Questions tagged [inverse-problem]
The inverse-problem tag has no usage guidance.
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Deconvolution using Toeplitz matrices [duplicate]
Lets say I have 2 vectors (1D signals that are sigmoids): $s$ and $m$, both related through the relation: $m = s * r$, my goal here is to recover the vector $r$ (should look like a gaussian).
I tried ...
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What does the $H$ operator exactly do in the deconvolution process and why is it needed?
According to Deblurring Dynamic Scenes via Spatially Varying Recurrent Neural Networks, given a 2D sharp image $x(m, n)$ and a blur kernel $h(k, l)$, the blurred image is obtained as
$$
y(m, n) = (x*h)...
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Solving Inverse Problem of Multiple Pulses Over Multiple Channels with Convolution Kernel and Cross Channel Mix
Before I start, let me note that I have 0 experience with signal processing, so please bear with me:
My System
My system can be represented as an $m \times n$ matrix $X$ (input) where each
column ...
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Consistent reconstruction of image from partial images
I am given a set of $N = 649$ color PNG images, each of size $W \times H \times 3 = 586 \times 689 \times 3$. The corresponding pixels in each image represent the same object. Many of the pixels in ...
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Solving inverse problem using black box implementation of the kernel
My question is related to Solving regularized least squares problem using black-box computation of $\mathbf{A}\mathbf{x}$ and $\mathbf{A}^T\mathbf{x}$.
In case, the problem is formulated as:
\begin{...
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Matrix-vector multiplication representation of Total Variation function
I'm reading a paper - Total Variation Superiorized Conjugate Gradient Method for Image Reconstruction on total variation regularization and conjugate gradients. In page $3$, the authors define the ...
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Solving regularized least squares problem using black-box computation of $\mathbf{A}\mathbf{x}$ and $\mathbf{A}^T\mathbf{x}$
Let $\mathbf{A} \in \mathbb{R}^{n \times n}$. I'm working in a problem where I have a black-box algorithmic solution to compute the products $\mathbf{A}\mathbf{x}$ and $\mathbf{A}^T \mathbf{x}$ given ...
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The Different Solutions for Filter Coefficients Estimation for Periodic Convolution and Full Convolution
As a continuation of the question Least Squares Solution Using the DFT vs Wiener Hopf Equations raised by Dan Boschen.
The question is, given the model:
$$ \boldsymbol{y} = \boldsymbol{h} * \...
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Academic reference for a specific type of regularized inverse filtering
Let $y(t) = (h * x)(t) + n(t)$ be some observed signal where $h(t)$ is some filter / impulse response, $x(t)$ is some input signal we are interested in, and $n(t)$ is noise.
In order to recover $x$ ...
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Get the inverse transfer function from the measured response
What is a numerically stable way to obtain the inverse transfer function from a measured response?
I have a system that shows a low-pass behavior. I would like to increase the bandwidth by some form ...
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Reconstructing a signal from a Nyquist plot
I have a system which is like a blackbox which has just one input which could be a sinusoidal wave which is a sum of a range of frequencies, now the problem is that I dont have the time-domain output ...
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How to compute the inverse Z-transform
How to compute the inverse Z-transform of the form $$ G(z)=\frac{z^{2n}}{a(z^{2n})+b(z^n)+c} $$
I started by taking
$$ F(z^n)=G(z) $$
so
$$ \frac{F(z)}{z}=\frac{z}{az^2+bz+c}$$
This can be solved ...
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Is the system $y\left(t\right)=\int _{t^3-1}^{t^3}x\left(s\right)ds\:$ invertible? [duplicate]
I have the following system: $$y\left(t\right)=\int _{t^3-1}^{t^3}x\left(s\right)ds\:$$
I was told to determine if it's invertible system, casual system, memoryless system and linear system. I was ...
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How to use inverse 2D Fourier transform to reconstruct the original image?
I have managed to get the forward Fourier transform of an image to the frequency space like so:
But I cannot for the life of me reconstruct the original image from the inverse Fourier transform of ...
user60606
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Why does sign ambiguity occur in ICA?
I do not really understand the source of sign ambiguity in ICA. First, my understanding that If I apply ICA on a signal $X$ and I got 3 ICs which are represented by a set $IC^1$. Then, applying ICA on ...
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Image / Video Upscaling (Super Resolution) Algorithm Explanation (Image and Video Upscaling from Local Self Examples)
So, I'm trying to implement the classical algorithm described in this paper Image and Video Upscaling from Local Self-Examples and this presentation to serve as a baseline for comparison with AI/NN-...
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Blind source separation for asynchronously observed mixture channels
Given your practical and theoretical expertise: Does ICA work reliably when applied to a multidimensional mixture (observation) $X = (X^1, \cdots, X^d)$ if the different channels $X^i$ of the ...
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Is a neural network an adaptive filter?
I am confused as to the difference between neural networks and adaptive filters: As far as I understand it, "neural networks" are largely used for solving inverse problems, where an unknown ...
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Identifying a signal by its power spectrum
Background: There is a method in optics for determining the electric field of light (both intensity and phase) via a three step process:
Add a known phase shift to the light (called the "...
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Reconstruction of a Signal from Sub Sampled Spectrum by Compressed Sensing
Context
Attempting to reproduce an illustrative example of compressive sampling from Candes-Wakin 2008. Specifically, the L1 recovery of a sparse signal shown on pg 5 in Fig. 2.
Using my code (below), ...
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Estimate the Image Using Multi Realizations of Its Convolution with a Known Filters Using Wiener Filter
Suppose we have a corrupted image $Y = H*X + \epsilon$ formed by taking an image $X$, convolving it with a point-spread function $H$, and adding gaussian noise $\epsilon$. Then we know that the Wiener ...
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Cocktail Party Problem with a Single Signal of Data (Single Mic)
I have been doing some multimodal signal analysis, and sometimes ICA is used for detecting statistically independent components.
From my understanding, say if you have 2 sources and 2 receivers/...
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Solve Efficiently the 1D Total Variation Regularized Least Squares Problem (Denoising / Deblurring)
How to solve a 1D Least Squares with Total Variation Regularization?
I know gradient based methods, I wonder how much faster / efficient I can get.
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How Could One Accelerate the Convergence of the Least Mean Squares (LMS) Filter?
How can the convergence of an LMS filter be accelerated?
Can we do better than the Vanilla algorithm?
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Sparse FFT of ramp using zero padding
I would like to sparsely represent a linear function/ramp in the Fourier domain. In an attempt to improve the sparsity, I have zero padded it. With this padding, it is possible in the example I tried ...
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Tikhonov Regularization for Complex Matrices
Tikhonov regularization is used to regularize ill-posed inverse problems if the matrix $A \in \mathbb{R}^{n,m}$ to be inversed has a high condition number. For example
$$
A=\begin{bmatrix}1&1\\
1&...
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Inverse Fourier transform: where am I going wrong?
I am studying a course in signal processing, currently we are examining Fourier transforms. I got stuck on an exercise with an inverse Fourier transform.
I am supposed to find the inverse Fourier ...
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Regularization for inverse filter design
Given a $2 \times 2$ matrix, $C$, suppose I want to compute a filter matrix $H = C^{-1}$ and that I need to add regularization for practical purposes (e.g., for an audio filter, regularization is ...
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Explain the relationship between Tikhonov regularization, SVD , least squres, and the Wiener filter
I found in the
Wikipedia site for Tikhonov regularization
that the SVD for a Tikhonov regularized problem take us to the least squares regularized solution:
\begin{eqnarray}
\hat{x}= V D U^T b,
\end{...
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Stochastic Methods for Image Deconvolution Problem
If we convolve an image with a point spread function and from the resulting image to find the input image, can we use any stochastic approaches? I feel like we will not be able to. A single image ...
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Inverse system of sinc?
I'm doing some self-study for an important exam I'll have in late March and came across the following question:
So, using the convolution properties, if I want to find an identity system so that the ...
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Can Principal Component Analysis (PCA) Solve the Cocktail Party Problem?
I'm looking into the cocktail party problem and trying to figure out whether something like Principal Component Analysis is enough to separate out all the various voices at the cocktail party into its ...
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Deconvolution of a 1D Time Domain Wave Signal Convolved with Series of Rect Signals
I have a synthesized signal (the bottom of the following figure), which is the convolution of the input signal (at the top) and the objective function (in the middle). The intention is to retrieve the ...
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Adapting Richardson Lucy (RL) Deconvolution for Shot Noise Limited Coherent Imaging
I am an experimental physicist who is collecting a series of coherent imaging of trapped gas. If you are familiar with phase contrast imaging, you may understand what I mean by coherent imaging. The ...
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An invertible system with memory
Suppose $\mathcal{L}$ be invertible system with memory. Does $\mathcal{L}^{-1}$ have memory necessarily?
Intuitively I think the answer is "yes". There are many examples showing that. For ...
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In computed tomography (CT), why is 'Inverse Problem of Radon transform' studied?
As I know, the Radon transform is a very important tool in CT by Beer's law. Thus, finding $f(x)$ of Radon transform $Rf(L):=\int_{L} f(x)dl(x)$ is helpful in CT. Nowaday, the Filtered back-projection ...
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Regularization for Inverse Problems using the Singular Value Decomposition (SVD)
I am reading these lecture notes on Optimisation and Inverse Problems in Imaging, and I have difficulties understanding how figures on page 20 (Figure 3.2) or page 21 (Figure 3.3).
Precisely, I don't ...
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Deconvolving a 1d Signal Using a Lookup Table
assuming I measure a signal that has different PSFs per position in time.
for example:
...
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Is Differentiation as a system, is an invertible system?
is the following system invertible?
as I understand it, invertible means finding an inverse function which should return back the original input from an output of the given system.
if so I ...
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Estimating Convolution Input Under the Assumption of Sparsity and Constant Non Zero Values Using Compressive Sensing Approach
I was wondering about if there is compressive sensing algorithm to estimate the sparse vector where the number of non-zeros values and amplitude of every non-zeros value are known. For example, assume ...
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Whats the Correct Approach to Estimate the PSF of a Moving Detector
In lab, I did a bunch of scans using a radiation source and a detector. My source emits a gaussian beam (whose dimensions I know), and my detector is a uniform 7mmx7mm square. These are stationary and ...
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Estimating the Signal by Deconvolution with a Prior on the Filter Coefficients and the Signal Samples
Assume I have signal $y[n]$ which is a result of convolution between channel $h[n]$ and signal $x[n]$. which means:
$$y[n] = h[n] \ast x[n]$$ where $\ast$ is the convolution operation
The signal $...
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Deconvolution of an Image Acquired by a Square Uniform Detector
So, I acquired some images by scanning a radiation source with a square detector like in the following gif.
Where the dashed grid represents reality, the 3x3 square my detector, and the 4x4 my ...
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Are all LTI systems invertible? If not, what is a good counterexample?
I have been trying to figure this out for a while now. Everywhere I have looked I could easily find examples of invertible LTI systems, but I could not find any counterexamples. Can anybody shed some ...
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Solving equation with convolution
I have the measured signal $y(t)$ that can be modeled in the frequency domain as $Y(f)$:
$$Y(f) = X(f)\cdot A(f) - [X(f)\cdot B(f)] \ast C(f)$$
where $\ast$ is the convolution.
I know $A(f)$, $B(f)$,...
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Extracting 'structure' post permutation
I have particle activity as shown in the left pane of animation below. The activity is clustered and it moves slowly. Sometimes these clusters merges together. On the right side of it, its shuffled ...
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What Is the Relation Between Deblurring and Deconvolution in Computer Vision and Image Processing?
The deblurring problem can be modelled as follows
$$
f = \phi u + \epsilon, \; \epsilon \sim N(0, \sigma)
$$
where $\phi$ is a filter (e.g. a low-pass filter) and $\epsilon$ is a Gaussian noise.
In ...
user40095
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Partial Fractions
Attached is image with solution and my attempt.
I am trying to calculate the coefficients for partial fractions expansion of the following:
$$ H(e^{j\omega}) = \frac{ \frac{1}{3} e^{-j2\omega} }{(1-\...
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Deconvolution of Synthetic 1D Signals - How To?
I convolved a square wave with a Gaussian wave using linear convolution. Can I get the original square wave back by deconvolving my output with the Gaussian function?
I took the FFT of both signals, ...
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Deconvolution of a 1D Signal with Known Kernel (Square Wave)
I have a signal measured from a radiation detector in a narrow beam of radiation. The peaks I get are quasi-gaussian in shape, see attached picture.
The signal is not a function of time, rather a ...
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