Questions tagged [computer-vision]
Mathematical methods used in widely understood computer vision.
232
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Velocity field and deformation flow in diffeomorphic registration
I am trying to implement a diffeomorphic registration of two images using an unsupervised algorithm$^\color{magenta}{\star}$ and one of the steps is a generation of velocity field $\bf z$ which is ...
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Choosing number of subdivisions for approximating double integrals in context of image processing
I have a few double integrals that I have to approximate that are based on getting motion parameters from a sequence of images, and I have chosen to use Riemann sums for that purpose. These integrals ...
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how erosion works in mathematical morphology?
I have these definitions:
Let $A \subset E^N$ and $b \in E^N$. The translation of $A$ by $b$, denoted by $(A)_{b r}$ is defined as
$$
(A)_b=\left\{c \in E^N \mid c=a+b \quad \text { for some } a \in A\...
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Pinhole camera model definitions
As I were implementing the pinhole camera model based on Tsai, I noticed that there were two conventions used. For opencv and others, the following is used:
$$
\begin{pmatrix}
u_{c} \\\ v_{c} \\\ h_{...
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Deriving equations for estimating the motion parameters of a rigid body [closed]
I am referencing the book "Robot Vision" by Dr Horn. There, he provides the following equations for estimating the general motion parameters of a rigid body:-
∫∫((u-ur)β - (v - vr)α)(-xyβ + (...
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Need help with deriving equations for estimating the motion parameters of a rigid body
I am referencing the book "Robot Vision" by Dr Horn. There, he provides the following equations for estimating the general motion parameters of a rigid body:-
$∫∫((u-u_r)β - (v - v_r)α)(-xyβ ...
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33
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Reversing perspective projection of cricles
I have a set of real-world images containing circles, but due to the images not being taken parallel to the circle plane, the circles appear as ellipses in the images. I need to crop out the inside of ...
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Vanishing points are collinear
I am studying projective geometry where I came across a definition of vanishing line, "The line containing the vanishing point of 2 or more sets of parallel lines on a plane form the vanishing ...
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Need help in understanding the proof for Non-invariance of DLT Algorithm in the book by Richard hartely and Andrew Zisserman.
Background:
In section 4.4.2 of the book titled, "Multiple View Geometry for computer vision" on page 106, the author writes the proof, $$\tilde{A_i}\tilde{h}=(\tilde{\epsilon_{i1}},\tilde{\...
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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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Min-Sum Belief Propagation not working on a chain model with equal unary potentials
Given is a chain factor graph as presented in the image below with the following properties:
Each node can take values 0 or 1
All unary potentials are equal (e.g. $U(a) = 0$ for every node)
All ...
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1
answer
34
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Neural Networks Miscalibration Measure
I have read two papers related on a neural networks miscalibration problem (first: "On Calibration of Modern Neural Networks", link: https://arxiv.org/abs/1706.04599 ; second: "...
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Self-Calibration in Aerial Computer Vision
I have read that there are 4 common procedures for auto-calibration in Computer Vision:
Mendonca & Cipolla's Method
Classical Kruppa's Method
Simplified Kruppa's Method
Dual Absolute Quadric ...
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Coordinate transform with known camera pose pairs
Let's say that we have two $3D$ coordinate systems, $A$ and $B$. Now I want to get the coordinate transform matrix $T_{AB}$ such that for any points $p_A$ in $A$, we can get its coordinate in $B$ by $...
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Effect of changing focal length in general/skew camera case (Hartley, Zisserman)
This result and derivation seem very strange (and wrong). The most general form (6.10-p157) contains $f_x, f_y$ and skew $s$, but these are not explicitly included/accounted for in the result somehow. ...
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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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Question about 3D homogenous line segments from a textbook
I'm reading about homogeneous coordinates from a computer vision textbook. I'm not sure what the author means by "instead of the four degrees that a 3D line truly has." I think in the first ...
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For two square matrices $A$ and $B$, if $\left | A - B \right |^2_F$ is a small scalar number then can we assume $A \approx B$?
When researching a way to evaluate if two square matrices are equal or (very close to being equal) for a computer vision localization problem, I came across this Math Exchange post
Minimize the ...
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How do I calculate the inverse function of this distortion function?
Assume I have the following lens distortion function: $$ x' = x (1 + k_1 r^2 + k_2 r^4 + k_3r^6) + [2*p_1*x*y+p_2*(r^2+2*x^2)] \\ y' = y (1 + k_1 r^2 + k_2 r^4+ k_3r^6)+[p_1*(r^2+2*y^2)+2*p_2*x*y] $$ ...
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Prerequisits on eulerian video magnification
From this article http://people.csail.mit.edu/mrub/papers/vidmag.pdf I became extremely interested in the subject, but I couldn't find too much content that starts from the "basics". This ...
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Understanding distance from point to line in homogeneous coordinates
I would like to find the distance between A 2d point given as the homogeneous $P=(x, y, z)$, and a 2d line given as $L=(a, b, c)$.
I found an algorithm that does this, I can't figure out why it is ...
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How to calculate the projective transformation matrix given two planes?
When projecting an image from one plane $\left\{ A \right\}$ to another plane $\left\{ B \right\}$, we have
$X' = \mathbf{H} \cdot X$
Namely,
$\begin{bmatrix}x'\\ y'\\ 1\end{bmatrix} =
\begin{bmatrix}...
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1
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Place the camera to match a given camera - object transformation
I know the pose (in camera frame) of an object related to a camera in real world (results are coming from a 6d-pose estimator).
In the simulation I want to replicate the same view and same position (...
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59
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Small doubt about Constrained Least Squares solution
I was watching this video about Camera Calibration and I have a doubt regarding the solution of the following Constrained Least Squares problem
$$
\underset{\mathbf{p}}{\operatorname{min}}\Vert A\...
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Effect of a flipped image on projection matrix and the intrinsic/extrinsic calibration matrices
With the direct linear transform, one can calibrate a camera and receive the projection matrix, which can be further decomposed in intrinsic and extrinsic camera parameters.
For this, at least 6 ...
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Composing a Homography matrix from camera parameters
I am having an issue with composing a homography matrix using the following formula from this lecture on camera Homography.
$$
\begin{pmatrix}
u\\
v\\
w\\
\end{pmatrix} =
\begin{pmatrix}
\frac{1}{p_u}...
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How to determine if an object is within a camera's frame and its position within said frame knowing the camera properties and distance?
I will list camera and lens properties that may be of use.
Camera resolution: 1280×1024 ((0,0) top left and (1279, 1023) bottom
right)
Lens Focal Length: 3.5 mm
Sensor size and lens format: 1/2&...
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Does there exist a structuring element B such that $X \oplus B \supseteq X \ominus B$ isn't true?
In general, if B contains the origin of $\mathbb{E}$ then, the set erosion and dilation by $B$ become, respectively, anti-extensive and extensive; i.e., for all $X$:
$ \begin{equation} X \ominus B \...
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SVD and identifying a projectivity upto similarity
This is from Hartley-Zisserman's Multiple View Geometry in Computer Vision pg 55-56.
We are given a conic $$C_\infty^* = \begin{bmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0& 0 & 0 \end{...
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Decomposition of a projective transformation of $\mathbb{P}^2$
On page 42 of Hartley-Zisserman's Multiple View geometry in Computer Vision, it states that a projective transformation of the plane (which can be represented by a nonsingular matrix $H$) can be ...
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Why is homography matrix $H$ further minimized using Levenberg-Marquardt algorithm?
A homography matrix for a 2d plane is a 3x3 matrix which can be calculated using LH = 0 where L is a 2Nx9 matrix and H is vector of all 9 variables. N is number of points = 4. This matrix is solved ...
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Trouble with chessboard set definition
The following equation is supposed to define a set of points that produces a square grid (like a chessboard).
s is the distance between each square in the grid
w and h are the dimensions of the board ...
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Camera Projection Matrix and Inverse Perspective Mapping
In computer vision, the camera projection matrix $P$ is a 3x4 matrix that translates world coordinates $(X, Y, Z)$ to the pixel coordinate $(u, v)$. This can be written as follows -
$\begin{bmatrix}u, ...
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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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How to determine the pose of a 3D ellipse given two projections to 2D of the ellipse?
This question is about whether I'm doing an optimization problem the right way, or if it can be done in a simpler way. I'll start with my problem scenario and then move into my proposed solution. I ...
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Prove: affine transformation maps "line at infinity" to "line at infinity"
I'm studying Computer Vision and my lecturer stated that:
The affine transformation maps "line at infinity" to "line at infinity".
I'm trying to prove it as part of my ...
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Minimization of residual in Shi-Tomasi feature extraction
I don't understand how in the following paper https://users.cs.duke.edu/~tomasi/papers/shi/TR_93-1399_Cornell.pdf (Good features to track) the residual (Eq. 3.1) is differentiated so that Equations 3....
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Transform matrix calculation for image registration known the relative camera poses
I'm working on a problem where I define some positions of a chaser around an target object and take pictures of the same target with 2 cameras separated by a baseline.
Example of the described problem
...
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Why can't I estimate the Fundamental Matrix from a coplanar set of points?
I am learning how to estimate the Fundamental Matrix via the 8-point algorithm:
From N > 8 corresponding points that satisfy $xFx' = 0$, we need to solve the LS problem:
$kron(x,x')f = 0$
Where $...
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Back projecting a 2D pixel from an image to its corresponding 3D point.
I have been trying to understand how to project a 3D point to 2D image and vice versa. I had images of a building taken by a drone, I reconstructed the building using COLMAP(an SFM tool). For those ...
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Camera Calibration - Calculate Rotation and Translation
I have a calibration problem between two cameras, my setup contains two static cameras. Both cameras capture the same scene but from a different viewpoint.
I estimate the individual cameras poses ...
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How to change coordinate system from one plane to another?
I have two planes plane1 and plane2 which equations are known. Given a point c with coordinates (x,y,z) of the plane plane1, is there any way to get the correspondent point c' on plane2?
I've tried ...
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1
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Projective Geometry: line representation in P3
I am trying to understand how lines are represented in P3 with the null-space / span representation as described in [1]. The book defines a $2 \times 4$ matrix
$$W = \left[
\begin{matrix}
\mathbf{A}^T ...
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Recovering 3D from inverse disparity and instinct matrix
I am facing the challenge to recover the Z coordinate from an inverse disparity map and the instinct matrix K. The Z coordinate does not have to be metric but it has to be scale aware.
$$
K_{3\times ...
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Gauss-Newton normal equations with norm of residual
The Wiki definition of Gauss-Newton has the following scalar cost function:
${\displaystyle S({\boldsymbol {\beta }})=\sum _{i=1}^{m}r_{i}^{2}({\boldsymbol {\beta }}).}$
where $r_i(\beta)$ are scalar ...
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How to calculate 3D depth reconstruction of an image using a set of 3 similar images?
I have tried asking this in computer science SE, but sadly get no answers. Since this is fundamentally mathematical, I hope someone could give pointers to an answer for this, please.
From a set of 2 ...
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Understanding focal length (Camera intrinsics)
I'd like to know if my understanding is correct regarding camera focal length.
The intrinsics of the camera consists the focal length (and of course other parameters, not focusing on them here..) ...
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answer
331
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Back-Projection of points to rays
I'd like if someone could elaborate on the definition of the following formula, X(λ) = pinv(P)x + λC.
According to Zisserman book (multiple view geometry) page 162 (6.13), and to my understanding (...
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1
answer
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A proper definition for homogeneous coordinates in geometric image formation
In my computer vision course, we apply homogenous coordinates to represent points in 2D or 3D space. However, in my course it lacks a proper definition of what homogeneous coordinates are.
I have ...
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Why if I fit non-specific(without constraint on a type) conic to a noisy data, I get ellypse or hyperbola when ground truth is parabola?
Thank you for attention, my question is:
I try to estimate a general, non-specified conic, with a type-estimation from a noisy data.
I generate a data, points, which lay on the conic, by using ...
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