Questions tagged [inverse-problem]

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2
votes
3answers
144 views

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 $...
2
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2answers
71 views

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 ...
7
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4answers
2k views

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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0answers
42 views

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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0answers
26 views

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 ...
1
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2answers
119 views

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 ...
0
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2answers
50 views

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-\...
5
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3answers
400 views

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, ...
1
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1answer
610 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 ...
3
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1answer
629 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?
2
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3answers
703 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] ...
3
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2answers
274 views

Auto-correlation function, an inverse problem

$x[n]$ 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+...
1
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1answer
418 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
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1answer
98 views

Inverse FFT - synch the phase

Is there any way to synchronise phase of output of inverse DFT in each buffer? When I send the output of inverse DFT to the speaker it sounds nasty. Of course I know the „windowing functions” but it ...
1
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1answer
713 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\...
2
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4answers
193 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
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0answers
179 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 ...
3
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1answer
444 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 blur image $B$ is obtained as follows $$B = X * K$$ I want to solve the following general regularization problem $$...
4
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1answer
158 views

Underdetermined deconvolution of windowed output

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 ...
2
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1answer
747 views

Is this system invertible or not?

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 ...
0
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1answer
106 views

Phase error correction for Fourier transform basis vectors

I was wondering what is the best way to account for phase noise in a set of basis vectors? Example: I have a measured signal, say $f'(x)$ that should be related to a desired spectrum $f(\nu)$ through ...
2
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1answer
298 views

What Are the Types of Deconvolution?

I am totally new to image processing and wanted to ask you if you could confirm what I understood. It is about deconvolution: From what I read we find 2 main types of deconvolution: 1. Analytical <...
1
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1answer
329 views

Best (Perceptually / Objectively) Super Resolution Methods Out There?

I'm curious about the advances in the area of image super-resolution (SR) that have given the best results to date, both perceptually (visually pleasing) and objectively (e.g. PSNR, SSIM criterias). I'...
1
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2answers
266 views

Inverse Problem / Deconvolution with Pink Noise

Hi I dived somewhat into deconvolution of systems which can be described as: $s(t) = o(t) * h(t) + n(t)$ where $s$ is my measured 1D time resolved signal, $o$ is the original signal $h$ is the ...
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3answers
5k views

Deriving the Convolution Kernel of the Inverse of a Signal

Let $y$ be the inverse (in the sense of convolution) of $x$, i.e. $$x \star y = \delta$$ Context: $x[n]$ is a discrete signal defined for $n = 0,\ldots, N$. We can assume $x[n] = 0$ if $n \not\in [...
5
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3answers
2k views

Estimating the Impulse Response of the Room Using Sweep Signal Microphone Recorded Signal (Input & Output of a Convolution)

I played this signal A (a 20Hz to 20000Hz sinusoidal sweep in 10 seconds) with a studio monitor speaker in a big church, and I recorded the result B with good microphones. The result is very reverb-...
2
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1answer
308 views

Why Sparse Priors Like Total Variation Opts to Concentrate Derivatives at a Small Number of Pixels?

When performing image deconvolution (deblurring), people often make use of priors to get rid of the illness of the problem. One very common prior is total variation, a sparse prior. Compared to ...
1
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1answer
137 views

What are Local and Global Inpainting Techniques in Image Processing?

Is Diffusion-based inpaiting Local or Global? Is Pixel-based inpaiting Local or Global? Is Patch-based inpaiting Local or Global? Can Local-diffusion be used inside Patch-based in-painting problems of ...
4
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1answer
2k views

Use MATLAB to Restore a Signal from a Given Degraded Signal Using Tikhonov Regularization

Anyone could share how to develop an application in MATLAB to restore the signal from a given degraded signal using Tikhonov regularization i.e restoring the signal $f$ via solving $$ \min || g - f ...
2
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1answer
386 views

Deconvolution Using Response to an Heavy Side

I'm measuring a "charge" signal in function of time from an amplifier. Here is a measured signal (x-axis is the time in some arbitrary units, y-axis is the charge in ADU): I would like to get the "...
2
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1answer
112 views

How to Increase the Resolution of a Video from a Sequence of Photos?

I have a video filmed in a relatively low quality and resolution and a sequence of photos of higher quality taken of the same scene at the rate of about one image every two seconds or so. Could those ...
4
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1answer
3k views

1D Deconvolution with Gaussian Kernel (MATLAB)

Suppose that I know the output and the transfer functions of a system and I would like to calculate the input function using deconvolution. To get a grasp of the idea I have created a simple ...
1
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1answer
325 views

Sharpen Defocused Image (Deconvolution / Image Restoration)

Using OCR, I want to extract text from product packages using Google Glass. However, because of the fixed focus of the camera the package pictures are blurred. Is there a way to sharpen the image? ...
6
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4answers
3k views

Deconvolution by Convolution

This is now a second time I am attempting to ask this very important but simple question here. What I want to know is can you do deconvolution by convolving a signal. It is often stated that, for ...
4
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2answers
3k views

Estimate the Transfer Function of an Unknown System

Suppose you have a system, H, that you want to estimate its transfer function. You have a finite number of complex input samples, x, and noisy complex (magnitude and phase) output samples, y: In ...
4
votes
3answers
311 views

Deconvolution (Linear Convolution) with an Under Determined System of Equations?

If I have a measured signal $\mathbf{y}$ which is the result of the true signal $\mathbf{x}$ convolved with another function (linear and not circular convolution), I always seem to get an ...
2
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1answer
86 views

Why Does Image Deconvolution Still Work with Image without Sharp Edges?

Why does image deconvolution still work with image without sharp edges? Take this image for example:
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2answers
2k views

Differences Using Maximum Likelihood or Maximum a Posteriori for Deconvolution / Deblur

Are there any differences if you use Maximum Likelihood or Maximum a Posteriori to estimate the Point Spread Function for image deconvolution?
1
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2answers
159 views

Why Is Super Resolution (SR) Possible?

I've been reading about Super Resolution Image reconstruction (Reconstruction of high resolution image from multiple low resolution aliased images contain sub pixel shifts), and i want to know why SR ...
3
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2answers
2k views

How Is the Formula for the Wiener Deconvolution Derived?

Wikipedia shows this formula: $$ \ G(f) = \frac{H^*(f)S(f)}{ |H(f)|^2 S(f) + N(f) } $$ But how is this Wiener deconvolution formula derived and where does the squaring ($|H(f)|^2$) come from?
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2answers
520 views

Deconvolution of Images - How To?

I'd like to do a deconvolution of image. For example for convolution I'm using a $3\times 3$ mask with all elements $= 1$: $$\begin{bmatrix}1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1\...
8
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1answer
1k views

Deconvolution - Richardson Lucy vs. Wiener Filter

I am studying some deconvolution techniques, In order to remove motion blur, like: Richardson-Lucy Wiener Are there any pros / cons of using one versus another? For example which are the pros / cons ...
12
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3answers
352 views

Increasing Image Resolution

I know of some oscilloscopes (DSA8300) that repeatedly sample at a few hundred kS/s to reconstruct a few GHz signal. I was wondering if this could be extended to 2D signals (photographs). Can I take a ...
6
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1answer
384 views

Deconvolution Question on Article “Deriving Intrinsic Images from Image Sequences” by Yair Weiss

there are n derivative filters: $f_i$, and denote $f_i^r$ as $f_i$'s reverse filter such that $$f_i(x,y)=f_i^r(-x, -y)$$ $r_i, f_i$ given, to find $r$ from the equations: $$f_i * r = r_i, (1 \leq i \...
8
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2answers
5k views

Using the Inverse Filter to Correct a Spatially Convolved Image (Deconvolution)

As part of a homework assignment, we are implementing the Inverse Filter. Degrade an image then recover with an Inverse Filter. I convolve the image in the spatial domain with a 5x5 box filter. I FFT ...