pinvprob3

📝 a short paper written in Japanese

🐧 pinvprob3 is a Python 3 package for the linear inverse problem including generalized inverse matrix, truncated SVD, Tikhonov regularization, L-curve criterion, Bayesian linear inverse problem. Originally, I wrote Fortran code for the following two papers, Kawahara & Fujii (2011), Fujii & Kawahara (2012), , which I converted for python. I then added the code for the Bayesian version for the following paper: Kawahara & Masuda (2020).

The algorithm of the L-curve criterion is based on the brilliant book:

📗 Hansen, P. C. 2010, Discrete Inverse Problems: Insight and Algorithms (the Society for Industrial and Applied Mathema tics).

"random_light.py" is a sample code. This code retrieves a small png image from a collection of summation of random rectangle parts of the image. If you use Japanese, see invprov.pdf, otherwise, see the figure below. I hope you will understand the problem.

Note that these codes are inefficient when the image size is large because the codes directly use the singular value decomposition. The sample image che.png was taken from Wikipedia and was compressed to a small image.

Requirements

• python 3
• scipy
• pylab
• scikit-learn (only for random_light_fast.py)

Install

python setup.py install

Tutorial

• Solve the problem by the Natural Generalized Inverse Matrix (NGIM) with no noise.

 ./random_light.py -f che.png -n 1000 -l 0.0 -p 0.7 -w 20.0

• Solve the problem by the Natural Generalized Inverse Matrix (NGIM) with an additional noise. The retrieved map is very unstable.

 ./random_light.py -f che.png -n 2500 -l 0.0 -p 0.7 -s 1.0

• Solve the problem by the Tiknov regularization with an additional noise.

 ./random_light.py -f che.png -n 2500 -l 3.0 -p 0.7 -s 1.0

• Solve the problem by the Truncated Singular Value Decomposition (TSVD) with an additional noise.

 ./random_light.py -f che.png -n 2500 -l 0.0 -p 0.7 -s 1.0 -lim 1.0

• Use the L-curve criterion (Hansen 2010) to look for an appropriate regularization parameter.

 ./random_light.py -f che.png -n 2500 -L 0.01 100.0 -p 0.7 -s 1.0

• Fast version of the Tikhonov regularization using scikit-learn (linear_model.Ridge)

This example uses the Ridge Regression in scikit-learn linear_model. It's about ten times faster than np.linalg.svd+tikhonov.py.

 ./random_light_fast.py -f che.png

If you want to perform the cross-validation (CV), provide a list of lambda. In this case, linear_model.RidgeCV is used for the selection of lambda (but usually, smaller than I expect for the image retrieval).

./random_light_fast.py -f che.png -l 1.e-9 1.e-8 1.e-7 1.e-6 1.e-5 -s 1.0

You can also try LASSO with or w/o CV... but... anyway try.

./random_light_fast.py -f che.png -l 0.001 -solver lasso

License

GPL. See "License" in detail.