1answer
55 views

Instrinsic Camera Parameters

I am trying to understand intrinsic camera parameters. Specifically I not able to understand the skew factor and pixel scaling derivation in it. Example: Most of the literature explanation starts ...
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1answer
137 views

Make a square matrix singular through differentiation

I have the following $4 \times 4$ matrix $$ M = \sum_{i=1}^n P_i^T \left(I_{3x3} - \frac{x_ix_i^T}{x_i^Tx_i}\right)P_i $$ where $P_i$ and $x_i$ are $3 \times 4$ and $3 \times 1$, respectively. The $...
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1answer
46 views

Find correct 2D position of a point after rotation

Suppose there is a white rectangle $R$ with a red point $P$ drawn on it. Suppose there is a camera looking at $R$. $R$ center is in the position $(0,0)$ of the camera (its center) $P$ is in position ...
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1answer
39 views

How to convert an affine matrix to a homography matrix?

The title is quite self-explanatory, given an affine matrix, is there a way to convert it to a homography matrix?
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2answers
135 views

Solving a system of linear equations on the form $a^\top X \, b = c$

I'm implementing a computer vision algorithm for 3D point reconstruction from photos (Tomasi-Kanade). For the last day I've been stuck on a linear algebra problem. The system that I want to solve has $...
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0answers
14 views

Projection-Matrix to Y-Plane

If it even is a projection to the Y-Plane. I'm having difficulties understanding this matrix right here: When using this in computer graphics, it helps projecting a shadow. But I don't quite ...
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1answer
48 views

Projective transform

I have a set of n ($n\geq4$) or more unique point correspondences $[x_i, y_i] \to [x_i',y_i']$ and want to find the projective transform matrix A that relates each point as follows: $$\begin{pmatrix}...
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22 views

How to get the bounding ellipsoid of a three-dimensional points data-set?

I have a three-dimensional point cluster and I have to calculate its height, width and depth. Using an axis aligned bounding box is not an accurate way of doing this, because the object given by the ...
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1answer
219 views

Reprojection error formula

In a paper,the pinhole camera model is defined as follows: $$ w\small \left(\begin{array}{c} u\\ v \\ 1 \end{array}\right)=\left( \begin{array}{ccc} f_{u} & 0 & u_{c}\\ 0 & f_{v}...
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Why is the equation of line through 2 points in projective space is their cross product

Can anyone give intuition that why the equation of line passing through two points in projective space is given by their cross product?
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0answers
21 views

Depth Estimation from two points

How can i find depth under this condition. 1) Two 2D points with (x,y) and angle of rotation is know in Y axis with each other. 2) Center of rotation is know of turn table. Process is for 2D ...
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0answers
239 views

Approximation of Hessian=$J^TJ$ for general non-linear optimization problems

My question is: when is the aprroximation of Hessian matrix $H=J^TJ$ reasonable? One truth is that it is reasonable to approximate Hessian with first order derivatives (jacobian), i.e., $H=J^TJ$ ...
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0answers
24 views

Understanding cost function description

I'm trying to understand this paper, but I'm struggling on page 7 on: To determine point correspondences we use a cost function that is the sum of the squared distance to the epipolar line and ...
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0answers
78 views

Pixel error bound for SVD image compression [closed]

Given an image I, I can perform an SVD on the image to obtain $I=USV^T$. Taking the first K singular values, I can reconstruct a compressed image $I_{c}$. How can I find an expression that bounds the ...
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0answers
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What if 0 is one of the entry of a rotated point in $R^3$

If a point $P^T=(X,Y,Z) $ is rotated by a rotation matrix R and if one of the entry of $RP$ is 0, that is $P’=RP=(0,X’,Y’)$ then what can be said about atleast one of the row of R. I started with the ...
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0answers
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Subtracting 1 from a generic low pass filter

I am taking a Computer Vision class and the following question about Signal Processing and Filter Kernels is asked on the provided slides: You are given a generic normalized low pass Filter kernel. ...
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1answer
59 views

Even numbers combinatorial game

We have $2$ piles of coins-one of them with $X$ number of sheets and the other with $Y$ number of sheets, $X$ and $Y$ are larger than $0$. $2$ players are playing against each others , every player ...
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Scaling coordinate systems of different scales

I am trying to perform a classification algorithm on a set of images by obtaining various points from the image and feeding them to a classifier.The problem is that many of those images are of varying ...
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1answer
33 views

combinatorial game of sheets [closed]

We have an odd number of sheets organized in a pile. We have two players, and every player can remove $1,2,5$ or $6$ sheets, and keep them by their side. The winner is the player(s) that have even ...
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0answers
111 views

Comparison of estimated transformation matrix with ground truth transformation matrix.

I am studying the effect of smoothing an image before passing it to SFM pipeline. The SFM pipeline provides me the camera intrinsic and extrinsic matrix, and as ground truth, I have the intrinsic and ...
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1answer
106 views

Calculating the polar of a given pole relative to a conic (with NO Calculus)

Let $M$ be the matrix of the conic $$Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0$$ so we have: $$M=\left[\begin{array}{ccc} A&\displaystyle{\frac{B}{2}}&\displaystyle{\frac{D}{2}}\\ \displaystyle{\frac{B}{2}}...
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1answer
109 views

Visualization of Projective Space

Currently, I am learning 3D Vision. I am trying to visualize projective space. I am not Mathematician so please pardon me if I have missed out subtle concreteness. Q-1 In $P^2$ we can visualize each ...