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wiimatch is a package that provides core computational algorithms for optimal “matching” of weighted N-dimensional image intensity data using (multivariate) polynomials.

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LSQ Image Intensity Matching

A module that provides main API for optimal (LSQ) “matching” of weighted N-dimensional image intensity data using (multivariate) polynomials.

Author:Mihai Cara (contact: help@stsci.edu)
License:LICENSE
wiimatch.match.match_lsq(images, masks=None, sigmas=None, degree=0, center=None, image2world=None, center_cs='image', ext_return=False, solver='RLU')[source]

Compute coefficients of (multivariate) polynomials that once subtracted from input images would provide image intensity matching in the least squares sense.

Parameters:
images : list of numpy.ndarray

A list of 1D, 2D, etc. numpy.ndarray data array whose “intensities” must be “matched”. All arrays must have identical shapes.

masks : list of numpy.ndarray, None

A list of numpy.ndarray arrays of same length as images. Non-zero mask elements indicate valid data in the corresponding images array. Mask arrays must have identical shape to that of the arrays in input images. Default value of None indicates that all pixels in input images are valid.

sigmas : list of numpy.ndarray, None

A list of numpy.ndarray data array of same length as images representing the uncertainties of the data in the corresponding array in images. Uncertainty arrays must have identical shape to that of the arrays in input images. The default value of None indicates that all pixels will be assigned equal weights.

degree : iterable, int

A list of polynomial degrees for each dimension of data arrays in images. The length of the input list must match the dimensionality of the input images. When a single integer number is provided, it is assumed that the polynomial degree in each dimension is equal to that integer.

center : iterable, None, optional

An iterable of length equal to the number of dimensions in image_shape that indicates the center of the coordinate system in image coordinates when center_cs is 'image' otherwise center is assumed to be in world coordinates (when center_cs is 'world'). When center is None then center is set to the middle of the “image” as center[i]=image_shape[i]//2. If image2world is not None and center_cs is 'image', then supplied center will be converted to world coordinates.

image2world : function, None, optional

Image-to-world coordinates transformation function. This function must be of the form f(x,y,z,...) and accept a number of arguments numpy.ndarray arguments equal to the dimensionality of images.

center_cs : {‘image’, ‘world’}, optional

Indicates whether center is in image coordinates or in world coordinates. This parameter is ignored when center is set to None: it is assumed to be False. center_cs cannot be 'world' when image2world is None unless center is None.

ext_return : bool, optional

Indicates whether this function should return additional values besides optimal polynomial coefficients (see bkg_poly_coeff return value below) that match image intensities in the LSQ sense. See Returns section for more details.

solver : {‘RLU’, ‘PINV’}, optional

Specifies method for solving the system of equations.
Returns:
bkg_poly_coeff : numpy.ndarray

When nimages is None, this function returns a 1D numpy.ndarray that holds the solution (polynomial coefficients) to the system.

When nimages is not None, this function returns a 2D numpy.ndarray that holds the solution (polynomial coefficients) to the system. The solution is grouped by image.

a : numpy.ndarray

A 2D numpy.ndarray that holds the coefficients of the linear system of equations. This value is returned only when ext_return is True.

b : numpy.ndarray

A 1D numpy.ndarray that holds the free terms of the linear system of equations. This value is returned only when ext_return is True.

coord_arrays : list

A list of numpy.ndarray coordinate arrays each of image_shape shape. This value is returned only when ext_return is True.

eff_center : tuple

A tuple of coordinates of the effective center as used in generating coordinate arrays. This value is returned only when ext_return is True.

coord_system : {‘image’, ‘world’}

Coordinate system of the coordinate arrays and returned center value. This value is returned only when ext_return is True.

Notes

match_lsq() builds a system of linear equations

\[a \cdot c = b\]

whose solution \(c\) is a set of coefficients of (multivariate) polynomials that represent the “background” in each input image (these are polynomials that are “corrections” to intensities of input images) such that the following sum is minimized:

\[L = \sum^N_{n,m=1,n \neq m} \sum_k \frac{\left[I_n(k) - I_m(k) - P_n(k) + P_m(k)\right]^2} {\sigma^2_n(k) + \sigma^2_m(k)}.\]

In the above equation, index \(k=(k_1,k_2,...)\) labels a position in input image’s pixel grid [NOTE: all input images share a common pixel grid].

“Background” polynomials \(P_n(k)\) are defined through the corresponding coefficients as:

\[P_n(k_1,k_2,...) = \sum_{d_1=0,d_2=0,...}^{D_1,D_2,...} c_{d_1,d_2,...}^n \cdot k_1^{d_1} \cdot k_2^{d_2} \cdot \ldots .\]

Coefficients \(c_{d_1,d_2,...}^n\) are arranged in the vector \(c\) in the following order:

\[(c_{0,0,\ldots}^1,c_{1,0,\ldots}^1,\ldots,c_{0,0,\ldots}^2, c_{1,0,\ldots}^2,\ldots).\]

match_lsq() returns coefficients of the polynomials that minimize L.

Examples

>>> import wiimatch
>>> import numpy as np
>>> im1 = np.zeros((5, 5, 4), dtype=np.float)
>>> cbg = 1.32 * np.ones_like(im1)
>>> ind = np.indices(im1.shape, dtype=np.float)
>>> im3 = cbg + 0.15 * ind[0] + 0.62 * ind[1] + 0.74 * ind[2]
>>> mask = np.ones_like(im1, dtype=np.int8)
>>> sigma = np.ones_like(im1, dtype=np.float)
>>> wiimatch.match.match_lsq([im1, im3], [mask, mask], [sigma, sigma],
... degree=(1,1,1), center=(0,0,0))   # doctest: +FLOAT_CMP
array([[-6.60000000e-01, -7.50000000e-02, -3.10000000e-01,
        -6.96331881e-16, -3.70000000e-01, -1.02318154e-15,
        -5.96855898e-16,  2.98427949e-16],
       [ 6.60000000e-01,  7.50000000e-02,  3.10000000e-01,
         6.96331881e-16,  3.70000000e-01,  1.02318154e-15,
         5.96855898e-16, -2.98427949e-16]])

LSQ Equation Construction and Solving

A module that provides core algorithm for optimal matching of backgrounds of N-dimensional images using (multi-variate) polynomials.

Author:Mihai Cara (contact: help@stsci.edu)
License:LICENSE
wiimatch.lsq_optimizer.build_lsq_eqs(images, masks, sigmas, degree, center=None, image2world=None, center_cs='image')[source]

Build system of linear equations whose solution would provide image intensity matching in the least squares sense.

Parameters:
images : list of numpy.ndarray

A list of 1D, 2D, etc. numpy.ndarray data array whose “intensities” must be “matched”. All arrays must have identical shapes.

masks : list of numpy.ndarray

A list of numpy.ndarray arrays of same length as images. Non-zero mask elements indicate valid data in the corresponding images array. Mask arrays must have identical shape to that of the arrays in input images.

sigmas : list of numpy.ndarray

A list of numpy.ndarray data array of same length as images representing the uncertainties of the data in the corresponding array in images. Uncertainty arrays must have identical shape to that of the arrays in input images.

degree : iterable

A list of polynomial degrees for each dimension of data arrays in images. The length of the input list must match the dimensionality of the input images.

center : iterable, None, optional

An iterable of length equal to the number of dimensions of images in images parameter that indicates the center of the coordinate system in image coordinates when center_cs is 'image' otherwise center is assumed to be in world coordinates (when center_cs is 'world'). When center is None then center is set to the middle of the “image” as center[i]=image.shape[i]//2. If image2world is not None and center_cs is 'image', then supplied center will be converted to world coordinates.

image2world : function, None, optional

Image-to-world coordinates transformation function. This function must be of the form f(x,y,z,...) and accept a number of arguments numpy.ndarray arguments equal to the dimensionality of images.

center_cs : {‘image’, ‘world’}, optional

Indicates whether center is in image coordinates or in world coordinates. This parameter is ignored when center is set to None: it is assumed to be False. center_cs cannot be 'world' when image2world is None unless center is None.
Returns:
a : numpy.ndarray

A 2D numpy.ndarray that holds the coefficients of the linear system of equations.

b : numpy.ndarray

A 1D numpy.ndarray that holds the free terms of the linear system of equations.

coord_arrays : list

A list of numpy.ndarray coordinate arrays each of images[0].shape shape.

eff_center : tuple

A tuple of coordinates of the effective center as used in generating coordinate arrays.

coord_system : {‘image’, ‘world’}

Coordinate system of the coordinate arrays and returned center value.

Notes

build_lsq_eqs() builds a system of linear equations

\[a \cdot c = b\]

whose solution \(c\) is a set of coefficients of (multivariate) polynomials that represent the “background” in each input image (these are polynomials that are “corrections” to intensities of input images) such that the following sum is minimized:

\[L = \sum^N_{n,m=1,n \neq m} \sum_k \frac{\left[I_n(k) - I_m(k) - P_n(k) + P_m(k)\right]^2} {\sigma^2_n(k) + \sigma^2_m(k)}.\]

In the above equation, index \(k=(k_1,k_2,...)\) labels a position in input image’s pixel grid [NOTE: all input images share a common pixel grid].

“Background” polynomials \(P_n(k)\) are defined through the corresponding coefficients as:

\[P_n(k_1,k_2,...) = \sum_{d_1=0,d_2=0,...}^{D_1,D_2,...} c_{d_1,d_2,...}^n \cdot k_1^{d_1} \cdot k_2^{d_2} \cdot \ldots .\]

Coefficients \(c_{d_1,d_2,...}^n\) are arranged in the vector \(c\) in the following order:

\[(c_{0,0,\ldots}^1,c_{1,0,\ldots}^1,\ldots,c_{0,0,\ldots}^2, c_{1,0,\ldots}^2,\ldots).\]

Examples

>>> import wiimatch
>>> import numpy as np
>>> im1 = np.zeros((5, 5, 4), dtype=np.float)
>>> cbg = 1.32 * np.ones_like(im1)
>>> ind = np.indices(im1.shape, dtype=np.float)
>>> im3 = cbg + 0.15 * ind[0] + 0.62 * ind[1] + 0.74 * ind[2]
>>> mask = np.ones_like(im1, dtype=np.int8)
>>> sigma = np.ones_like(im1, dtype=np.float)
>>> a, b, ca, ef, cs = wiimatch.lsq_optimizer.build_lsq_eqs([im1, im3],
... [mask, mask], [sigma, sigma], degree=(1, 1, 1), center=(0, 0, 0))
>>> print(a)
[[   50.   100.   100.   200.    75.   150.   150.   300.   -50.  -100.
   -100.  -200.   -75.  -150.  -150.  -300.]
 [  100.   300.   200.   600.   150.   450.   300.   900.  -100.  -300.
   -200.  -600.  -150.  -450.  -300.  -900.]
 [  100.   200.   300.   600.   150.   300.   450.   900.  -100.  -200.
   -300.  -600.  -150.  -300.  -450.  -900.]
 [  200.   600.   600.  1800.   300.   900.   900.  2700.  -200.  -600.
   -600. -1800.  -300.  -900.  -900. -2700.]
 [   75.   150.   150.   300.   175.   350.   350.   700.   -75.  -150.
   -150.  -300.  -175.  -350.  -350.  -700.]
 [  150.   450.   300.   900.   350.  1050.   700.  2100.  -150.  -450.
   -300.  -900.  -350. -1050.  -700. -2100.]
 [  150.   300.   450.   900.   350.   700.  1050.  2100.  -150.  -300.
   -450.  -900.  -350.  -700. -1050. -2100.]
 [  300.   900.   900.  2700.   700.  2100.  2100.  6300.  -300.  -900.
   -900. -2700.  -700. -2100. -2100. -6300.]
 [  -50.  -100.  -100.  -200.   -75.  -150.  -150.  -300.    50.   100.
    100.   200.    75.   150.   150.   300.]
 [ -100.  -300.  -200.  -600.  -150.  -450.  -300.  -900.   100.   300.
    200.   600.   150.   450.   300.   900.]
 [ -100.  -200.  -300.  -600.  -150.  -300.  -450.  -900.   100.   200.
    300.   600.   150.   300.   450.   900.]
 [ -200.  -600.  -600. -1800.  -300.  -900.  -900. -2700.   200.   600.
    600.  1800.   300.   900.   900.  2700.]
 [  -75.  -150.  -150.  -300.  -175.  -350.  -350.  -700.    75.   150.
    150.   300.   175.   350.   350.   700.]
 [ -150.  -450.  -300.  -900.  -350. -1050.  -700. -2100.   150.   450.
    300.   900.   350.  1050.   700.  2100.]
 [ -150.  -300.  -450.  -900.  -350.  -700. -1050. -2100.   150.   300.
    450.   900.   350.   700.  1050.  2100.]
 [ -300.  -900.  -900. -2700.  -700. -2100. -2100. -6300.   300.   900.
    900.  2700.   700.  2100.  2100.  6300.]]
>>> print(b)
[ -198.5  -412.   -459.   -948.   -344.   -710.5  -781.  -1607.    198.5
   412.    459.    948.    344.    710.5   781.   1607. ]
wiimatch.lsq_optimizer.pinv_solve(matrix, free_term, nimages, tol=None)[source]

Solves a system of linear equations

\[a \cdot c = b.\]

using Moore-Penrose pseudoinverse.

Parameters:
matrix : numpy.ndarray

A 2D array containing coefficients of the system.

free_term : numpy.ndarray

A 1D array containing free terms of the system of the equations.

nimages : int

Number of images for which the system is being solved.

tol : float, None, optional

Cutoff for small singular values for Moore-Penrose pseudoinverse. When provided, singular values smaller (in modulus) than tol * |largest_singular_value| are set to zero. When tol is None (default), cutoff value is determined based on the type of the input matrix argument.
Returns:
bkg_poly_coeff : numpy.ndarray

A 2D numpy.ndarray that holds the solution (polynomial coefficients) to the system. The solution is grouped by image.

Examples

>>> import wiimatch
>>> import numpy as np
>>> im1 = np.zeros((5, 5, 4), dtype=np.float)
>>> cbg = 1.32 * np.ones_like(im1)
>>> ind = np.indices(im1.shape, dtype=np.float)
>>> im3 = cbg + 0.15 * ind[0] + 0.62 * ind[1] + 0.74 * ind[2]
>>> mask = np.ones_like(im1, dtype=np.int8)
>>> sigma = np.ones_like(im1, dtype=np.float)
>>> a, b, _, _, _ = wiimatch.lsq_optimizer.build_lsq_eqs([im1, im3],
... [mask, mask], [sigma, sigma], degree=(1, 1, 1), center=(0, 0, 0))
>>> wiimatch.lsq_optimizer.pinv_solve(a, b, 2) # doctest: +FLOAT_CMP
array([[-6.60000000e-01, -7.50000000e-02, -3.10000000e-01,
        -4.44089210e-15, -3.70000000e-01, -7.66053887e-15,
         3.69704267e-14,  8.37108161e-14],
       [ 6.60000000e-01,  7.50000000e-02,  3.10000000e-01,
         3.55271368e-15,  3.70000000e-01,  4.32986980e-15,
         4.88498131e-14,  7.87148124e-14]])
wiimatch.lsq_optimizer.rlu_solve(matrix, free_term, nimages)[source]

Computes solution of a “reduced” system of linear equations

\[a' \cdot c' = b'.\]

using LU-decomposition. If the original system contained a set of linearly-dependent equations, then the “reduced” system is formed by dropping equations and unknowns related to the first image. The unknowns corresponding to the first image initially are assumed to be 0. Upon solving the reduced system, these unknowns are recomputed so that mean corection coefficients for all images are 0. This function uses lu_solve and lu_factor functions.

Parameters:
matrix : numpy.ndarray

A 2D array containing coefficients of the system.

free_term : numpy.ndarray

A 1D array containing free terms of the system of the equations.

nimages : int

Number of images for which the system is being solved.
Returns:
bkg_poly_coeff : numpy.ndarray

A 2D numpy.ndarray that holds the solution (polynomial coefficients) to the system. The solution is grouped by image.

Examples

>>> import wiimatch
>>> import numpy as np
>>> im1 = np.zeros((5, 5, 4), dtype=np.float)
>>> cbg = 1.32 * np.ones_like(im1)
>>> ind = np.indices(im1.shape, dtype=np.float)
>>> im3 = cbg + 0.15 * ind[0] + 0.62 * ind[1] + 0.74 * ind[2]
>>> mask = np.ones_like(im1, dtype=np.int8)
>>> sigma = np.ones_like(im1, dtype=np.float)
>>> a, b, _, _, _ = wiimatch.lsq_optimizer.build_lsq_eqs([im1, im3],
... [mask, mask], [sigma, sigma], degree=(1, 1, 1), center=(0, 0, 0))
>>> wiimatch.lsq_optimizer.rlu_solve(a, b, 2)   # doctest: +FLOAT_CMP
array([[-6.60000000e-01, -7.50000000e-02, -3.10000000e-01,
        -6.96331881e-16, -3.70000000e-01, -1.02318154e-15,
        -5.96855898e-16,  2.98427949e-16],
       [ 6.60000000e-01,  7.50000000e-02,  3.10000000e-01,
         6.96331881e-16,  3.70000000e-01,  1.02318154e-15,
         5.96855898e-16, -2.98427949e-16]])

Utilities used by wiimatch

This module provides utility functions for use by wiimatch module.

Author:Mihai Cara (contact: help@stsci.edu)
License:LICENSE
wiimatch.utils.create_coordinate_arrays(image_shape, center=None, image2world=None, center_cs='image')[source]

Create a list of coordinate arrays/grids for each dimension in the image shape. This function is similar to numpy.indices except it returns the list of arrays in reversed order. In addition, it can center image coordinates to a provided center and also convert image coordinates to world coordinates using provided image2world function.

Parameters:
image_shape : sequence of int

The shape of the image/grid.

center : iterable, None, optional

An iterable of length equal to the number of dimensions in image_shape that indicates the center of the coordinate system in image coordinates when center_cs is 'image' otherwise center is assumed to be in world coordinates (when center_cs is 'world'). When center is None then center is set to the middle of the “image” as center[i]=image_shape[i]//2. If image2world is not None and center_cs is 'image', then supplied center will be converted to world coordinates.

image2world : function, None, optional

Image-to-world coordinates transformation function. This function must be of the form f(x,y,z,...) and accept a number of arguments numpy.ndarray arguments equal to the dimensionality of images.

center_cs : {‘image’, ‘world’}, optional

Indicates whether center is in image coordinates or in world coordinates. This parameter is ignored when center is set to None: it is assumed to be False. center_cs cannot be 'world' when image2world is None unless center is None.
Returns:
coord_arrays : list

A list of numpy.ndarray coordinate arrays each of image_shape shape.

eff_center : tuple

A tuple of coordinates of the effective center as used in generating coordinate arrays.

coord_system : {‘image’, ‘world’}

Coordinate system of the coordinate arrays and returned center value.

Examples

>>> import wiimatch
>>> wiimatch.utils.create_coordinate_arrays((3, 5, 4))   # doctest: +FLOAT_CMP
((array([[[-1.,  0.,  1.,  2.],
        [-1.,  0.,  1.,  2.],
        [-1.,  0.,  1.,  2.],
        [-1.,  0.,  1.,  2.],
        [-1.,  0.,  1.,  2.]],
       [[-1.,  0.,  1.,  2.],
        [-1.,  0.,  1.,  2.],
        [-1.,  0.,  1.,  2.],
        [-1.,  0.,  1.,  2.],
        [-1.,  0.,  1.,  2.]],
       [[-1.,  0.,  1.,  2.],
        [-1.,  0.,  1.,  2.],
        [-1.,  0.,  1.,  2.],
        [-1.,  0.,  1.,  2.],
        [-1.,  0.,  1.,  2.]]]),
array([[[-2., -2., -2., -2.],
        [-1., -1., -1., -1.],
        [ 0.,  0.,  0.,  0.],
        [ 1.,  1.,  1.,  1.],
        [ 2.,  2.,  2.,  2.]],
       [[-2., -2., -2., -2.],
        [-1., -1., -1., -1.],
        [ 0.,  0.,  0.,  0.],
        [ 1.,  1.,  1.,  1.],
        [ 2.,  2.,  2.,  2.]],
       [[-2., -2., -2., -2.],
        [-1., -1., -1., -1.],
        [ 0.,  0.,  0.,  0.],
        [ 1.,  1.,  1.,  1.],
        [ 2.,  2.,  2.,  2.]]]),
array([[[-2., -2., -2., -2.],
        [-2., -2., -2., -2.],
        [-2., -2., -2., -2.],
        [-2., -2., -2., -2.],
        [-2., -2., -2., -2.]],
       [[-1., -1., -1., -1.],
        [-1., -1., -1., -1.],
        [-1., -1., -1., -1.],
        [-1., -1., -1., -1.],
        [-1., -1., -1., -1.]],
       [[ 0.,  0.,  0.,  0.],
        [ 0.,  0.,  0.,  0.],
        [ 0.,  0.,  0.,  0.],
        [ 0.,  0.,  0.,  0.],
        [ 0.,  0.,  0.,  0.]]])), (1.0, 2.0, 2.0), 'image')

LICENSE

Copyright (C) 2019, Association of Universities for Research in Astronomy

Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met:

  1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.
  2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution.
  3. Neither the name of the copyright holder nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission.

THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS “AS IS” AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.

Development Notes

Release Notes

0.2.0 (07-August-2019)

Added

  • Added a new, more stable, solver rlu_solve(). match_lsq() now takes a new parameter solver which, by default, is set to 'LU' - the new solver. [#1]

Fixed

  • Updated package structure, setup, docs. [#1]

0.1.2 (12-June-2017)

Added

  • Several functions now return more values that can be used to analyse returned results:
    • wiimatch.utils.create_coordinate_arrays() now returns effective center values used in generating coordinate array and coordinate system type ('image' or 'world');
    • wiimatch.lsq_optimizer.build_lsq_eqs() now returns coordinate arrays, effective center values used in generating coordinate array, and the coordinate system type of coordinates in addition to coefficients of linear equations;
    • wiimatch.match.match_lsq() now optionally returns coefficients of linear equations, coordinate arrays, effective center values used in generating coordinate array, and the coordinate system type of coordinates in addition to optimal solution to the matching problem. New parameter ext_return indicates to return extended information.

0.1.1 (06-June-2017)

Added

Fixed

0.1.0 (09-May-2017)

Initial release.

Indices and tables