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)

jwst.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

\[\begin{split}a \\cdot c = b\end{split}\]

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:

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

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:

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

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

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

Examples

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

Return type

numpy.ndarray

Examples

>>> from jwst.wiimatch.lsq_optimizer import build_lsq_eqs, pinv_solve
>>> 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, _, _, _ = build_lsq_eqs([im1, im3], [mask, mask],
... [sigma, sigma], degree=(1,1,1), center=(0,0,0))
>>> pinv_solve(a, b, 2) 
array([[-6.60000000e-01, -7.50000000e-02, -3.10000000e-01,
         7.10542736e-15, -3.70000000e-01,  8.88178420e-15,
         9.21485110e-15, -2.77555756e-15],
       [ 6.60000000e-01,  7.50000000e-02,  3.10000000e-01,
        -6.43929354e-15,  3.70000000e-01, -7.77156117e-15,
        -9.32587341e-15,  2.99760217e-15]])
jwst.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 – A 2D numpy.ndarray that holds the solution (polynomial coefficients) to the system. The solution is grouped by image.

Return type

numpy.ndarray

Examples

>>> from jwst.wiimatch.lsq_optimizer import build_lsq_eqs, rlu_solve
>>> 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, _, _, _ = build_lsq_eqs([im1, im3], [mask, mask],
... [sigma, sigma], degree=(1, 1, 1), center=(0, 0, 0))
>>> rlu_solve(a, b, 2)   
array([[-6.60000000e-01, -7.50000000e-02, -3.10000000e-01,
        -1.19371180e-15, -3.70000000e-01, -1.62003744e-15,
        -1.10844667e-15,  5.11590770e-16],
       [ 6.60000000e-01,  7.50000000e-02,  3.10000000e-01,
         1.19371180e-15,  3.70000000e-01,  1.62003744e-15,
         1.10844667e-15, -5.11590770e-16]])