Hot answers tagged

7 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 ...
Laurent Duval's user avatar
4 votes
Accepted

Why is the high-pass filter result in a discrete wavelet transform (DWT) downsampled?

Assume the original sampling rate is $F_s$ and that the filters are perfect brickwall Halfband filters. After the low pass filtering the frequency content is from $0 \rightarrow \frac{F_s}{4}$. ...
David's user avatar
  • 2,861
3 votes

Real-time wavelet decomposition and reconstruction for ECG feature extraction

Right now, I read the live ECG data into a 5 second buffer, and then perform the wavelet filter on the buffer (as described above). This seems rather inefficient, because I have to filter the entire ...
A_A's user avatar
  • 10.7k
3 votes

Wavelet image denoising: dual-tree versus double-density

The formalism of the continuous wavelet transform is relatively flexible. To make a practical tool out of it, you ought to discretize it, and here come the pain. It is quite easy to discretize it in a ...
Laurent Duval's user avatar
2 votes
Accepted

Why do Dyadic filterbanks downsample the high pass signal portions?

[Good question, that made me rethink of stuff I believed natural. I shall incorporate them in future lectures] Downsampling the highpass (and the lowpass) provide you with a critically sampled ...
Laurent Duval's user avatar
2 votes
Accepted

sparsifying an ECG signal using wavelet

As illustrated by the simplified figure and the Matlab code below, such a graph can be obtained with a standard discrete wavelet decomposition, possibly with a periodic extension to preserve the size, ...
Laurent Duval's user avatar
2 votes

Should I ever pick the continuous wavelet transform over the discrete one? DWT vs CWT vs STFT

Standard 2-band discrete wavelet transforms have some subsampling, thus they tend to be much less precise in localization. Plus, their shapes are limited, and the ones with finite support are slightly ...
Laurent Duval's user avatar
2 votes
Accepted

Looking for a analytical formula to compute the central frequency of a signal analyzed by a discrete wavelet at a given scale

They do mean the pseudo-frequency of the wavelet which is not dependent on the signal being analyzed. The misleading terminology that they use seems to come from from one of the references, Han, P. (...
Olli Niemitalo's user avatar
2 votes

Wavelet transform of a spatial convolution

I cannot say I have a clear understanding of this at this time. However, a few pointers. I'd love to see somebody provide a detailed account. Others bits at: Multiplication in the wavelet domain, ...
Laurent Duval's user avatar
2 votes

Some questions about the intuition of the DWT

First, one should be cautious about processing short signals like these. I am unsure about the length of 3 for a3 and d3. Now, ...
Laurent Duval's user avatar
2 votes

Is there an equivalent of Parseval's theorem for wavelets?

Yes. The instantaneous frequency, multiplied by the square modulus of the transformed function, upon integration, yields a Parseval theorem result for the original signal - provided that it has been ...
NinjaDarth's user avatar
2 votes
Accepted

What is the importance of the translational invariance of the CWT?

First, translational invariance makes signal processing somehow independent on the time origin of the recording. The term "invariance" might be debated, as done here: What is the difference ...
Laurent Duval's user avatar
2 votes

What is the importance of the translational invariance of the CWT?

CWT is translation-invariant in feature sense: translating a pattern translates its representation but not modify it. In coefficient sense, it is translation equivariant: shift signal $\Leftrightarrow$...
OverLordGoldDragon's user avatar
1 vote

Manual DWT vs Python pywt

Here is what is really happening. pywt first extends the signal based on padding mode. Default mode is symmetric. It then performs convolution and returns only the relevant samples for the output. ...
Shailesh Kumar's user avatar
1 vote

Manual DWT vs Python pywt

This might be related to the extension/periodization mode, or how the signal is treated outside its natural boundaries (left and right). The standard extension of ...
Laurent Duval's user avatar
1 vote

Detecting and removing interferences from a signal

I'd recommend using a median filter to smooth your signal. A median filter will get rid of the outliers, or the spikes in your signal. The length of the median filter will have to be determined by how ...
orchi_d's user avatar
  • 567
1 vote

DWT on short signals

With such short (known) stimuli, does it make sense to use generic analysis methods, or is it «better» (less work, results that are more usable) to tailor your processing to the transmitted processing ...
Knut Inge's user avatar
  • 3,350
1 vote
Accepted

Lifting scheme versus filter banks

Traditional (analysis) filter banks are banks of linear filters followed by downsampling. This is not so efficient, because you convolve and then subsample: possibly a waste of operations. People ...
Laurent Duval's user avatar
1 vote
Accepted

wavelet packet transform and lifting scheme?

A lifting scheme is a method for splitting a sequence of discrete samples into downsampled subsequences, so that you can predict a subsequence from the other, and update the former later, using ...
Laurent Duval's user avatar
1 vote
Accepted

Time location of the DWT detail coefficients using MATLAB

This question has multiple facets (after comments), so I will focus on the principal. First, regarding coefficient localization: a discrete wavelet coefficient depends on several signal samples. ...
Laurent Duval's user avatar
1 vote
Accepted

Discrete Wavelet Transform output: coefficients or FIR-filtered signals?

In one level DWT, each output of the low-pass or a high-pass can indeed be considered as signals. Thus each of those signals are subsampled by a factor of 2, and the same two-filter-subsampling is ...
Laurent Duval's user avatar
1 vote
Accepted

At what stage do we compute the approximations and details while performing a DWT?

This expression is more a discretization of a continuous wavelet transform than an actual DWT (discrete wavelet transform), provided $\psi$ is a genuine wavelet. It only computes the wavelet ...
Laurent Duval's user avatar
1 vote
Accepted

Difference between "Discrete Wavelet Transform" and "Discrete Wavelet Decomposition"

The discrete wavelet transform should denote "the operations" that, applied to some data, yield a discrete wavelet decomposition. The first one can be seen as a matrix operator, while the second ...
Laurent Duval's user avatar
1 vote

Why is the high-pass filter result in a discrete wavelet transform (DWT) downsampled?

The classical discrete wavelet transform is critically decimated. In other words, it should preserve the quantity of "samples". In other words, apart from data border effects, the number of wavelet ...
Laurent Duval's user avatar
1 vote

Why is the high-pass filter result in a discrete wavelet transform (DWT) downsampled?

The whole idea is that every level splits the image information in two equal halves, which can be represented by half the number of coefficient as the previous level, each. Otherwise, you'd not be ...
Marcus Müller's user avatar
1 vote

Other time-frequency-plane tiling than STFT, DWT, ConstantQ-Transform: multiresolution STFT?

Your time-frequency grid is Mondrian-shaped: essentially, rectangles supported on dyadic splits of the time or the frequency axes. Hence, you can easily start for any reversible time-frequency tiling, ...
Laurent Duval's user avatar
1 vote

Programming the IDWT for image processing

here is one version of the Haar analysis/synthesis filter bank pair. Oddly, the synthesis filters, up to a scale factor, are mirrors of the analysis filters. Other options are: $\pm 1/\sqrt{2}$ ...
Laurent Duval's user avatar
1 vote

DWT versus band-pass filter

With high probability, the band-pass filter. DWT are invertible non-redundant discrete transformations that decompose data onto iterated low- and high-pass filters. Hence, the bands limits are mostly ...
Laurent Duval's user avatar
1 vote
Accepted

Implementing the DWT

Treating boundaries correctly is non trivial for wavelets and filter banks in general. Symmetric/anti-symmetric filters (such as with biorthogonal wavelets as the 5/3 or the 9/7) help a little. Yet, ...
Laurent Duval's user avatar

Only top scored, non community-wiki answers of a minimum length are eligible