HLSVDPRO provides code to fit a model function (sum of lorentzians) to time-domain data via a 'black box' state space approach. This is often used in clinical MRS to remove residual water from time domain signals.

# HLSVDPRO

## Overview - Black box fitting of Time Domain Signals

The HLSVDPRO package provides code to fit a model function (sum of lorentzians) to time-domain data via a 'black box' state space approach (see references below). One frequent use for this is by the clinical MRS community for residual water removal from MRS signals in the time domain.

Internally, we use the PROPACK library which contains a set of functions for computing the singular value decomposition (SVD) of large and sparse matrices. The SVD routines are based on the Lanczos bidiagonalization algorithm with partial reorthogonalization (BPRO).

The calculated singular values and column vectors are subsequently processed into lists of parameters that describe the sum of lorentzians that approximate the data based on the paper by Laudadio (see below). Parameters are numpy arrays of: frequencies, damping factors, amplitudes, and phases.

Example:

import hlsvdpro
import numpy as np
import matplotlib.pyplot as plt

data = hlsvdpro.get_testdata()
npts = len(data)
indat = hlsvdpro.TESTDATA   		# this is a built-in dict with test data
dwell = float(indat['step_size'])
nsv_sought = indat['n_singular_values']

result = hlsvdpro.hlsvd(data, nsv_sought, dwell)

nsv_found, singvals, freq, damp, ampl, phas = result

print("np.allclose(freq, indat['freq0']) = ", np.allclose(freq, np.array(indat['freq0'])) )

fid = hlsvdpro.create_hlsvd_fids(result, npts, dwell, sum_results=True, convert=False)

chop = ((((np.arange(len(fid)) + 1) % 2) * 2) - 1)
dat = data * chop
fit = fid * chop
dat[0] *= 0.5
fit[0] *= 0.5

plt.plot(np.fft.fft(dat).real, color='r')
plt.plot(np.fft.fft(fit).real, color='b')
plt.plot(np.fft.fft(dat-fit).real, color='g')
plt.show()



## HLSVDPRO Methods

• hlsvdpro.hlsvdpro(data, nsv_sought, m=None, sparse=True) - the main method for running the hlsvdpro algorithm. It does not require the dwell time of the time domain data, but it also does not convert the results to standard units. It does allow the user to specify the dimensions of the Hankel matrix, and whether a sparse SVD is performed or not.

• hlsvdpro.hlsvd(data, nsv_sought, dwell_time) - provides backwards compatibility to the API for HLSVDPRO version 1.x. It calls the hlsvdpro() method with default values corresponding to the algorithm used in version 1.x. See docstring for more information on the default values used.

## HLSVDPRO Utility Methods

• hlsvdpro.create_hlsvd_fids(result, npts, dwell, sum_results=False, convert=True) - can be used to create FIDs from the results tuple from either the hlsvd() or the hlsvdpro() methods. It can return either individual FIDs or a sum of all FIDs as a result.

• hlsvdpro.convert_hlsvd_result(result, dwell) - uses the dwell time to convert the hlsvdpro() result tuple to more standard units. Frequencies convert to [kHz], and damping factors to [ms]. Phases convert to [degrees]. Singular values, amplitudes and row and column matrices are maintained at their same values and output tuple locations. Note - the hlsvd() method automatically calls this internally, so you don't have to convert values if you use that method.

• hlsvdpro.get_testdata() - returns a numpy array of 1024 complex data points that represents a real world short TE single voxel PRESS data set. This function converts the base64 encoded string saved in the TESTDATA dict into a numpy array for you. Additional information about the data and the known values for fitting it via the hlsvd() method can be retrieved from the TESTDATA dict. See 'Example' for more usage information.

As of version 2.0.0, HLSVDPRO also provides access to internal PROPACK SVD functions via the 'propack' module. These have been compiled from PROPACK version 2.1, and include:

• hlsvdpro.propack.lansvd() - matrix-vector product via user defined Python callback vector/matrix multiplication
• hlsvdpro.propack.lansvd_irl() - matrix-vector product via user defined Python callback
• hlsvdpro.propack.lansvd_aprod() - matrix-vector product via fixed internal Fortran call
• hlsvdpro.propack.lansvd_irl_aprod() - matrix-vector product via fixed internal Fortran call

## Technical Overview and References

The PROPACK SVD Fortran code was written by Rasmus Munk Larsen. Teresa Laudadio and Diana Sima adapted it into HLSVDPRO algorithm. Brian J Soher adapted the algorithm for PROPACK v2.1 use and the python wrappers used to access the code.

The Python wrapper is compatible with Python 2.7 and 3.x. The binaries that the Python wrapper calls are 64-bit, so this code is not compatible with 32-bit Python.

The state space approach is described in: S.Y. Kung, K.S. Arun and D.V. Bhaskar Rao, J. Opt. Soc. Am. 73, 1799 (1983).

HLSVDPRO version 1.0.x made use of code from PROPACK version 1.x and was implemented based on the paper by: W.W.F. Pijnappel, A. van den Boogaart, R. de Beer, D. van Ormondt, J. Magn. Reson. 97, 122 (1992).

HLSVDPRO version 2.0.0 was adaptated to use PROPACK library version 2.1 to implement the HLSVDPRO algorithm as described in: T. Laudadio, N. Mastronardi L. Vanhamme, P. Van Hecke and S. Van Huffel, "Improved Lanczos algorithms for blackbox MRS data quantitation", Journal of Magnetic Resonance, Volume 157, pages 292-297, 2002.

The Python wrapper used to access the PROPACK library was adapted from the pypropack repository create by Jake van der Plas on Github: https://github.com/jakevdp/pypropack with extensions by Brian J Soher to include the matrix-vector product calculation within the Fortran code.

## Acknowlegements and Kudos

We (the HLSVDPRO team) have made some minor modifications to this version of HLSVDPRO. Most significantly, we have updated the PROPACK library to version 2.1, added Makefiles and Python wrappers for both the HLSVDPRO and PROPACK methods. We are very grateful to all of the above scientists for sharing their work and agreeing to release it under a BSD license.

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