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# Questions tagged [inverse-problem]

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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 ...
55 views

### 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 ...
1 vote
253 views

### 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 ... 124 views

### 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-...
44 views

### 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 ...
1 vote
180 views

### 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 ...
38 views

### 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 "...
431 views

### 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), ...
66 views

### 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 ...
55 views

### 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/...
126 views

### 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.
76 views

### 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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### How do I represent any signal in the form x[n]u[n] and x[n]u[-n-1]?

For eg-: If I have the signal x[n]={1,a,a²...}I can represent it as a^n * u[n]. Similarly if I havethe sequence as x[-1]=-a^-1 x[-2]=-a^-2 Then I can represent it as x[n]=-a^n*u[-n-1].
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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 ...
168 views

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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, ...
1k views

### 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 ...
1k views

### What Are the Differences between Super Resolution, Denoising and Deblurring?

In the fields of computer vision and image processing, what are the differences between Super Resolution, Denoising and Deblurring?
2k views

### Deconvolution Using Complex Division in The Frequency Domain

Consider these two signals: a = [1 1 0 0 0 0 0 0] b = [1 0 1 0 0 0 0 0] their convolution is c = a * b = [1 1 1 1 0 0 0 0] ...
1k views

$x[n]$ is a complex function $n=0,1,2,\cdots,L-1$ we assume $x[n]$ is periodic in its index: $x[n+L]=x[n]$ Its auto-correlation function $C[n]$ is uniquely defined as: $$C[n]=\sum_{i=0}^{L-1} x[i+... 5 votes 1 answer 589 views ### Intuitive Meaning of Regularization in Imaging Inverse Problems Hello Every one I have been trying to understand the intuitive meaning of using a regularizer in images. Specifically what does the Total Variation regularizer do in images and how is it able to ... 0 votes 1 answer 152 views ### Inverse FFT - Synch the Phase Is there any way to synchronize phase of output of inverse DFT in each buffer? When I send the output of inverse DFT to the ... 1 vote 1 answer 1k views ### How to find inverse of convolution integral? If x^{-1}(t) and  y^{-1}(t) denote the integrals of x(t) and y(t) defined by x^{-1}(t)=\displaystyle\int_{-\infty}^{t} x(\lambda)d\lambda y^{-1}(t)=\displaystyle\int_{-\infty}^{t}y(\lambda)d\... 6 votes 4 answers 312 views ### Estimate Filter Coefficients from the Result of Linear Convolution with a Known Signal If I have samples of input say x(1:500) and it passes through FIR filter with 9 taps and some unknown coefficients. The output y(1:508) is also known. The aim is to estimate the filter response in ... 3 votes 0 answers 252 views ### Room Impulse Response Inverse Problem This article here talks about a room impulse response generator which takes the source and destination coordinates, sound speed, sampling rate, room dimension and wall reflection coefficients as input ... 7 votes 1 answer 843 views ### How to Use the DFT (FFT) to Solve a Least Squares Regularization Problem (Inverse Problem)? Let X and K be an image and a Point Spread Function (PSF), respectively. The blurred image B is obtained as follows$$B = X * K$$I want to solve the following general regularization problem$$\...
Consider a discrete 'blurred' output $h[t]$ given by the convolution of filter $f[t]$ and signal $g[t]$. This question considers recovering $g[t]$ from a window (subset) of $h[t]$. This causes the ...
Prove that the following system is invertible. $$y(t) = \mathcal{T}\{x(t)\} = \int_{-\infty}^{3t} x(\tau) \,\mathrm d \tau$$ Answer: yes, the system is invertible. I need some hint here, not the full ...