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author | Sebastien Jodogne <s.jodogne@gmail.com> |
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date | Thu, 24 Jun 2021 14:19:36 +0200 |
parents | 9ac2a65d4172 |
children | 3889ae96d2e9 |
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#!/usr/bin/python # Stone of Orthanc # Copyright (C) 2012-2016 Sebastien Jodogne, Medical Physics # Department, University Hospital of Liege, Belgium # Copyright (C) 2017-2021 Osimis S.A., Belgium # # This program is free software: you can redistribute it and/or # modify it under the terms of the GNU Lesser General Public License # as published by the Free Software Foundation, either version 3 of # the License, or (at your option) any later version. # # This program is distributed in the hope that it will be useful, but # WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU # Lesser General Public License for more details. # # You should have received a copy of the GNU Lesser General Public # License along with this program. If not, see # <http://www.gnu.org/licenses/>. from sympy import * from sympy.solvers import solve import pprint import sys init_printing(use_unicode=True) # Create a test 3D vector using homogeneous coordinates x, y, z, w = symbols('x y z w') p = Matrix([ x, y, z, w ]) # Create a shear matrix, and a scale/shift "T * S" transform as in # Lacroute's thesis (Equation A.16, page 209) ex, ey, ew = symbols('ex ey ew') sx, sy, tx, ty = symbols('sx sy tx ty') TS = Matrix([[ sx, 0, 0, tx ], [ 0, sy, 0, ty ], [ 0, 0, 1, 0 ], [ 0, 0, 0, 1 ]]) pureShear = Matrix([[ 1, 0, ex, 0 ], [ 0, 1, ey, 0 ], [ 0, 0, 1, 0 ], [ 0, 0, ew, 1 ]]) # Create a general warp matrix, that corresponds to "M_warp" in # Equation (A.17) of Lacroute's thesis: ww11, ww12, ww13, ww14, ww21, ww22, ww23, ww24, ww31, ww32, ww33, ww34, ww41, ww42, ww43, ww44 = symbols('ww11 ww12 ww13 ww14 ww21 ww22 ww23 ww24 ww31 ww32 ww33 ww34 ww41 ww42 ww43 ww44') WW = Matrix([[ ww11, ww12, ww13, ww14 ], [ ww21, ww22, ww23, ww24 ], [ ww31, ww32, ww33, ww34 ], [ ww41, ww43, ww43, ww44 ]]) # Create the matrix of intrinsic parameters of the camera k11, k22, k14, k24 = symbols('k11 k22 k14 k24') K = Matrix([[ k11, 0, 0, k14 ], [ 0, k22, 0, k24 ], [ 0, 0, 0, 1 ]]) # The full decomposition is: M_shear = TS * pureShear M_warp = K * WW * TS.inv() AA = M_warp * M_shear # Check that the central component "M_warp == K * WW * TS.inv()" that # is the left part of "A" is another general warp matrix (i.e. no # exception is thrown about incompatible matrix sizes): M_warp * p if (M_warp.cols != 4 or M_warp.rows != 3): raise Exception('Invalid matrix size') # We've just shown that "M_warp" is a general 3x4 matrix. Let's call # it W: w11, w12, w13, w14, w21, w22, w23, w24, w41, w42, w43, w44 = symbols('w11 w12 w13 w14 w21 w22 w23 w24 w41 w42 w43 w44') W = Matrix([[ w11, w12, w13, w14 ], [ w21, w22, w23, w24 ], [ w41, w43, w43, w44 ]]) # This shows that it is sufficient to study a decomposition of the # following form: A = W * M_shear print('\nA = W * M_shear =') pprint.pprint(A) sys.stdout.write('\nW = ') pprint.pprint(W) sys.stdout.write('\nM_shear = ') pprint.pprint(M_shear) # Let's consider one fixed 2D point (i,j) in the intermediate # image. The 3D points (x,y,z,1) that are mapped to (i,j) must satisfy # the equation "(i,j) == M_shear * (x,y,z,w)". As "M_shear" is # invertible, we solve "(x,y,z,w) == inv(M_shear) * (i,j,k,1)". i, j, k = symbols('i j k') l = M_shear.inv() * Matrix([ i, j, k, 1 ]) print('\nLocus for points imaged to some fixed (i,j,k,l) point in the intermediate image:') print('x = %s' % l[0]) print('y = %s' % l[1]) print('z = %s' % l[2]) print('w = %s' % l[3]) # By inspecting the 4 equations above, we see that the locus entirely # depends upon the "k" value that encodes the Z-axis print('\nGlobal effect of the shear-warp transform on this locus:') q = expand(A * l) pprint.pprint(q) print("\nWe can arbitrarily fix the value of 'k', so let's choose 'k=0':") pprint.pprint(q.subs(k, 0)) print("\nThis gives the warp transform.") print("QED: line after Equation (A.17) on page 209.\n")