12
votes
Are all LTI systems invertible? If not, what is a good counterexample?
You need to define what you mean by "invertible". Do you mean invertible by a causal and stable system? If yes, then any system that is not minimum-phase is not invertible (because the inverse system ...
- 92.5k
8
votes
Accepted
Auto-correlation function, an inverse problem
Let's look at the case $x[n] \in \mathbb{R}$, where $x[n]$ is real.
Autocorrelation is basically convolution of the signal with it's time inverse. This can be easily expressed in the frequency domain....
- 48.2k
8
votes
Are all LTI systems invertible? If not, what is a good counterexample?
A necessary condition for invertibility is that any output has only one possible input (or injectivity, as proposed in comments). Since we are looking at counterexamples, we can look at when this ...
- 32.3k
7
votes
Accepted
Deconvolution of Synthetic 1D Signals - How To?
You cant't recover the original signal through deconvolution.
A Gaussian kernel is in essence a lowpass filter, i.e. it will remove information at higher frequencies from the signal. Once it's gone, ...
- 48.2k
6
votes
Accepted
Can Principal Component Analysis (PCA) Solve the Cocktail Party Problem?
The Cocktail Party Problem is a Blind Source Separation (BSS) problem.
Given a linear mixture of signals:
$$ \boldsymbol{y} \left[ n \right] = A \boldsymbol{x} \left[ n \right] $$
We're trying to ...
- 20.5k
5
votes
How Is the Formula for the Wiener Deconvolution Derived?
The Wiener Filter can also be derived by another (Easier) way.
Let's assume the following model:
$$ y = h \ast x + n $$
Namely the data is a result of a linear combination (Convolution) of $ x $ with ...
- 20.5k
5
votes
Auto-correlation function, an inverse problem
There is in general, as @Hilmar's answer points out, no unique solution to the question of a sequence that has the given perodic autocorrelation function. In the simplest case, that a shifted ...
- 20.9k
5
votes
Accepted
Deconvolution of a 1D Time Domain Wave Signal Convolved with Series of Rect Signals
Solving a deconvolution isn't easy even in simulated environment not to mention in practice.
The main trick to solve it is using the proper model / prior for the problem and very good measurements (...
- 20.5k
4
votes
Accepted
Estimating the Signal by Deconvolution with a Prior on the Filter Coefficients and the Signal Samples
I would take approach based on Blind Deconvolution.
Since we're dealing with ill posed problem some assumptions should be made.
The intuitive approach would be using the information as a prior for ...
- 20.5k
4
votes
Accepted
Estimating Convolution Input Under the Assumption of Sparsity and Constant Non Zero Values Using Compressive Sensing Approach
Basically your problem is called Blind Deconvolution.
It means we want to estimate both the operator and the input given the output.
You model is Linear Time Invariant Operator so we have LTI Blind ...
- 20.5k
4
votes
Solving Inverse Problem of Multiple Pulses Over Multiple Channels with Convolution Kernel and Cross Channel Mix
Crosstalk between channels is small and well-conditioned, at least in the example you provided. Your matrix A has a condition number of 1.85, which is really good. It means you can invert it and it ...
- 531
3
votes
Accepted
Intuitive Meaning of Regularization in Imaging Inverse Problems
One general form of Inverse Problem in Imaging which assumes Linear Operator is given by:
$$ \arg \min_{x} \frac{1}{2} {\left\| A x - b \right\|}_{2}^{2} + \lambda R \left( x \right) $$
Where $ R \...
- 20.5k
3
votes
Inverse Problem / Deconvolution with Pink Noise
I think you are looking for Wiener Deconvolution: It does not require the noise to be white, but you can assume any correlation function/PSD of the noise. It gives you the optimal estimate of $O(v)$ ...
- 6,238
3
votes
Accepted
How to Use the DFT (FFT) to Solve a Least Squares Regularization Problem (Inverse Problem)?
The question really depends on $ f \left( \cdot \right) $.
Yet in order to show how to use FFT we can even use 1D signals.
Let's rewrite the problem:
$$ \hat{x} = \arg \min_{x} \frac{1}{2} \left\| K ...
- 20.5k
3
votes
What Is the Relation Between Deblurring and Deconvolution in Computer Vision and Image Processing?
In the context of image processing (and machine vision as well), blurring is an operation that reduces the sharpness of an image by some lowpass filtering applied on it.
There are different causes of ...
- 28.4k
3
votes
What Is the Relation Between Deblurring and Deconvolution in Computer Vision and Image Processing?
Let me present the following Diagram:
So, both Deblurring and Deconvolution are operations within the family of Image Restoration (Which is a subset of Inverse Problem set).
Basically we build the ...
- 20.5k
3
votes
Are all LTI systems invertible? If not, what is a good counterexample?
Whether LTI or not all systems are invertible if
unique (distinct) inputs produce unique (distinct) outputs
Causality and stability are later concerns for making sense of the obtained inverse ...
- 28.4k
3
votes
Are all LTI systems invertible? If not, what is a good counterexample?
In addition to all the answers that are correct in a mathematical sense, in a practical sense, a system whose frequency response goes below some finite but small-enough value will not be usefully ...
- 13.3k
3
votes
Accepted
Deconvolution of an Image Acquired by a Square Uniform Detector
Your model is exactly a Convolution with Uniform Kernel where the output is what is called the Valid Part of the Convolution.
In MATLAB lingo it will be using ...
- 20.5k
3
votes
Deconvolution of a 1D Time Domain Wave Signal Convolved with Series of Rect Signals
Actually I have a similar problem like yours. But in mine, the objective function is not like rectangular pulse but just spikes as shown below. I work in ultrasoinc testing field. So, this example is ...
- 71
3
votes
Can Principal Component Analysis (PCA) Solve the Cocktail Party Problem?
Speech Source Separation (SSS) or Audio Source Separation (ASS) can be seen as a specialized version of source separation. I mention these expressions under which one can find additional works. One ...
- 32.3k
3
votes
Regularization for inverse filter design
This alternative approach seems fishy to me. Consider the two-dimensional case with $C$ having eigenvalues $\lambda_0, \lambda_1$ that are real and positive and $\lambda_0 \ll \lambda_1$. You have
$$
\...
- 1,269
3
votes
Accepted
Tikhonov Regularization for Complex Matrices
Usually Tikhonov Regularization is applied in the following form:
$$ \arg \min_{\boldsymbol{x}} \frac{1}{2} {\left\| A \boldsymbol{x} - \boldsymbol{y} \right\|}_{2}^{2} + \frac{\lambda}{2} {\left\| \...
- 20.5k
3
votes
Solve Efficiently the 1D Total Variation Regularized Least Squares Problem (Denoising / Deblurring)
I will answer Total Variation Regularization:
$$ \arg \min_{\boldsymbol{x}} f \left( \boldsymbol{x} \right) = \arg \min_{\boldsymbol{x}} \frac{1}{2} {\left\| A \boldsymbol{x} - \boldsymbol{y} \right\|}...
- 20.5k
3
votes
Is a neural network an adaptive filter?
An adaptive filter is a special case of a neural network (NN). They have in common that they multiply an input x[n] with weights w[n], the result y[n]=x[n]w[n] is compared to the target t[n] (e.g. the ...
- 31
3
votes
Accepted
Blind source separation for asynchronously observed mixture channels
Based on a few quick experiments, ICA fails when the channels are delayed relative to each other. It's fairly easy to test in MATLAB (or no doubt other software packages) with FastICA (or Robust ICA, ...
- 2,064
3
votes
Solving inverse problem using black box implementation of the kernel
I will try to illustrate a solution.
The general form of the problem is:
$$ \arg \min_{\boldsymbol{x}} \frac{1}{2} {\left\| \boldsymbol{h} \ast \boldsymbol{x} - \boldsymbol{y} \right\|}_{2}^{2} + \...
- 20.5k
3
votes
Remove Wave Patterns from an Image Using Inpainting
If it is just «amplitude modulation» can you identify the vertical periodic pattern and multiply with the inverse? Doing inpainting seems excessive as there seems to be real detail in the dark wave ...
- 3,595
3
votes
General solution that have the rectangular function as amplitude of the Fourier transform?
No matter what phase function you choose, you can always represent the final transfer function as the cascade of a real rectangle and an allpass filter.
There are quite a few different types of ...
- 48.2k
3
votes
Inquiry About the Nyström Method and Matrix Inversion
I would use it as follows. First, assume that $K\in\mathbb{R}^{p\times p}$ is of rank $r$ where $r<<p$. Let's pick $B=\Lambda^{-1}$, $C=K_{21}U_1$ to write:
$$
(\epsilon I + K_{21}U_1\Lambda^{-1}...
- 5,515
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