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1252 lines
40 KiB
1252 lines
40 KiB
/////////////////////////////////////////////////////////////////////////// |
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// |
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// Copyright (c) 2002, Industrial Light & Magic, a division of Lucas |
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// Digital Ltd. LLC |
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// |
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// All rights reserved. |
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// |
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// Redistribution and use in source and binary forms, with or without |
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// modification, are permitted provided that the following conditions are |
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// met: |
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// * Redistributions of source code must retain the above copyright |
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// notice, this list of conditions and the following disclaimer. |
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// * Redistributions in binary form must reproduce the above |
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// copyright notice, this list of conditions and the following disclaimer |
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// in the documentation and/or other materials provided with the |
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// distribution. |
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// * Neither the name of Industrial Light & Magic nor the names of |
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// its contributors may be used to endorse or promote products derived |
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// from this software without specific prior written permission. |
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// |
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// THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS |
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// "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT |
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// LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR |
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// A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT |
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// OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, |
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// SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT |
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// LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, |
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// DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY |
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// THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT |
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// (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE |
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// OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. |
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// |
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/////////////////////////////////////////////////////////////////////////// |
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//---------------------------------------------------------------------------- |
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// |
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// Implementation of non-template items declared in ImathMatrixAlgo.h |
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// |
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//---------------------------------------------------------------------------- |
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#include "ImathMatrixAlgo.h" |
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#include <cmath> |
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#include <algorithm> // for std::max() |
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#if defined(OPENEXR_DLL) |
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#define EXPORT_CONST __declspec(dllexport) |
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#else |
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#define EXPORT_CONST const |
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#endif |
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namespace Imath { |
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EXPORT_CONST M33f identity33f ( 1, 0, 0, |
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0, 1, 0, |
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0, 0, 1); |
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EXPORT_CONST M33d identity33d ( 1, 0, 0, |
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0, 1, 0, |
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0, 0, 1); |
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EXPORT_CONST M44f identity44f ( 1, 0, 0, 0, |
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0, 1, 0, 0, |
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0, 0, 1, 0, |
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0, 0, 0, 1); |
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EXPORT_CONST M44d identity44d ( 1, 0, 0, 0, |
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0, 1, 0, 0, |
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0, 0, 1, 0, |
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0, 0, 0, 1); |
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namespace |
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{ |
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class KahanSum |
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{ |
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public: |
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KahanSum() : _total(0), _correction(0) {} |
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void |
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operator+= (const double val) |
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{ |
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const double y = val - _correction; |
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const double t = _total + y; |
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_correction = (t - _total) - y; |
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_total = t; |
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} |
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double get() const |
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{ |
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return _total; |
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} |
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private: |
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double _total; |
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double _correction; |
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}; |
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} |
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template <typename T> |
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M44d |
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procrustesRotationAndTranslation (const Vec3<T>* A, const Vec3<T>* B, const T* weights, const size_t numPoints, const bool doScale) |
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{ |
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if (numPoints == 0) |
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return M44d(); |
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// Always do the accumulation in double precision: |
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V3d Acenter (0.0); |
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V3d Bcenter (0.0); |
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double weightsSum = 0.0; |
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if (weights == 0) |
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{ |
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for (int i = 0; i < numPoints; ++i) |
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{ |
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Acenter += (V3d) A[i]; |
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Bcenter += (V3d) B[i]; |
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} |
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weightsSum = (double) numPoints; |
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} |
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else |
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{ |
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for (int i = 0; i < numPoints; ++i) |
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{ |
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const double w = weights[i]; |
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weightsSum += w; |
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Acenter += w * (V3d) A[i]; |
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Bcenter += w * (V3d) B[i]; |
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} |
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} |
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if (weightsSum == 0) |
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return M44d(); |
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Acenter /= weightsSum; |
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Bcenter /= weightsSum; |
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// |
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// Find Q such that |Q*A - B| (actually A-Acenter and B-Bcenter, weighted) |
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// is minimized in the least squares sense. |
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// From Golub/Van Loan, p.601 |
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// |
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// A,B are 3xn |
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// Let C = B A^T (where A is 3xn and B^T is nx3, so C is 3x3) |
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// Compute the SVD: C = U D V^T (U,V rotations, D diagonal). |
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// Throw away the D part, and return Q = U V^T |
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M33d C (0.0); |
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if (weights == 0) |
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{ |
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for (int i = 0; i < numPoints; ++i) |
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C += outerProduct ((V3d) B[i] - Bcenter, (V3d) A[i] - Acenter); |
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} |
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else |
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{ |
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for (int i = 0; i < numPoints; ++i) |
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{ |
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const double w = weights[i]; |
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C += outerProduct (w * ((V3d) B[i] - Bcenter), (V3d) A[i] - Acenter); |
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} |
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} |
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M33d U, V; |
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V3d S; |
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jacobiSVD (C, U, S, V, Imath::limits<double>::epsilon(), true); |
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// We want Q.transposed() here since we are going to be using it in the |
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// Imath style (multiplying vectors on the right, v' = v*A^T): |
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const M33d Qt = V * U.transposed(); |
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double s = 1.0; |
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if (doScale && numPoints > 1) |
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{ |
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// Finding a uniform scale: let us assume the Q is completely fixed |
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// at this point (solving for both simultaneously seems much harder). |
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// We are trying to compute (again, per Golub and van Loan) |
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// min || s*A*Q - B ||_F |
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// Notice that we've jammed a uniform scale in front of the Q. |
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// Now, the Frobenius norm (the least squares norm over matrices) |
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// has the neat property that it is equivalent to minimizing the trace |
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// of M^T*M (see your friendly neighborhood linear algebra text for a |
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// derivation). Thus, we can expand this out as |
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// min tr (s*A*Q - B)^T*(s*A*Q - B) |
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// = min tr(Q^T*A^T*s*s*A*Q) + tr(B^T*B) - 2*tr(Q^T*A^T*s*B) by linearity of the trace |
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// = min s^2 tr(A^T*A) + tr(B^T*B) - 2*s*tr(Q^T*A^T*B) using the fact that the trace is invariant |
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// under similarity transforms Q*M*Q^T |
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// If we differentiate w.r.t. s and set this to 0, we get |
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// 0 = 2*s*tr(A^T*A) - 2*tr(Q^T*A^T*B) |
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// so |
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// 2*s*tr(A^T*A) = 2*s*tr(Q^T*A^T*B) |
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// s = tr(Q^T*A^T*B) / tr(A^T*A) |
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KahanSum traceATA; |
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if (weights == 0) |
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{ |
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for (int i = 0; i < numPoints; ++i) |
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traceATA += ((V3d) A[i] - Acenter).length2(); |
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} |
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else |
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{ |
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for (int i = 0; i < numPoints; ++i) |
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traceATA += ((double) weights[i]) * ((V3d) A[i] - Acenter).length2(); |
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} |
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KahanSum traceBATQ; |
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for (int i = 0; i < 3; ++i) |
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for (int j = 0; j < 3; ++j) |
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traceBATQ += Qt[j][i] * C[i][j]; |
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s = traceBATQ.get() / traceATA.get(); |
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} |
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// Q is the rotation part of what we want to return. |
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// The entire transform is: |
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// (translate origin to Bcenter) * Q * (translate Acenter to origin) |
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// last first |
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// The effect of this on a point is: |
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// (translate origin to Bcenter) * Q * (translate Acenter to origin) * point |
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// = (translate origin to Bcenter) * Q * (-Acenter + point) |
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// = (translate origin to Bcenter) * (-Q*Acenter + Q*point) |
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// = (translate origin to Bcenter) * (translate Q*Acenter to origin) * Q*point |
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// = (translate Q*Acenter to Bcenter) * Q*point |
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// So what we want to return is: |
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// (translate Q*Acenter to Bcenter) * Q |
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// |
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// In block form, this is: |
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// [ 1 0 0 | ] [ 0 ] [ 1 0 0 | ] [ 1 0 0 | ] [ | ] [ ] |
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// [ 0 1 0 tb ] [ s*Q 0 ] [ 0 1 0 -ta ] = [ 0 1 0 tb ] [ s*Q -s*Q*ta ] = [ Q tb-s*Q*ta ] |
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// [ 0 0 1 | ] [ 0 ] [ 0 0 1 | ] [ 0 0 1 | ] [ | ] [ ] |
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// [ 0 0 0 1 ] [ 0 0 0 1 ] [ 0 0 0 1 ] [ 0 0 0 1 ] [ 0 0 0 1 ] [ 0 0 0 1 ] |
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// (ofc the whole thing is transposed for Imath). |
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const V3d translate = Bcenter - s*Acenter*Qt; |
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return M44d (s*Qt.x[0][0], s*Qt.x[0][1], s*Qt.x[0][2], T(0), |
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s*Qt.x[1][0], s*Qt.x[1][1], s*Qt.x[1][2], T(0), |
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s*Qt.x[2][0], s*Qt.x[2][1], s*Qt.x[2][2], T(0), |
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translate.x, translate.y, translate.z, T(1)); |
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} // procrustesRotationAndTranslation |
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template <typename T> |
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M44d |
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procrustesRotationAndTranslation (const Vec3<T>* A, const Vec3<T>* B, const size_t numPoints, const bool doScale) |
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{ |
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return procrustesRotationAndTranslation (A, B, (const T*) 0, numPoints, doScale); |
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} // procrustesRotationAndTranslation |
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template M44d procrustesRotationAndTranslation (const V3d* from, const V3d* to, const size_t numPoints, const bool doScale); |
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template M44d procrustesRotationAndTranslation (const V3f* from, const V3f* to, const size_t numPoints, const bool doScale); |
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template M44d procrustesRotationAndTranslation (const V3d* from, const V3d* to, const double* weights, const size_t numPoints, const bool doScale); |
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template M44d procrustesRotationAndTranslation (const V3f* from, const V3f* to, const float* weights, const size_t numPoints, const bool doScale); |
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namespace |
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{ |
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// Applies the 2x2 Jacobi rotation |
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// [ c s 0 ] [ 1 0 0 ] [ c 0 s ] |
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// [ -s c 0 ] or [ 0 c s ] or [ 0 1 0 ] |
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// [ 0 0 1 ] [ 0 -s c ] [ -s 0 c ] |
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// from the right; that is, computes |
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// J * A |
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// for the Jacobi rotation J and the matrix A. This is efficient because we |
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// only need to touch exactly the 2 columns that are affected, so we never |
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// need to explicitly construct the J matrix. |
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template <typename T, int j, int k> |
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void |
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jacobiRotateRight (Imath::Matrix33<T>& A, |
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const T c, |
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const T s) |
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{ |
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for (int i = 0; i < 3; ++i) |
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{ |
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const T tau1 = A[i][j]; |
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const T tau2 = A[i][k]; |
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A[i][j] = c * tau1 - s * tau2; |
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A[i][k] = s * tau1 + c * tau2; |
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} |
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} |
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template <typename T> |
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void |
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jacobiRotateRight (Imath::Matrix44<T>& A, |
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const int j, |
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const int k, |
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const T c, |
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const T s) |
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{ |
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for (int i = 0; i < 4; ++i) |
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{ |
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const T tau1 = A[i][j]; |
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const T tau2 = A[i][k]; |
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A[i][j] = c * tau1 - s * tau2; |
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A[i][k] = s * tau1 + c * tau2; |
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} |
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} |
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// This routine solves the 2x2 SVD: |
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// [ c1 s1 ] [ w x ] [ c2 s2 ] [ d1 0 ] |
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// [ ] [ ] [ ] = [ ] |
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// [ -s1 c1 ] [ y z ] [ -s2 c2 ] [ 0 d2 ] |
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// where |
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// [ w x ] |
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// A = [ ] |
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// [ y z ] |
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// is the subset of A consisting of the [j,k] entries, A([j k], [j k]) in |
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// Matlab parlance. The method is the 'USVD' algorithm described in the |
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// following paper: |
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// 'Computation of the Singular Value Decomposition using Mesh-Connected Processors' |
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// by Richard P. Brent, Franklin T. Luk, and Charles Van Loan |
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// It breaks the computation into two steps: the first symmetrizes the matrix, |
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// and the second diagonalizes the symmetric matrix. |
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template <typename T, int j, int k, int l> |
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bool |
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twoSidedJacobiRotation (Imath::Matrix33<T>& A, |
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Imath::Matrix33<T>& U, |
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Imath::Matrix33<T>& V, |
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const T tol) |
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{ |
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// Load everything into local variables to make things easier on the |
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// optimizer: |
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const T w = A[j][j]; |
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const T x = A[j][k]; |
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const T y = A[k][j]; |
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const T z = A[k][k]; |
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// We will keep track of whether we're actually performing any rotations, |
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// since if the matrix is already diagonal we'll end up with the identity |
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// as our Jacobi rotation and we can short-circuit. |
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bool changed = false; |
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// The first step is to symmetrize the 2x2 matrix, |
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// [ c s ]^T [ w x ] = [ p q ] |
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// [ -s c ] [ y z ] [ q r ] |
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T mu_1 = w + z; |
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T mu_2 = x - y; |
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T c, s; |
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if (std::abs(mu_2) <= tol*std::abs(mu_1)) // Already symmetric (to tolerance) |
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{ // Note that the <= is important here |
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c = T(1); // because we want to bypass the computation |
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s = T(0); // of rho if mu_1 = mu_2 = 0. |
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const T p = w; |
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const T r = z; |
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mu_1 = r - p; |
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mu_2 = x + y; |
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} |
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else |
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{ |
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const T rho = mu_1 / mu_2; |
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s = T(1) / std::sqrt (T(1) + rho*rho); // TODO is there a native inverse square root function? |
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if (rho < 0) |
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s = -s; |
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c = s * rho; |
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mu_1 = s * (x + y) + c * (z - w); // = r - p |
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mu_2 = T(2) * (c * x - s * z); // = 2*q |
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changed = true; |
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} |
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// The second stage diagonalizes, |
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// [ c2 s2 ]^T [ p q ] [ c2 s2 ] = [ d1 0 ] |
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// [ -s2 c2 ] [ q r ] [ -s2 c2 ] [ 0 d2 ] |
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T c_2, s_2; |
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if (std::abs(mu_2) <= tol*std::abs(mu_1)) |
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{ |
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c_2 = T(1); |
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s_2 = T(0); |
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} |
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else |
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{ |
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const T rho_2 = mu_1 / mu_2; |
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T t_2 = T(1) / (std::abs(rho_2) + std::sqrt(1 + rho_2*rho_2)); |
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if (rho_2 < 0) |
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t_2 = -t_2; |
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c_2 = T(1) / std::sqrt (T(1) + t_2*t_2); |
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s_2 = c_2 * t_2; |
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changed = true; |
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} |
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const T c_1 = c_2 * c - s_2 * s; |
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const T s_1 = s_2 * c + c_2 * s; |
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if (!changed) |
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{ |
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// We've decided that the off-diagonal entries are already small |
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// enough, so we'll set them to zero. This actually appears to result |
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// in smaller errors than leaving them be, possibly because it prevents |
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// us from trying to do extra rotations later that we don't need. |
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A[k][j] = 0; |
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A[j][k] = 0; |
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return false; |
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} |
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const T d_1 = c_1*(w*c_2 - x*s_2) - s_1*(y*c_2 - z*s_2); |
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const T d_2 = s_1*(w*s_2 + x*c_2) + c_1*(y*s_2 + z*c_2); |
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// For the entries we just zeroed out, we'll just set them to 0, since |
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// they should be 0 up to machine precision. |
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A[j][j] = d_1; |
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A[k][k] = d_2; |
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A[k][j] = 0; |
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A[j][k] = 0; |
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// Rotate the entries that _weren't_ involved in the 2x2 SVD: |
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{ |
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// Rotate on the left by |
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// [ c1 s1 0 ]^T [ c1 0 s1 ]^T [ 1 0 0 ]^T |
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// [ -s1 c1 0 ] or [ 0 1 0 ] or [ 0 c1 s1 ] |
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// [ 0 0 1 ] [ -s1 0 c1 ] [ 0 -s1 c1 ] |
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// This has the effect of adding the (weighted) ith and jth _rows_ to |
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// each other. |
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const T tau1 = A[j][l]; |
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const T tau2 = A[k][l]; |
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A[j][l] = c_1 * tau1 - s_1 * tau2; |
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A[k][l] = s_1 * tau1 + c_1 * tau2; |
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} |
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{ |
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// Rotate on the right by |
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// [ c2 s2 0 ] [ c2 0 s2 ] [ 1 0 0 ] |
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// [ -s2 c2 0 ] or [ 0 1 0 ] or [ 0 c2 s2 ] |
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// [ 0 0 1 ] [ -s2 0 c2 ] [ 0 -s2 c2 ] |
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// This has the effect of adding the (weighted) ith and jth _columns_ to |
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// each other. |
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const T tau1 = A[l][j]; |
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const T tau2 = A[l][k]; |
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A[l][j] = c_2 * tau1 - s_2 * tau2; |
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A[l][k] = s_2 * tau1 + c_2 * tau2; |
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} |
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// Now apply the rotations to U and V: |
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// Remember that we have |
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// R1^T * A * R2 = D |
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// This is in the 2x2 case, but after doing a bunch of these |
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// we will get something like this for the 3x3 case: |
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// ... R1b^T * R1a^T * A * R2a * R2b * ... = D |
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// ----------------- --------------- |
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// = U^T = V |
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// So, |
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// U = R1a * R1b * ... |
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// V = R2a * R2b * ... |
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jacobiRotateRight<T, j, k> (U, c_1, s_1); |
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jacobiRotateRight<T, j, k> (V, c_2, s_2); |
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return true; |
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} |
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template <typename T> |
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bool |
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twoSidedJacobiRotation (Imath::Matrix44<T>& A, |
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int j, |
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int k, |
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Imath::Matrix44<T>& U, |
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Imath::Matrix44<T>& V, |
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const T tol) |
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{ |
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// Load everything into local variables to make things easier on the |
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// optimizer: |
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const T w = A[j][j]; |
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const T x = A[j][k]; |
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const T y = A[k][j]; |
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const T z = A[k][k]; |
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|
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// We will keep track of whether we're actually performing any rotations, |
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// since if the matrix is already diagonal we'll end up with the identity |
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// as our Jacobi rotation and we can short-circuit. |
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bool changed = false; |
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|
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// The first step is to symmetrize the 2x2 matrix, |
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// [ c s ]^T [ w x ] = [ p q ] |
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// [ -s c ] [ y z ] [ q r ] |
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T mu_1 = w + z; |
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T mu_2 = x - y; |
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|
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T c, s; |
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if (std::abs(mu_2) <= tol*std::abs(mu_1)) // Already symmetric (to tolerance) |
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{ // Note that the <= is important here |
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c = T(1); // because we want to bypass the computation |
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s = T(0); // of rho if mu_1 = mu_2 = 0. |
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|
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const T p = w; |
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const T r = z; |
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mu_1 = r - p; |
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mu_2 = x + y; |
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} |
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else |
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{ |
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const T rho = mu_1 / mu_2; |
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s = T(1) / std::sqrt (T(1) + rho*rho); // TODO is there a native inverse square root function? |
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if (rho < 0) |
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s = -s; |
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c = s * rho; |
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|
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mu_1 = s * (x + y) + c * (z - w); // = r - p |
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mu_2 = T(2) * (c * x - s * z); // = 2*q |
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|
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changed = true; |
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} |
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|
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// The second stage diagonalizes, |
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// [ c2 s2 ]^T [ p q ] [ c2 s2 ] = [ d1 0 ] |
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// [ -s2 c2 ] [ q r ] [ -s2 c2 ] [ 0 d2 ] |
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T c_2, s_2; |
|
if (std::abs(mu_2) <= tol*std::abs(mu_1)) |
|
{ |
|
c_2 = T(1); |
|
s_2 = T(0); |
|
} |
|
else |
|
{ |
|
const T rho_2 = mu_1 / mu_2; |
|
T t_2 = T(1) / (std::abs(rho_2) + std::sqrt(1 + rho_2*rho_2)); |
|
if (rho_2 < 0) |
|
t_2 = -t_2; |
|
c_2 = T(1) / std::sqrt (T(1) + t_2*t_2); |
|
s_2 = c_2 * t_2; |
|
|
|
changed = true; |
|
} |
|
|
|
const T c_1 = c_2 * c - s_2 * s; |
|
const T s_1 = s_2 * c + c_2 * s; |
|
|
|
if (!changed) |
|
{ |
|
// We've decided that the off-diagonal entries are already small |
|
// enough, so we'll set them to zero. This actually appears to result |
|
// in smaller errors than leaving them be, possibly because it prevents |
|
// us from trying to do extra rotations later that we don't need. |
|
A[k][j] = 0; |
|
A[j][k] = 0; |
|
return false; |
|
} |
|
|
|
const T d_1 = c_1*(w*c_2 - x*s_2) - s_1*(y*c_2 - z*s_2); |
|
const T d_2 = s_1*(w*s_2 + x*c_2) + c_1*(y*s_2 + z*c_2); |
|
|
|
// For the entries we just zeroed out, we'll just set them to 0, since |
|
// they should be 0 up to machine precision. |
|
A[j][j] = d_1; |
|
A[k][k] = d_2; |
|
A[k][j] = 0; |
|
A[j][k] = 0; |
|
|
|
// Rotate the entries that _weren't_ involved in the 2x2 SVD: |
|
for (int l = 0; l < 4; ++l) |
|
{ |
|
if (l == j || l == k) |
|
continue; |
|
|
|
// Rotate on the left by |
|
// [ 1 ] |
|
// [ . ] |
|
// [ c2 s2 ] j |
|
// [ 1 ] |
|
// [ -s2 c2 ] k |
|
// [ . ] |
|
// [ 1 ] |
|
// j k |
|
// |
|
// This has the effect of adding the (weighted) ith and jth _rows_ to |
|
// each other. |
|
const T tau1 = A[j][l]; |
|
const T tau2 = A[k][l]; |
|
A[j][l] = c_1 * tau1 - s_1 * tau2; |
|
A[k][l] = s_1 * tau1 + c_1 * tau2; |
|
} |
|
|
|
for (int l = 0; l < 4; ++l) |
|
{ |
|
// We set the A[j/k][j/k] entries already |
|
if (l == j || l == k) |
|
continue; |
|
|
|
// Rotate on the right by |
|
// [ 1 ] |
|
// [ . ] |
|
// [ c2 s2 ] j |
|
// [ 1 ] |
|
// [ -s2 c2 ] k |
|
// [ . ] |
|
// [ 1 ] |
|
// j k |
|
// |
|
// This has the effect of adding the (weighted) ith and jth _columns_ to |
|
// each other. |
|
const T tau1 = A[l][j]; |
|
const T tau2 = A[l][k]; |
|
A[l][j] = c_2 * tau1 - s_2 * tau2; |
|
A[l][k] = s_2 * tau1 + c_2 * tau2; |
|
} |
|
|
|
// Now apply the rotations to U and V: |
|
// Remember that we have |
|
// R1^T * A * R2 = D |
|
// This is in the 2x2 case, but after doing a bunch of these |
|
// we will get something like this for the 3x3 case: |
|
// ... R1b^T * R1a^T * A * R2a * R2b * ... = D |
|
// ----------------- --------------- |
|
// = U^T = V |
|
// So, |
|
// U = R1a * R1b * ... |
|
// V = R2a * R2b * ... |
|
jacobiRotateRight (U, j, k, c_1, s_1); |
|
jacobiRotateRight (V, j, k, c_2, s_2); |
|
|
|
return true; |
|
} |
|
|
|
template <typename T> |
|
void |
|
swapColumns (Imath::Matrix33<T>& A, int j, int k) |
|
{ |
|
for (int i = 0; i < 3; ++i) |
|
std::swap (A[i][j], A[i][k]); |
|
} |
|
|
|
template <typename T> |
|
T |
|
maxOffDiag (const Imath::Matrix33<T>& A) |
|
{ |
|
T result = 0; |
|
result = std::max (result, std::abs (A[0][1])); |
|
result = std::max (result, std::abs (A[0][2])); |
|
result = std::max (result, std::abs (A[1][0])); |
|
result = std::max (result, std::abs (A[1][2])); |
|
result = std::max (result, std::abs (A[2][0])); |
|
result = std::max (result, std::abs (A[2][1])); |
|
return result; |
|
} |
|
|
|
template <typename T> |
|
T |
|
maxOffDiag (const Imath::Matrix44<T>& A) |
|
{ |
|
T result = 0; |
|
for (int i = 0; i < 4; ++i) |
|
{ |
|
for (int j = 0; j < 4; ++j) |
|
{ |
|
if (i != j) |
|
result = std::max (result, std::abs (A[i][j])); |
|
} |
|
} |
|
|
|
return result; |
|
} |
|
|
|
template <typename T> |
|
void |
|
twoSidedJacobiSVD (Imath::Matrix33<T> A, |
|
Imath::Matrix33<T>& U, |
|
Imath::Vec3<T>& S, |
|
Imath::Matrix33<T>& V, |
|
const T tol, |
|
const bool forcePositiveDeterminant) |
|
{ |
|
// The two-sided Jacobi SVD works by repeatedly zeroing out |
|
// off-diagonal entries of the matrix, 2 at a time. Basically, |
|
// we can take our 3x3 matrix, |
|
// [* * *] |
|
// [* * *] |
|
// [* * *] |
|
// and use a pair of orthogonal transforms to zero out, say, the |
|
// pair of entries (0, 1) and (1, 0): |
|
// [ c1 s1 ] [* * *] [ c2 s2 ] [* *] |
|
// [-s1 c1 ] [* * *] [-s2 c2 ] = [ * *] |
|
// [ 1] [* * *] [ 1] [* * *] |
|
// When we go to zero out the next pair of entries (say, (0, 2) and (2, 0)) |
|
// then we don't expect those entries to stay 0: |
|
// [ c1 s1 ] [* *] [ c2 s2 ] [* * ] |
|
// [-s1 c1 ] [ * *] [-s2 c2 ] = [* * *] |
|
// [ 1] [* * *] [ 1] [ * *] |
|
// However, if we keep doing this, we'll find that the off-diagonal entries |
|
// converge to 0 fairly quickly (convergence should be roughly cubic). The |
|
// result is a diagonal A matrix and a bunch of orthogonal transforms: |
|
// [* * *] [* ] |
|
// L1 L2 ... Ln [* * *] Rn ... R2 R1 = [ * ] |
|
// [* * *] [ *] |
|
// ------------ ------- ------------ ------- |
|
// U^T A V S |
|
// This turns out to be highly accurate because (1) orthogonal transforms |
|
// are extremely stable to compute and apply (this is why QR factorization |
|
// works so well, FWIW) and because (2) by applying everything to the original |
|
// matrix A instead of computing (A^T * A) we avoid any precision loss that |
|
// would result from that. |
|
U.makeIdentity(); |
|
V.makeIdentity(); |
|
|
|
const int maxIter = 20; // In case we get really unlucky, prevents infinite loops |
|
const T absTol = tol * maxOffDiag (A); // Tolerance is in terms of the maximum |
|
if (absTol != 0) // _off-diagonal_ entry. |
|
{ |
|
int numIter = 0; |
|
do |
|
{ |
|
++numIter; |
|
bool changed = twoSidedJacobiRotation<T, 0, 1, 2> (A, U, V, tol); |
|
changed = twoSidedJacobiRotation<T, 0, 2, 1> (A, U, V, tol) || changed; |
|
changed = twoSidedJacobiRotation<T, 1, 2, 0> (A, U, V, tol) || changed; |
|
if (!changed) |
|
break; |
|
} while (maxOffDiag(A) > absTol && numIter < maxIter); |
|
} |
|
|
|
// The off-diagonal entries are (effectively) 0, so whatever's left on the |
|
// diagonal are the singular values: |
|
S.x = A[0][0]; |
|
S.y = A[1][1]; |
|
S.z = A[2][2]; |
|
|
|
// Nothing thus far has guaranteed that the singular values are positive, |
|
// so let's go back through and flip them if not (since by contract we are |
|
// supposed to return all positive SVs): |
|
for (int i = 0; i < 3; ++i) |
|
{ |
|
if (S[i] < 0) |
|
{ |
|
// If we flip S[i], we need to flip the corresponding column of U |
|
// (we could also pick V if we wanted; it doesn't really matter): |
|
S[i] = -S[i]; |
|
for (int j = 0; j < 3; ++j) |
|
U[j][i] = -U[j][i]; |
|
} |
|
} |
|
|
|
// Order the singular values from largest to smallest; this requires |
|
// exactly two passes through the data using bubble sort: |
|
for (int i = 0; i < 2; ++i) |
|
{ |
|
for (int j = 0; j < (2 - i); ++j) |
|
{ |
|
// No absolute values necessary since we already ensured that |
|
// they're positive: |
|
if (S[j] < S[j+1]) |
|
{ |
|
// If we swap singular values we also have to swap |
|
// corresponding columns in U and V: |
|
std::swap (S[j], S[j+1]); |
|
swapColumns (U, j, j+1); |
|
swapColumns (V, j, j+1); |
|
} |
|
} |
|
} |
|
|
|
if (forcePositiveDeterminant) |
|
{ |
|
// We want to guarantee that the returned matrices always have positive |
|
// determinant. We can do this by adding the appropriate number of |
|
// matrices of the form: |
|
// [ 1 ] |
|
// L = [ 1 ] |
|
// [ -1 ] |
|
// Note that L' = L and L*L = Identity. Thus we can add: |
|
// U*L*L*S*V = (U*L)*(L*S)*V |
|
// if U has a negative determinant, and |
|
// U*S*L*L*V = U*(S*L)*(L*V) |
|
// if V has a neg. determinant. |
|
if (U.determinant() < 0) |
|
{ |
|
for (int i = 0; i < 3; ++i) |
|
U[i][2] = -U[i][2]; |
|
S.z = -S.z; |
|
} |
|
|
|
if (V.determinant() < 0) |
|
{ |
|
for (int i = 0; i < 3; ++i) |
|
V[i][2] = -V[i][2]; |
|
S.z = -S.z; |
|
} |
|
} |
|
} |
|
|
|
template <typename T> |
|
void |
|
twoSidedJacobiSVD (Imath::Matrix44<T> A, |
|
Imath::Matrix44<T>& U, |
|
Imath::Vec4<T>& S, |
|
Imath::Matrix44<T>& V, |
|
const T tol, |
|
const bool forcePositiveDeterminant) |
|
{ |
|
// Please see the Matrix33 version for a detailed description of the algorithm. |
|
U.makeIdentity(); |
|
V.makeIdentity(); |
|
|
|
const int maxIter = 20; // In case we get really unlucky, prevents infinite loops |
|
const T absTol = tol * maxOffDiag (A); // Tolerance is in terms of the maximum |
|
if (absTol != 0) // _off-diagonal_ entry. |
|
{ |
|
int numIter = 0; |
|
do |
|
{ |
|
++numIter; |
|
bool changed = twoSidedJacobiRotation (A, 0, 1, U, V, tol); |
|
changed = twoSidedJacobiRotation (A, 0, 2, U, V, tol) || changed; |
|
changed = twoSidedJacobiRotation (A, 0, 3, U, V, tol) || changed; |
|
changed = twoSidedJacobiRotation (A, 1, 2, U, V, tol) || changed; |
|
changed = twoSidedJacobiRotation (A, 1, 3, U, V, tol) || changed; |
|
changed = twoSidedJacobiRotation (A, 2, 3, U, V, tol) || changed; |
|
if (!changed) |
|
break; |
|
} while (maxOffDiag(A) > absTol && numIter < maxIter); |
|
} |
|
|
|
// The off-diagonal entries are (effectively) 0, so whatever's left on the |
|
// diagonal are the singular values: |
|
S[0] = A[0][0]; |
|
S[1] = A[1][1]; |
|
S[2] = A[2][2]; |
|
S[3] = A[3][3]; |
|
|
|
// Nothing thus far has guaranteed that the singular values are positive, |
|
// so let's go back through and flip them if not (since by contract we are |
|
// supposed to return all positive SVs): |
|
for (int i = 0; i < 4; ++i) |
|
{ |
|
if (S[i] < 0) |
|
{ |
|
// If we flip S[i], we need to flip the corresponding column of U |
|
// (we could also pick V if we wanted; it doesn't really matter): |
|
S[i] = -S[i]; |
|
for (int j = 0; j < 4; ++j) |
|
U[j][i] = -U[j][i]; |
|
} |
|
} |
|
|
|
// Order the singular values from largest to smallest using insertion sort: |
|
for (int i = 1; i < 4; ++i) |
|
{ |
|
const Imath::Vec4<T> uCol (U[0][i], U[1][i], U[2][i], U[3][i]); |
|
const Imath::Vec4<T> vCol (V[0][i], V[1][i], V[2][i], V[3][i]); |
|
const T sVal = S[i]; |
|
|
|
int j = i - 1; |
|
while (std::abs (S[j]) < std::abs (sVal)) |
|
{ |
|
for (int k = 0; k < 4; ++k) |
|
U[k][j+1] = U[k][j]; |
|
for (int k = 0; k < 4; ++k) |
|
V[k][j+1] = V[k][j]; |
|
S[j+1] = S[j]; |
|
|
|
--j; |
|
if (j < 0) |
|
break; |
|
} |
|
|
|
for (int k = 0; k < 4; ++k) |
|
U[k][j+1] = uCol[k]; |
|
for (int k = 0; k < 4; ++k) |
|
V[k][j+1] = vCol[k]; |
|
S[j+1] = sVal; |
|
} |
|
|
|
if (forcePositiveDeterminant) |
|
{ |
|
// We want to guarantee that the returned matrices always have positive |
|
// determinant. We can do this by adding the appropriate number of |
|
// matrices of the form: |
|
// [ 1 ] |
|
// L = [ 1 ] |
|
// [ 1 ] |
|
// [ -1 ] |
|
// Note that L' = L and L*L = Identity. Thus we can add: |
|
// U*L*L*S*V = (U*L)*(L*S)*V |
|
// if U has a negative determinant, and |
|
// U*S*L*L*V = U*(S*L)*(L*V) |
|
// if V has a neg. determinant. |
|
if (U.determinant() < 0) |
|
{ |
|
for (int i = 0; i < 4; ++i) |
|
U[i][3] = -U[i][3]; |
|
S[3] = -S[3]; |
|
} |
|
|
|
if (V.determinant() < 0) |
|
{ |
|
for (int i = 0; i < 4; ++i) |
|
V[i][3] = -V[i][3]; |
|
S[3] = -S[3]; |
|
} |
|
} |
|
} |
|
|
|
} |
|
|
|
template <typename T> |
|
void |
|
jacobiSVD (const Imath::Matrix33<T>& A, |
|
Imath::Matrix33<T>& U, |
|
Imath::Vec3<T>& S, |
|
Imath::Matrix33<T>& V, |
|
const T tol, |
|
const bool forcePositiveDeterminant) |
|
{ |
|
twoSidedJacobiSVD (A, U, S, V, tol, forcePositiveDeterminant); |
|
} |
|
|
|
template <typename T> |
|
void |
|
jacobiSVD (const Imath::Matrix44<T>& A, |
|
Imath::Matrix44<T>& U, |
|
Imath::Vec4<T>& S, |
|
Imath::Matrix44<T>& V, |
|
const T tol, |
|
const bool forcePositiveDeterminant) |
|
{ |
|
twoSidedJacobiSVD (A, U, S, V, tol, forcePositiveDeterminant); |
|
} |
|
|
|
template void jacobiSVD (const Imath::Matrix33<float>& A, |
|
Imath::Matrix33<float>& U, |
|
Imath::Vec3<float>& S, |
|
Imath::Matrix33<float>& V, |
|
const float tol, |
|
const bool forcePositiveDeterminant); |
|
template void jacobiSVD (const Imath::Matrix33<double>& A, |
|
Imath::Matrix33<double>& U, |
|
Imath::Vec3<double>& S, |
|
Imath::Matrix33<double>& V, |
|
const double tol, |
|
const bool forcePositiveDeterminant); |
|
template void jacobiSVD (const Imath::Matrix44<float>& A, |
|
Imath::Matrix44<float>& U, |
|
Imath::Vec4<float>& S, |
|
Imath::Matrix44<float>& V, |
|
const float tol, |
|
const bool forcePositiveDeterminant); |
|
template void jacobiSVD (const Imath::Matrix44<double>& A, |
|
Imath::Matrix44<double>& U, |
|
Imath::Vec4<double>& S, |
|
Imath::Matrix44<double>& V, |
|
const double tol, |
|
const bool forcePositiveDeterminant); |
|
|
|
namespace |
|
{ |
|
|
|
template <int j, int k, typename TM> |
|
inline |
|
void |
|
jacobiRotateRight (TM& A, |
|
const typename TM::BaseType s, |
|
const typename TM::BaseType tau) |
|
{ |
|
typedef typename TM::BaseType T; |
|
|
|
for (unsigned int i = 0; i < TM::dimensions(); ++i) |
|
{ |
|
const T nu1 = A[i][j]; |
|
const T nu2 = A[i][k]; |
|
A[i][j] -= s * (nu2 + tau * nu1); |
|
A[i][k] += s * (nu1 - tau * nu2); |
|
} |
|
} |
|
|
|
template <int j, int k, int l, typename T> |
|
bool |
|
jacobiRotation (Matrix33<T>& A, |
|
Matrix33<T>& V, |
|
Vec3<T>& Z, |
|
const T tol) |
|
{ |
|
// Load everything into local variables to make things easier on the |
|
// optimizer: |
|
const T x = A[j][j]; |
|
const T y = A[j][k]; |
|
const T z = A[k][k]; |
|
|
|
// The first stage diagonalizes, |
|
// [ c s ]^T [ x y ] [ c -s ] = [ d1 0 ] |
|
// [ -s c ] [ y z ] [ s c ] [ 0 d2 ] |
|
const T mu1 = z - x; |
|
const T mu2 = 2 * y; |
|
|
|
if (std::abs(mu2) <= tol*std::abs(mu1)) |
|
{ |
|
// We've decided that the off-diagonal entries are already small |
|
// enough, so we'll set them to zero. This actually appears to result |
|
// in smaller errors than leaving them be, possibly because it prevents |
|
// us from trying to do extra rotations later that we don't need. |
|
A[j][k] = 0; |
|
return false; |
|
} |
|
const T rho = mu1 / mu2; |
|
const T t = (rho < 0 ? T(-1) : T(1)) / (std::abs(rho) + std::sqrt(1 + rho*rho)); |
|
const T c = T(1) / std::sqrt (T(1) + t*t); |
|
const T s = t * c; |
|
const T tau = s / (T(1) + c); |
|
const T h = t * y; |
|
|
|
// Update diagonal elements. |
|
Z[j] -= h; |
|
Z[k] += h; |
|
A[j][j] -= h; |
|
A[k][k] += h; |
|
|
|
// For the entries we just zeroed out, we'll just set them to 0, since |
|
// they should be 0 up to machine precision. |
|
A[j][k] = 0; |
|
|
|
// We only update upper triagnular elements of A, since |
|
// A is supposed to be symmetric. |
|
T& offd1 = l < j ? A[l][j] : A[j][l]; |
|
T& offd2 = l < k ? A[l][k] : A[k][l]; |
|
const T nu1 = offd1; |
|
const T nu2 = offd2; |
|
offd1 = nu1 - s * (nu2 + tau * nu1); |
|
offd2 = nu2 + s * (nu1 - tau * nu2); |
|
|
|
// Apply rotation to V |
|
jacobiRotateRight<j, k> (V, s, tau); |
|
|
|
return true; |
|
} |
|
|
|
template <int j, int k, int l1, int l2, typename T> |
|
bool |
|
jacobiRotation (Matrix44<T>& A, |
|
Matrix44<T>& V, |
|
Vec4<T>& Z, |
|
const T tol) |
|
{ |
|
const T x = A[j][j]; |
|
const T y = A[j][k]; |
|
const T z = A[k][k]; |
|
|
|
const T mu1 = z - x; |
|
const T mu2 = T(2) * y; |
|
|
|
// Let's see if rho^(-1) = mu2 / mu1 is less than tol |
|
// This test also checks if rho^2 will overflow |
|
// when tol^(-1) < sqrt(limits<T>::max()). |
|
if (std::abs(mu2) <= tol*std::abs(mu1)) |
|
{ |
|
A[j][k] = 0; |
|
return true; |
|
} |
|
|
|
const T rho = mu1 / mu2; |
|
const T t = (rho < 0 ? T(-1) : T(1)) / (std::abs(rho) + std::sqrt(1 + rho*rho)); |
|
const T c = T(1) / std::sqrt (T(1) + t*t); |
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const T s = c * t; |
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const T tau = s / (T(1) + c); |
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const T h = t * y; |
|
|
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Z[j] -= h; |
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Z[k] += h; |
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A[j][j] -= h; |
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A[k][k] += h; |
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A[j][k] = 0; |
|
|
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{ |
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T& offd1 = l1 < j ? A[l1][j] : A[j][l1]; |
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T& offd2 = l1 < k ? A[l1][k] : A[k][l1]; |
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const T nu1 = offd1; |
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const T nu2 = offd2; |
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offd1 -= s * (nu2 + tau * nu1); |
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offd2 += s * (nu1 - tau * nu2); |
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} |
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|
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{ |
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T& offd1 = l2 < j ? A[l2][j] : A[j][l2]; |
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T& offd2 = l2 < k ? A[l2][k] : A[k][l2]; |
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const T nu1 = offd1; |
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const T nu2 = offd2; |
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offd1 -= s * (nu2 + tau * nu1); |
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offd2 += s * (nu1 - tau * nu2); |
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} |
|
|
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jacobiRotateRight<j, k> (V, s, tau); |
|
|
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return true; |
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} |
|
|
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template <typename TM> |
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inline |
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typename TM::BaseType |
|
maxOffDiagSymm (const TM& A) |
|
{ |
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typedef typename TM::BaseType T; |
|
T result = 0; |
|
for (unsigned int i = 0; i < TM::dimensions(); ++i) |
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for (unsigned int j = i+1; j < TM::dimensions(); ++j) |
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result = std::max (result, std::abs (A[i][j])); |
|
|
|
return result; |
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} |
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|
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} // namespace |
|
|
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template <typename T> |
|
void |
|
jacobiEigenSolver (Matrix33<T>& A, |
|
Vec3<T>& S, |
|
Matrix33<T>& V, |
|
const T tol) |
|
{ |
|
V.makeIdentity(); |
|
for(int i = 0; i < 3; ++i) { |
|
S[i] = A[i][i]; |
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} |
|
|
|
const int maxIter = 20; // In case we get really unlucky, prevents infinite loops |
|
const T absTol = tol * maxOffDiagSymm (A); // Tolerance is in terms of the maximum |
|
if (absTol != 0) // _off-diagonal_ entry. |
|
{ |
|
int numIter = 0; |
|
do |
|
{ |
|
// Z is for accumulating small changes (h) to diagonal entries |
|
// of A for one sweep. Adding h's directly to A might cause |
|
// a cancellation effect when h is relatively very small to |
|
// the corresponding diagonal entry of A and |
|
// this will increase numerical errors |
|
Vec3<T> Z(0, 0, 0); |
|
++numIter; |
|
bool changed = jacobiRotation<0, 1, 2> (A, V, Z, tol); |
|
changed = jacobiRotation<0, 2, 1> (A, V, Z, tol) || changed; |
|
changed = jacobiRotation<1, 2, 0> (A, V, Z, tol) || changed; |
|
// One sweep passed. Add accumulated changes (Z) to singular values (S) |
|
// Update diagonal elements of A for better accuracy as well. |
|
for(int i = 0; i < 3; ++i) { |
|
A[i][i] = S[i] += Z[i]; |
|
} |
|
if (!changed) |
|
break; |
|
} while (maxOffDiagSymm(A) > absTol && numIter < maxIter); |
|
} |
|
} |
|
|
|
template <typename T> |
|
void |
|
jacobiEigenSolver (Matrix44<T>& A, |
|
Vec4<T>& S, |
|
Matrix44<T>& V, |
|
const T tol) |
|
{ |
|
V.makeIdentity(); |
|
|
|
for(int i = 0; i < 4; ++i) { |
|
S[i] = A[i][i]; |
|
} |
|
|
|
const int maxIter = 20; // In case we get really unlucky, prevents infinite loops |
|
const T absTol = tol * maxOffDiagSymm (A); // Tolerance is in terms of the maximum |
|
if (absTol != 0) // _off-diagonal_ entry. |
|
{ |
|
int numIter = 0; |
|
do |
|
{ |
|
++numIter; |
|
Vec4<T> Z(0, 0, 0, 0); |
|
bool changed = jacobiRotation<0, 1, 2, 3> (A, V, Z, tol); |
|
changed = jacobiRotation<0, 2, 1, 3> (A, V, Z, tol) || changed; |
|
changed = jacobiRotation<0, 3, 1, 2> (A, V, Z, tol) || changed; |
|
changed = jacobiRotation<1, 2, 0, 3> (A, V, Z, tol) || changed; |
|
changed = jacobiRotation<1, 3, 0, 2> (A, V, Z, tol) || changed; |
|
changed = jacobiRotation<2, 3, 0, 1> (A, V, Z, tol) || changed; |
|
for(int i = 0; i < 4; ++i) { |
|
A[i][i] = S[i] += Z[i]; |
|
} |
|
if (!changed) |
|
break; |
|
} while (maxOffDiagSymm(A) > absTol && numIter < maxIter); |
|
} |
|
} |
|
|
|
template <typename TM, typename TV> |
|
void |
|
maxEigenVector (TM& A, TV& V) |
|
{ |
|
TV S; |
|
TM MV; |
|
jacobiEigenSolver(A, S, MV); |
|
|
|
int maxIdx(0); |
|
for(unsigned int i = 1; i < TV::dimensions(); ++i) |
|
{ |
|
if(std::abs(S[i]) > std::abs(S[maxIdx])) |
|
maxIdx = i; |
|
} |
|
|
|
for(unsigned int i = 0; i < TV::dimensions(); ++i) |
|
V[i] = MV[i][maxIdx]; |
|
} |
|
|
|
template <typename TM, typename TV> |
|
void |
|
minEigenVector (TM& A, TV& V) |
|
{ |
|
TV S; |
|
TM MV; |
|
jacobiEigenSolver(A, S, MV); |
|
|
|
int minIdx(0); |
|
for(unsigned int i = 1; i < TV::dimensions(); ++i) |
|
{ |
|
if(std::abs(S[i]) < std::abs(S[minIdx])) |
|
minIdx = i; |
|
} |
|
|
|
for(unsigned int i = 0; i < TV::dimensions(); ++i) |
|
V[i] = MV[i][minIdx]; |
|
} |
|
|
|
template void jacobiEigenSolver (Matrix33<float>& A, |
|
Vec3<float>& S, |
|
Matrix33<float>& V, |
|
const float tol); |
|
template void jacobiEigenSolver (Matrix33<double>& A, |
|
Vec3<double>& S, |
|
Matrix33<double>& V, |
|
const double tol); |
|
template void jacobiEigenSolver (Matrix44<float>& A, |
|
Vec4<float>& S, |
|
Matrix44<float>& V, |
|
const float tol); |
|
template void jacobiEigenSolver (Matrix44<double>& A, |
|
Vec4<double>& S, |
|
Matrix44<double>& V, |
|
const double tol); |
|
|
|
template void maxEigenVector (Matrix33<float>& A, |
|
Vec3<float>& S); |
|
template void maxEigenVector (Matrix44<float>& A, |
|
Vec4<float>& S); |
|
template void maxEigenVector (Matrix33<double>& A, |
|
Vec3<double>& S); |
|
template void maxEigenVector (Matrix44<double>& A, |
|
Vec4<double>& S); |
|
|
|
template void minEigenVector (Matrix33<float>& A, |
|
Vec3<float>& S); |
|
template void minEigenVector (Matrix44<float>& A, |
|
Vec4<float>& S); |
|
template void minEigenVector (Matrix33<double>& A, |
|
Vec3<double>& S); |
|
template void minEigenVector (Matrix44<double>& A, |
|
Vec4<double>& S); |
|
|
|
} // namespace Imath
|
|
|