Questions tagged [inverse-problem]

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12
votes
3answers
327 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 ...
8
votes
2answers
5k views

Using the Inverse Filter to Correct a Spatially Convolved Image

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 ...
8
votes
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 ...
5
votes
3answers
266 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, ...
5
votes
1answer
370 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 \...
4
votes
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
votes
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
1answer
114 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 ...
3
votes
2answers
153 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+...
3
votes
3answers
240 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 ...
3
votes
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 ...
3
votes
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 ...
2
votes
2answers
1k 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?
2
votes
1answer
342 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
votes
1answer
312 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 $$...
2
votes
1answer
461 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 ...
2
votes
0answers
128 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 ...
1
vote
2answers
62 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 ...
1
vote
1answer
265 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
vote
2answers
471 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\...
1
vote
3answers
346 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] ...
1
vote
2answers
149 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 ...
1
vote
1answer
374 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 ...
1
vote
1answer
555 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\...
1
vote
1answer
254 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
vote
0answers
21 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 ...
0
votes
2answers
223 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 ...
0
votes
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?
0
votes
2answers
36 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-\...
0
votes
1answer
287 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
1answer
88 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 ...
0
votes
1answer
306 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'...
0
votes
1answer
128 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 ...
0
votes
1answer
88 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 ...
0
votes
1answer
274 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? ...
-1
votes
3answers
4k 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 [...