18
votes
Accepted
Synchrosqueezing Wavelet Transform explanation?
Synchrosqueezing is a powerful reassignment method. To grasp its mechanisms, we dissect the (continuous) Wavelet Transform, and how its pitfalls can be remedied. Physical and statistical ...
- 5,005
16
votes
Accepted
Wavelet Scattering explanation?
Wavelet Scattering is an equivalent deep convolutional network, formed by cascade of wavelets, modulus nonlinearities, and lowpass filters. It yields representations that are time-shift invariant, ...
- 5,005
13
votes
Accepted
What Is the Difference between Difference of Gaussian, Laplace of Gaussian, and Mexican Hat Wavelet?
Laplace of Gaussian
The Laplace of Gaussian (LoG) of image $f$ can be written as
$$
\nabla^2 (f * g) = f * \nabla^2 g
$$
with $g$ the Gaussian kernel and $*$ the convolution. That is, the Laplace ...
- 982
10
votes
Accepted
Feature extraction/reduction using DWT
I think it is kind'a similar to soft and hard thresholding using in wavelet de-noising. Have you come across this topic? pywt has already an in-built function for ...
- 10.5k
9
votes
Accepted
Which transform most closely mimics the human auditory system?
In designing such transformations, one should take into account competing interests:
fidelity to the human auditory system (that varies with people), including non-linear or even chaotic aspects (...
- 29.9k
9
votes
Factor $|a|^{-1/2}$ in definition of mother wavelets
My answer is for real scale $a$ and the fact that wavelet transform is usually defined in $L_2$ with norm
$$||\Psi(\tau)|| = \int_\mathbb{R} \Psi(\tau)\Psi^*(\tau)\mathrm{d}\tau $$
So
$$||\Psi_{a,t}...
- 5,760
8
votes
Accepted
When can we write Heisenberg uncertainty Principle as a equality?
It is important to define the time and frequency widths $\Delta_t$ and $\Delta_{\omega}$ of a signal before discussing any special forms of the uncertainty principle. There is no unique definition of ...
- 80.2k
8
votes
Discrete wavelet transform; how to interpret approximation and detail coefficients?
Wavelet transforms can be more difficult to interpret than FFT at face value due to the various representations, nomenclature and output formats. I had to study more than 15 resources to get a good ...
- 181
8
votes
Accepted
Advantage of STFT over wavelet transform
Wavelet transforms and short-term/short-time Fourier transforms are broad names for classes of transformations that are not totally distinct and may overlap (pun intended).
Both can be efficient for ...
- 29.9k
7
votes
Accepted
Why Wavelet based Transform Is More Suitable for Image Compression Compared to DCT?
Both JPEG and JPEG 2000 use the change of basis compression type.
Namely, we transform the data into a different representation assuming in this representation the number of parameters needed to ...
- 40.4k
6
votes
Accepted
Shift invariant in wavelet
Let's say you have a signal which is all zeros except for a spike at one point where x(8)=1 (total N=32, for example). If you perform the DWT on this signal and then calculate the total energy (by ...
- 113
6
votes
Accepted
Slow Down Music Playing While Maintaining Frequency
Yeah some of us can do it, you can speed up or slow down without affect the pitch, some guys call this applications of Time Stretch, there different ways to do it, you can do in frequency domain or ...
- 1,913
6
votes
Synchrosqueezing Wavelet Transform explanation?
Low-level intuition can be obtained by inspecting the phase transform, visually. Answer complements and is complemented by this one. (-- Answer code)
We consider a pure sinusoidal tone; ideas extend ...
- 5,005
6
votes
Accepted
Wavelet Scattering properties & implementation?
Scattering overview provided in this answer.
Computational structure
Fig 4, Deep Scattering Spectrum
In steps:
(First order begins) $x$ convolves with $\psi1_i$ --> $W1_i$
Modulus, $W1_i \...
- 5,005
5
votes
Accepted
Continuous Wavelet Transform with Scipy.signal: what is parameter "widths" in cwt() function? How do time-frequency?
complex morlet was added Aug 10, 2007
ricker and cwt were added Sep 20, 2011
There's no indication that cwt is meant to be compatible with ...
- 15k
5
votes
STFT and DWT (Wavelets)
The short-time Fourier transform is generally a redundant transformation, usually implemented with the same subsampling over every frequency. If the window is well chosen, it is complete: you can ...
- 29.9k
5
votes
Accepted
Convolution of Signal with a Wavelet
The Conjugation is part of the definition of the Convolution as an Inner Product operation.
So you wrote the operation correctly.
Usually people uses "Real" Wavelets hence no need for that.
...
- 40.4k
5
votes
How does a wavelet help in image compression?
A common wavelet based standard is JPEG 2000 and a common DCT based standard is JPEG.
JPEG 2000 uses wavelets, but a good portion of the better compression it achieves than JPEG is due to the fact ...
- 1,071
5
votes
Is R suitable for digital signal processing
Since the bulk of R’s DSP capability comes from the signal package which was ported over from the open source project Octave (itself influenced by MATLAB), there's no intrinsic limitation of R.
What ...
- 1,710
5
votes
Accepted
"Fourier Transform can localize signals in frequency domain, but not in time domain." -- What does it mean in layman's terms?
In the Fourier transform, the basis functions are complex exponentials. These functions are perfectly localized in the frequency domain, i.e., they exist at one frequency, but they have no time ...
- 80.2k
5
votes
What Is the Difference between Difference of Gaussian, Laplace of Gaussian, and Mexican Hat Wavelet?
The Ricker wavelet, the (isotropic) Marr wavelet, the Mexican hat or the Laplacian of Gaussians belong to be the same concept: continuous admissible wavelets (satisfying certain conditions). ...
- 29.9k
5
votes
Additive White Gaussian Noise (AWGN) and Undecimated DWT
One property of Orthogonal Transformations is that White Noise stays White (Uncorrelated) under Orthogonal Transformations (One could say it's a property of White Noise).
Usually this property ...
- 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
5
votes
Accepted
Orthonormal Dictionaries for Band Limited Signals
Why would you add the constraint of being Orthonormal Dictionary?
It doesn't make sense in the context of what you ask.
First we need to define resolution.
If you mean the grid to be denser than ...
- 40.4k
5
votes
Accepted
Is there an equivalent of Parseval's theorem for wavelets?
Yes indeed! In theory as long as the wavelet is orthogonal, the sum of the squares of all the coefficients should be equal to the energy of the signal. In practice, one should be careful that:
the ...
- 29.9k
5
votes
Image / Video Upscaling (Super Resolution) Algorithm Explanation (Image and Video Upscaling from Local Self Examples)
The concept of using different scales data was very promising in the pre Deep Learning era (Look for Michal Irani's work on the subject as well).
Indeed the filters are applies over the rows and ...
- 40.4k
4
votes
What's the difference between the Gabor and Morlet wavelets?
For those looking for a compact description, I found this nice docstring while inspecting the kymatio GitHub repository:
A Morlet filter is the sum of a Gabor ...
- 140
4
votes
Accepted
Denoise images with wavelets
Wavelets are not key to denoising. There are different ways to denoise an image, for example in the original signal domain or in the transform domain (i.e. Fourier or wavelet). Wavelets work best for ...
- 1,710
4
votes
Accepted
Compare between JPEG and JPEG2000
JPEG is far simpler. It divides the image into 8x8 pixel blocks, and processes each using a Discrete Cosine Transform. The results are quantised and then encoded. The quality is fixed by the ...
- 507
4
votes
Accepted
Daubechies wavelet transform
Looks like you need a general explanation of the discrete wavelet transform (DWT). DWT breaks a signal down into subbands distributed evenly in a logarithmic frequency scale, each subband sampled at a ...
- 12.4k
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