1253 lines
40 KiB
C++
1253 lines
40 KiB
C++
///////////////////////////////////////////////////////////////////////////
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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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{
|
|
// Load everything into local variables to make things easier on the
|
|
// optimizer:
|
|
const T w = A[j][j];
|
|
const T x = A[j][k];
|
|
const T y = A[k][j];
|
|
const T z = A[k][k];
|
|
|
|
// We will keep track of whether we're actually performing any rotations,
|
|
// since if the matrix is already diagonal we'll end up with the identity
|
|
// as our Jacobi rotation and we can short-circuit.
|
|
bool changed = false;
|
|
|
|
// The first step is to symmetrize the 2x2 matrix,
|
|
// [ c s ]^T [ w x ] = [ p q ]
|
|
// [ -s c ] [ y z ] [ q r ]
|
|
T mu_1 = w + z;
|
|
T mu_2 = x - y;
|
|
|
|
T c, s;
|
|
if (std::abs(mu_2) <= tol*std::abs(mu_1)) // Already symmetric (to tolerance)
|
|
{ // Note that the <= is important here
|
|
c = T(1); // because we want to bypass the computation
|
|
s = T(0); // of rho if mu_1 = mu_2 = 0.
|
|
|
|
const T p = w;
|
|
const T r = z;
|
|
mu_1 = r - p;
|
|
mu_2 = x + y;
|
|
}
|
|
else
|
|
{
|
|
const T rho = mu_1 / mu_2;
|
|
s = T(1) / std::sqrt (T(1) + rho*rho); // TODO is there a native inverse square root function?
|
|
if (rho < 0)
|
|
s = -s;
|
|
c = s * rho;
|
|
|
|
mu_1 = s * (x + y) + c * (z - w); // = r - p
|
|
mu_2 = T(2) * (c * x - s * z); // = 2*q
|
|
|
|
changed = true;
|
|
}
|
|
|
|
// The second stage diagonalizes,
|
|
// [ c2 s2 ]^T [ p q ] [ c2 s2 ] = [ d1 0 ]
|
|
// [ -s2 c2 ] [ q r ] [ -s2 c2 ] [ 0 d2 ]
|
|
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);
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namespace
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{
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template <int j, int k, typename TM>
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inline
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void
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jacobiRotateRight (TM& A,
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const typename TM::BaseType s,
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const typename TM::BaseType tau)
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{
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typedef typename TM::BaseType T;
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for (unsigned int i = 0; i < TM::dimensions(); ++i)
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{
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const T nu1 = A[i][j];
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const T nu2 = A[i][k];
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A[i][j] -= s * (nu2 + tau * nu1);
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A[i][k] += s * (nu1 - tau * nu2);
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}
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}
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template <int j, int k, int l, typename T>
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bool
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jacobiRotation (Matrix33<T>& A,
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Matrix33<T>& V,
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Vec3<T>& Z,
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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 x = A[j][j];
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const T y = A[j][k];
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const T z = A[k][k];
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// The first stage diagonalizes,
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// [ c s ]^T [ x y ] [ c -s ] = [ d1 0 ]
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// [ -s c ] [ y z ] [ s c ] [ 0 d2 ]
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const T mu1 = z - x;
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const T mu2 = 2 * y;
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if (std::abs(mu2) <= tol*std::abs(mu1))
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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[j][k] = 0;
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return false;
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}
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const T rho = mu1 / mu2;
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const T t = (rho < 0 ? T(-1) : T(1)) / (std::abs(rho) + std::sqrt(1 + rho*rho));
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const T c = T(1) / std::sqrt (T(1) + t*t);
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const T s = t * c;
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const T tau = s / (T(1) + c);
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const T h = t * y;
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// Update diagonal elements.
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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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// 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][k] = 0;
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// We only update upper triagnular elements of A, since
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// A is supposed to be symmetric.
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T& offd1 = l < j ? A[l][j] : A[j][l];
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T& offd2 = l < k ? A[l][k] : A[k][l];
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const T nu1 = offd1;
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const T nu2 = offd2;
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offd1 = nu1 - s * (nu2 + tau * nu1);
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offd2 = nu2 + s * (nu1 - tau * nu2);
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// Apply rotation to V
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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 <int j, int k, int l1, int l2, typename T>
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bool
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jacobiRotation (Matrix44<T>& A,
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Matrix44<T>& V,
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Vec4<T>& Z,
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const T tol)
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{
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const T x = A[j][j];
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const T y = A[j][k];
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const T z = A[k][k];
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const T mu1 = z - x;
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const T mu2 = T(2) * y;
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// Let's see if rho^(-1) = mu2 / mu1 is less than tol
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// This test also checks if rho^2 will overflow
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// when tol^(-1) < sqrt(limits<T>::max()).
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if (std::abs(mu2) <= tol*std::abs(mu1))
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{
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A[j][k] = 0;
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return true;
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}
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const T rho = mu1 / mu2;
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const T t = (rho < 0 ? T(-1) : T(1)) / (std::abs(rho) + std::sqrt(1 + rho*rho));
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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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|
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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|
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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|
}
|
|
|
|
jacobiRotateRight<j, k> (V, s, tau);
|
|
|
|
return true;
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|
}
|
|
|
|
template <typename TM>
|
|
inline
|
|
typename TM::BaseType
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|
maxOffDiagSymm (const TM& A)
|
|
{
|
|
typedef typename TM::BaseType T;
|
|
T result = 0;
|
|
for (unsigned int i = 0; i < TM::dimensions(); ++i)
|
|
for (unsigned int j = i+1; j < TM::dimensions(); ++j)
|
|
result = std::max (result, std::abs (A[i][j]));
|
|
|
|
return result;
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|
}
|
|
|
|
} // namespace
|
|
|
|
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];
|
|
}
|
|
|
|
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
|