Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [tomography]

for questions about tomography and tomographical reconstruction, the 2D representation of 3D objects.

Tomography is imaging by sections or sectioning, through the use of any kind of penetrating wave or mechanical method, with a mathematical base in tomographical reconstruction.

21 questions
26
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1 answer

Name for a body that can be completely described using its silhouettes

I'm shooting blind over here because I have no background in this field of mathematics. I assume that if you have a body (in $\mathbb{R}^3$), you can call it convex if any segment from one point inside of the body to another point inside of it is…
Luka Horvat
  • 2,668
23
votes
4 answers

Can we invert this transform?

Consider the following transform $g\mapsto f$, where $$ f(x) = \int_{0}^{\infty} \exp\left\{-\int_0^t\int_{x-\tau}^{x+\tau} g(y) \, dy \, d\tau\right\} \, dt $$ Assume $f,g>0$ are $C^\infty(\mathbb{R})$, and $f,g\in L^2(\mathbb{R})$. Assume also…
sam wolfe
  • 3,465
9
votes
2 answers

Open problems in Mathematical Tomography?

Since I feel that Tomography can be applied to a wide range of sciences, I was wondering what the current open problems in Tomographic Reconstruction are. Furthermore, I am curious as to how these conjectures/open problems might help Tomography,…
Max Lonysa Muller
  • 7,546
4
votes
0 answers

Reconstructing a matrix from random matrix vector products

I am looking for new ideas how to construct a guess for a (positive, hermitian) matrix A given some matrix-vector products Ax (with random vectors x). One such method would be to perform rank one updates to A each time a new Ax becomes available,…
uekstrom
  • 133
2
votes
2 answers

About Radon Transform

Recently I got to Know that, Radon Transformation has huge contribution to Computer Tomography. So I would like to know about Radon transformation in Mathematical point of view. Can any one suggest books/good materials for Radon Transformation.
prasad
  • 517
2
votes
0 answers

How many edges is sufficient to check to prove polyhedron convexity?

Consider the set $\{u_{1}, u_{2}, \ldots, u_{n}\}$ of points on the spere in $\mathbb{R}^{3}$ (i. e. $||u_{i}|| = 1$) and their convex hull C = $Hull(u_{1}, \ldots, u_{n})$. It's obvious that each $u_{i}$ is the vertex of $C$. Suppose that $C$ is…
Ilya Palachev
  • 464
2
votes
0 answers

Can the Kaczmarz Method as a solution for linear equations be optimized through a spiral-based approach?

The Kaczmarz method is an iterative algorithm used to solve systems of linear equations, particularly useful in tomography and image reconstruction. It's a row-action method, which means it iterates through the rows of the matrix, updating the…
Omar Adel
  • 29
2
votes
1 answer

Find the form of $f(x,y)$ knowing the form of contour lines in the $XZ$ and $YZ$ planes

I am trying to find the form of $z=f(x,y)$. I know that: Contour lines in the $XZ$ plane are of the form: $$z=A*ln(x)+B$$ (the $A$ and $B$ parameters vary with $y$) Contour lines in the $YZ$ plane are of the form: $$z=Cy^2+Dy+E$$ (the $C$, $D$ and…
bouvierr
  • 123
1
vote
2 answers

How to find out if it is possible to contruct a binary matrix with given row and column sums.

How to find out if it is possible to contruct a binary matrix with given row and column sums. Input : The first row of input contains two numbers 1≤m,n≤1000, the number of rows and columns of the matrix. The next row contains m numbers 0≤ri≤n – the…
Sushil
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  • 2
1
vote
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Inverse Problem for Electrical Impedance Tomography

I am working for an Electrical Impedance Tomography(EIT) project and my background is basically algebra and number theory, so I'm having some problems with this project. Firstly, what is EIT: Electrical Impedance Tomography (EIT) is a non-invasive…
S.Hakan
  • 43
1
vote
0 answers

Smoothly interpolating $\mathrm{Skel}(M)$

Consider a particular solution to a diffusion equation: $$\frac{\partial^2}{\partial t^2}\Phi(x,t)=-\frac{x}{t}\frac{\partial}{\partial x}\Phi(x,t)$$ which is: $$ \Phi(x,t)=\exp \frac{t}{\log x}. $$ I wish to construct a space curve, that over…
J. Zimmerman
  • 842
1
vote
0 answers

What can be said about multiple derivatives of delta function on the sphere?

For fixed $\beta$ in the sphere $\mathbb S^2\subset \mathbb R^3$, we can define a "zonal derivative" via the distribution $\delta'(\langle \cdot, \beta \rangle)$ on $\mathbb S^2$ -- here $\delta'$ is the derivative of the Delta function, and…
Nathanael Schilling
  • 462
1
vote
1 answer

Theorem 1.1 of The mathematics of computerized tomography

In the book "The mathematics of computerized tomography" by Natterer comes the theorem: Theorem 1.1 $f=A^+g$ is the unique solution of $A^*Af=A^*g$ in $range(A^*)$. where $A:H\rightarrow K$ is a linear bounded operator, $H$ and $K$ are Hilbert…
Luz
  • 599
1
vote
1 answer

Radon-Transform after scaling a function

Say I have a Radon transform of $f:\mathbb R^2 \to \mathbb R^+$ and denote it by $\mathcal Rf$. I now read that the Radon-Transform of $f_a(x,y):=a^2f(ax,ay)$ is then given by $$\mathcal Rf_a(t,\theta)=a\mathcal Rf(at,\theta), \quad t\in \mathbb R,…
Tesla
  • 1,460
1
vote
1 answer

Difference between NFFTs, NUFFTs, USFFTs (for CT purposes)

I am trying to understand for what cases and in what way NUFFTs would be useful for CT reconstruction. Therefore, I am trying to create an overview for myself that starts with the "easy" problems where we can apply FFT and ends with the hardest…
Richard Schoonhoven
  • 103
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