11
votes
Accepted
Accessing Maximum Value from a Singular Value Decomposed Matrix
The SVD Decompose the image into the (One way to look at it) many matrices.
For instance, given an Image $ I $ its SVD is given by:
$$ I = U S {V}^{T} = \sum_{i=1}^{\textrm{rank}(I)} {s}_{i} {u}_{i} {...
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10
votes
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How to Check Separability of 2D Filter / Signal / Matrix
Nilesh Padhi, Welcome to the DSP Community.
The classic definition of separable means the data (2D) given by $ X \in \mathbb{R}^{m \times n} $ can be written as:
$$ X = \sigma u {v}^{T} $$
Where $ \...
- 40.4k
9
votes
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How to Decompose a Separable Filter?
Indeed you can do that.
You may look on my answer to How to Prove a 2D Filter Is Separable?
By the SVD for any filter $ A $:
$$ A = \sum_{i = 1}^{n} {\sigma}_{i} {u}_{i} {v}_{i}^{T} $$
Since we'...
- 40.4k
7
votes
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Questions on the Generalized Tikhonov Regularization
One way to interpret the Tikhonov Regularization is using the Maximum A Posteriori (MAP) framework.
Lets' say we have a model of the form:
$$ \boldsymbol{y} = H \boldsymbol{x} + \boldsymbol{n} $$
...
- 40.4k
6
votes
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Regularization for Inverse Problems using the Singular Value Decomposition (SVD)
Similar to The Concepts Behind SVD Based Image Processing the horizontal axis are the samples index of the SVD basis.
The idea in the chapter you linked is generalizing the Wiener Filter.
While the ...
- 40.4k
5
votes
Accepted
Variational Regularization Method in Image Processing
This is an example of the Fidelity Term and Prior Term model.
In many Inverse Problems we assume some model on the additive noise. This part is modeled by the Fidelity Term ($ \mathcal{D} \left(A \...
- 40.4k
5
votes
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The Concepts Behind SVD Based Image Processing
You may think on the SVD as a generalization of the Discrete Fourier Transform.
Namely, it is generates an orthogonal basis to represent the data.
The nice thing about it, it generates the basis ...
- 40.4k
5
votes
Denoise Techniques When Clean Signal and Pure Noise Are Available
If the data is stationary and the noise is white then you should use the Wiener Filter.
If data isn't stationary you should look into the family of adaptive filters (LMS Filter and RLS Filter for ...
- 40.4k
4
votes
Accepted
What are the advantages and disadvantages separability of Gaussian 2D filter?
Assuming the simplest case with a square image $x[n,m]$ of size $N \times N$ and a square filter kernel $h[n,m]$ of size $M \times M$, the raw 2D convolution to produce the, cropped, output image $y[n,...
- 26.7k
4
votes
Accessing Maximum Value from a Singular Value Decomposed Matrix
[EDIT: some code made available] A common framework for (multivariate) image processing is to suppose that its useful features (edges, textures, spectral correlation) contain redundancy, while the ...
- 29.9k
4
votes
Accepted
How Does Mean Centering Affect the Result of Using SVD to Compress Images?
I will try to illustrate why it is important to remove the Mean from data when doing the PCA (SVD is the tool to basically do the PCA approach to dimensionality reduction -> Compression).
If the data ...
- 40.4k
4
votes
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SVD vs matched filter
That's not true, it's not better.
The thing is: the matched filter just implements the projection in the signal vector space, onto the signal vector itself (or a multiple thereof). (You'll find ...
- 26.1k
3
votes
why use svd() to invert a matrix?
The two methods differ, above all, by their applicability to matrix classes.
col (cholesky) decomposes Hermitian, positive-definite rectangular matrices into the ...
- 1,415
3
votes
Where is truncated-SVD image compression actually used?
A first difficulty with your question is that "usual instances" may not be fully publicized nor documented. A second one is related to the fact that "compression standard" are ...
- 29.9k
3
votes
Denoise Techniques When Clean Signal and Pure Noise Are Available
I'm not sure I understand the question, but if you have the exact waveform you want to recover, you can basically employ a matched filter to detect the existence of the signal in the acquired data.
...
- 185
3
votes
Accepted
How to express one image in terms of another one
I found an answer which is good-enough for me. As @Stanley Pawlukiewicz has pointed out in the comments, this is hard to do for a general case when there is little correlation between the images. I, ...
3
votes
Accepted
Could not construct original matrix using SVD
Don't have enough reputation to comment but you need to use the conjugate transpose in your formula for the result to be correct. So try stftb=U*S*V'; in the last ...
- 195
3
votes
Image compression using SVD in MATLAB
The typical thing to do is the low-rank approximation on separate channels. Assume that $C$ is a channel of the RGB image $I$:
...
- 5,207
2
votes
Problem with Covariance matrix using diagonal loading involved in calculation of eigenvalues
You can write
$$ R=YY^H $$
where $Y$ is a matrix of size $N\times N_f$ and $N$ is the dimension of $y_k$. $Y$ contains all the measured $y_k$ as its columns.
Then, the rank of $R$ is upper bounded ...
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2
votes
Where is truncated-SVD image compression actually used?
It has been said that the DCT reasonably closely matches the KLT for a representative set of images. KLT is essentially the same as PCA, I believe and SVD is only a different way to compute the same?
...
- 2,186
2
votes
Accepted
Subspace decomposition
You can rewrite the equation in a block matrix form:
\begin{eqnarray}X &=& U_s \Lambda_s V_s^H + U_n \Lambda_n V_n^H \\
&=& \left[ \begin{matrix} U_s & U_n\end{matrix} \right]
\...
- 2,686
1
vote
Accepted
Hankel Matrix SVD Denoising
A simple approach is to just take the last value of your time series and keep repeating it.
If I repeat the last value 100 times, then I still get the large drop at the end, but the repetition means ...
- 22.3k
1
vote
System Identification with a Limited Bandwidth Input Signal and Region of Interest
Can't you just use the pseudo-inverse? That'll mean instead of:
$$
\hat{\mathbf{h}} = (\mathbf{X}^{T}\mathbf{X})^{-1} \mathbf{X}^T \mathbf{y}
$$
you use
$$
\hat{\mathbf{h}}_{\tt pseudo} = (\mathbf{...
- 22.3k
1
vote
System Identification with a Limited Bandwidth Input Signal and Region of Interest
The reason x[n] must be white is because the solution will effectively spectrally weight the channel response based on the amount of energy present in each spectral frequency location. A white noise ...
- 37.1k
1
vote
Denoise Techniques When Clean Signal and Pure Noise Are Available
If noise is available and it is sure that noise will remain same/almost same for entire duration then one can go for Spectral Subtraction or Wiener filter techniques for noise reduction, which will ...
- 687
1
vote
How to extract features from nonstationary signal using SVD (singular value decomposition)?
What you're describing is a spectrum estimation based on signal representation in a vector space spanned by eigenvectors of a particular matrix generated from your signal observation.
That reminds me ...
- 26.1k
1
vote
Accepted
MUSIC algorithm derivation
The piece I was missing was the distribution of the initial phase values $\varphi_1$ and $\varphi_2$. It is standard to assume that these are uniformly distributed [^]. This leads to:
$$
\mathbf{R_x} =...
- 4,014
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