Synchrosqueezing Toolbox ported to Python, authored by Eugene Brevdo, Gaurav Thakur. Original: https://github.com/ebrevdo/synchrosqueezing

Reviewers needed; the repository is in a development stage - details below.

• Forward & inverse CWT- and STFT-based Synchrosqueezing
• Forward & inverse discretized Continuous Wavelet Transform (CWT)
• Forward & inverse discretized Short-Time Fourier Transform (STFT)
• Phase CWT & STFT

An eventual goal is a merged Pull Request to PyWavelets (relevant Issue). Points of review include:

1. Correctness; plots generated from CWT transforms resemble those in the publications, but not entirely
2. Completeness; parts of code in the original Github are missing or incorrect, and I'm unable to replace them all
3. Unit tests; I'm not familiar with Synchrosqueezing itself, so I don't know how to validate its various functionalities
4. Licensing; unsure how to proceed here; original's says to "Redistributions in binary form must reproduce the above copyright notice" - but I'm not "redistributing" it, I'm distributing my rewriting of it
5. Code style; I'm aware PyWavelets conforms with PEP8 (but I don't), so I'll edit PR code accordingly

Correctness:

• 1. Example 1
• 2. Example 2

Completeness:

• 1. freqband in synsq_cwt_inv and synsq_stft_inf is defaulted to an integer, but indexed into as a 2D array; the two have nearly identical docstrings, and reference the same equation, but the equation appears completely irrelevant to both.
• 2. quadgk has been ported as quadpy's quad (linked its wrapped function), which does not support infinite integration bounds, and has trouble with computing synsq_stft_inv's integral. Needs a workaround.
• 3. As seen in examples, the y-axis shows "scales", not frequencies - and the relation between the two is neither clear nor linear; it also isn't linear w.r.t. len(t), nv, or fs. Publications show frequencies instead.

Unit tests: Whatever suffices for PyWavelets will suffice for me

One checkmark = code written; two = reviewed

There are more unlisted (see original repo), but not all will be implemented, in particular GUI implementations.

• Renamed variables/functions; more Pythonic & readable
• Removed unused arguments / variables
• Improved nan/inf handling
• Added examples; original repo lacks any examples in README
• Indexing / var references; MATLAB is 1-indexed, and handles object reference / assignment, and 'range' ops, differently
• Edited docstrings; filled missing info, & few corrections
• Moved functions; each no longer has its own file, but is grouped with other relevant functions
• Code style; grouped parts of code as sub-functions for improved readability; indentation for vertical alignment; other
• Performance; this repo may work faster or slower, as Numpy arrays are faster than C arrays, but some of original funcs use MEX-optimized code with no Numpy equivalent. Also using dense instead of sparse matrices (see below).
• Performance; this repo will work 10x+ faster for some of the methods which were vectorized out of for-loops

Other:

• Dense instead of sparse matrices for stft_fwd in stft_transforms.py, as Numpy doesn't handle latter in ops involved

See examples.py. Links to: paper [1], paper[2]. _inv methods (reconstruction, inversion) have been omitted as they involve freqband.

EX 1: Paper [1], pg. 1086

Only real components shown; imaginary are nearly identical, sometimes sign-opposite.

synsq-CWT (synsq_cwt_fwd) appears to produce strongest agreement with paper (FIG 4), while none of STFT yield any resemblance of anything in the papers. It's also unclear whether synsq_squeeze was used for "synsq" in FIG 4 instead.

EX 2: Paper [2], pg. 18

Similar situation as EX 1; again CWT has close resemblance, and STFT is in a separate reality. The two apparent discrepancies w/ CWT are: (1) slope of the forking incline, steeper in FIG. 3; (2) position of horizontal line, lower in FIG. 3. As for the black lines in FIG 3, they seem to be the (manual) "markings" mentioned under the figure in the paper.