From 5ef6668c3d2bc0a4e7c364119a99fcd6c6e43017 Mon Sep 17 00:00:00 2001 From: Mike Turvey Date: Sun, 26 Mar 2017 19:08:44 -0700 Subject: Adding svd --- redist/svd.h | 450 +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 450 insertions(+) create mode 100644 redist/svd.h (limited to 'redist') diff --git a/redist/svd.h b/redist/svd.h new file mode 100644 index 0000000..41708da --- /dev/null +++ b/redist/svd.h @@ -0,0 +1,450 @@ +/************************************************************************** +** +** svd3 +** +** Quick singular value decomposition as described by: +** A. McAdams, A. Selle, R. Tamstorf, J. Teran and E. Sifakis, +** "Computing the Singular Value Decomposition of 3x3 matrices +** with minimal branching and elementary floating point operations", +** University of Wisconsin - Madison technical report TR1690, May 2011 +** +** OPTIMIZED CPU VERSION +** Implementation by: Eric Jang +** +** 13 Apr 2014 +** +** This file originally retrieved from: +** https://github.com/ericjang/svd3/blob/master/svd3.h 3/26/2017 +** +** Original licesnse is MIT per: +** https://github.com/ericjang/svd3/blob/master/LICENSE.md +** +** Ported from C++ to C by Mike Turvey. All modifications also released +** under an MIT license. +**************************************************************************/ + + +#ifndef SVD3_H +#define SVD3_H + +#define _gamma 5.828427124 // FOUR_GAMMA_SQUARED = sqrt(8)+3; +#define _cstar 0.923879532 // cos(pi/8) +#define _sstar 0.3826834323 // sin(p/8) +#define EPSILON 1e-6 + +#include + +/* This is a novel and fast routine for the reciprocal square root of an +IEEE float (single precision). +http://www.lomont.org/Math/Papers/2003/InvSqrt.pdf +http://playstation2-linux.com/download/p2lsd/fastrsqrt.pdf +http://www.beyond3d.com/content/articles/8/ +*/ +inline float rsqrt(float x) { + // int ihalf = *(int *)&x - 0x00800000; // Alternative to next line, + // float xhalf = *(float *)&ihalf; // for sufficiently large nos. + float xhalf = 0.5f*x; + int i = *(int *)&x; // View x as an int. + // i = 0x5f3759df - (i >> 1); // Initial guess (traditional). + i = 0x5f375a82 - (i >> 1); // Initial guess (slightly better). + x = *(float *)&i; // View i as float. + x = x*(1.5f - xhalf*x*x); // Newton step. + // x = x*(1.5008908 - xhalf*x*x); // Newton step for a balanced error. + return x; +} + +/* This is rsqrt with an additional step of the Newton iteration, for +increased accuracy. The constant 0x5f37599e makes the relative error +range from 0 to -0.00000463. +You can't balance the error by adjusting the constant. */ +inline float rsqrt1(float x) { + float xhalf = 0.5f*x; + int i = *(int *)&x; // View x as an int. + i = 0x5f37599e - (i >> 1); // Initial guess. + x = *(float *)&i; // View i as float. + x = x*(1.5f - xhalf*x*x); // Newton step. + x = x*(1.5f - xhalf*x*x); // Newton step again. + return x; +} + +inline float accurateSqrt(float x) +{ + return x * rsqrt1(x); +} + +inline void condSwap(bool c, float *X, float *Y) +{ + // used in step 2 + float Z = *X; + *X = c ? *Y : *X; + *Y = c ? Z : *Y; +} + +inline void condNegSwap(bool c, float *X, float *Y) +{ + // used in step 2 and 3 + float Z = -*X; + *X = c ? *Y : *X; + *Y = c ? Z : *Y; +} + +// matrix multiplication M = A * B +inline void multAB(float a11, float a12, float a13, + float a21, float a22, float a23, + float a31, float a32, float a33, + // + float b11, float b12, float b13, + float b21, float b22, float b23, + float b31, float b32, float b33, + // + float *m11, float *m12, float *m13, + float *m21, float *m22, float *m23, + float *m31, float *m32, float *m33) +{ + + *m11 = a11*b11 + a12*b21 + a13*b31; *m12 = a11*b12 + a12*b22 + a13*b32; *m13 = a11*b13 + a12*b23 + a13*b33; + *m21 = a21*b11 + a22*b21 + a23*b31; *m22 = a21*b12 + a22*b22 + a23*b32; *m23 = a21*b13 + a22*b23 + a23*b33; + *m31 = a31*b11 + a32*b21 + a33*b31; *m32 = a31*b12 + a32*b22 + a33*b32; *m33 = a31*b13 + a32*b23 + a33*b33; +} + +// matrix multiplication M = Transpose[A] * B +inline void multAtB(float a11, float a12, float a13, + float a21, float a22, float a23, + float a31, float a32, float a33, + // + float b11, float b12, float b13, + float b21, float b22, float b23, + float b31, float b32, float b33, + // + float *m11, float *m12, float *m13, + float *m21, float *m22, float *m23, + float *m31, float *m32, float *m33) +{ + *m11 = a11*b11 + a21*b21 + a31*b31; *m12 = a11*b12 + a21*b22 + a31*b32; *m13 = a11*b13 + a21*b23 + a31*b33; + *m21 = a12*b11 + a22*b21 + a32*b31; *m22 = a12*b12 + a22*b22 + a32*b32; *m23 = a12*b13 + a22*b23 + a32*b33; + *m31 = a13*b11 + a23*b21 + a33*b31; *m32 = a13*b12 + a23*b22 + a33*b32; *m33 = a13*b13 + a23*b23 + a33*b33; +} + +inline void quatToMat3(const float * qV, + float *m11, float *m12, float *m13, + float *m21, float *m22, float *m23, + float *m31, float *m32, float *m33 +) +{ + float w = qV[3]; + float x = qV[0]; + float y = qV[1]; + float z = qV[2]; + + float qxx = x*x; + float qyy = y*y; + float qzz = z*z; + float qxz = x*z; + float qxy = x*y; + float qyz = y*z; + float qwx = w*x; + float qwy = w*y; + float qwz = w*z; + + *m11 = 1 - 2 * (qyy + qzz); *m12 = 2 * (qxy - qwz); *m13 = 2 * (qxz + qwy); + *m21 = 2 * (qxy + qwz); *m22 = 1 - 2 * (qxx + qzz); *m23 = 2 * (qyz - qwx); + *m31 = 2 * (qxz - qwy); *m32 = 2 * (qyz + qwx); *m33 = 1 - 2 * (qxx + qyy); +} + +inline void approximateGivensQuaternion(float a11, float a12, float a22, float *ch, float *sh) +{ + /* + * Given givens angle computed by approximateGivensAngles, + * compute the corresponding rotation quaternion. + */ + *ch = 2 * (a11 - a22); + *sh = a12; + bool b = _gamma* (*sh)*(*sh) < (*ch)*(*ch); + // fast rsqrt function suffices + // rsqrt2 (https://code.google.com/p/lppython/source/browse/algorithm/HDcode/newCode/rsqrt.c?r=26) + // is even faster but results in too much error + float w = rsqrt((*ch)*(*ch) + (*sh)*(*sh)); + *ch = b ? w*(*ch) : (float)_cstar; + *sh = b ? w*(*sh) : (float)_sstar; +} + +inline void jacobiConjugation(const int x, const int y, const int z, + float *s11, + float *s21, float *s22, + float *s31, float *s32, float *s33, + float * qV) +{ + float ch, sh; + approximateGivensQuaternion(*s11, *s21, *s22, &ch, &sh); + + float scale = ch*ch + sh*sh; + float a = (ch*ch - sh*sh) / scale; + float b = (2 * sh*ch) / scale; + + // make temp copy of S + float _s11 = *s11; + float _s21 = *s21; float _s22 = *s22; + float _s31 = *s31; float _s32 = *s32; float _s33 = *s33; + + // perform conjugation S = Q'*S*Q + // Q already implicitly solved from a, b + *s11 = a*(a*_s11 + b*_s21) + b*(a*_s21 + b*_s22); + *s21 = a*(-b*_s11 + a*_s21) + b*(-b*_s21 + a*_s22); *s22 = -b*(-b*_s11 + a*_s21) + a*(-b*_s21 + a*_s22); + *s31 = a*_s31 + b*_s32; *s32 = -b*_s31 + a*_s32; *s33 = _s33; + + // update cumulative rotation qV + float tmp[3]; + tmp[0] = qV[0] * sh; + tmp[1] = qV[1] * sh; + tmp[2] = qV[2] * sh; + sh *= qV[3]; + + qV[0] *= ch; + qV[1] *= ch; + qV[2] *= ch; + qV[3] *= ch; + + // (x,y,z) corresponds to ((0,1,2),(1,2,0),(2,0,1)) + // for (p,q) = ((0,1),(1,2),(0,2)) + qV[z] += sh; + qV[3] -= tmp[z]; // w + qV[x] += tmp[y]; + qV[y] -= tmp[x]; + + // re-arrange matrix for next iteration + _s11 = *s22; + _s21 = *s32; _s22 = *s33; + _s31 = *s21; _s32 = *s31; _s33 = *s11; + *s11 = _s11; + *s21 = _s21; *s22 = _s22; + *s31 = _s31; *s32 = _s32; *s33 = _s33; + +} + +inline float dist2(float x, float y, float z) +{ + return x*x + y*y + z*z; +} + +// finds transformation that diagonalizes a symmetric matrix +inline void jacobiEigenanlysis( // symmetric matrix + float *s11, + float *s21, float *s22, + float *s31, float *s32, float *s33, + // quaternion representation of V + float * qV) +{ + qV[3] = 1; qV[0] = 0; qV[1] = 0; qV[2] = 0; // follow same indexing convention as GLM + for (int i = 0; i<4; i++) + { + // we wish to eliminate the maximum off-diagonal element + // on every iteration, but cycling over all 3 possible rotations + // in fixed order (p,q) = (1,2) , (2,3), (1,3) still retains + // asymptotic convergence + jacobiConjugation(0, 1, 2, s11, s21, s22, s31, s32, s33, qV); // p,q = 0,1 + jacobiConjugation(1, 2, 0, s11, s21, s22, s31, s32, s33, qV); // p,q = 1,2 + jacobiConjugation(2, 0, 1, s11, s21, s22, s31, s32, s33, qV); // p,q = 0,2 + } +} + + +inline void sortSingularValues(// matrix that we want to decompose + float *b11, float *b12, float *b13, + float *b21, float *b22, float *b23, + float *b31, float *b32, float *b33, + // sort V simultaneously + float *v11, float *v12, float *v13, + float *v21, float *v22, float *v23, + float *v31, float *v32, float *v33) +{ + float rho1 = dist2(*b11, *b21, *b31); + float rho2 = dist2(*b12, *b22, *b32); + float rho3 = dist2(*b13, *b23, *b33); + bool c; + c = rho1 < rho2; + condNegSwap(c, b11, b12); condNegSwap(c, v11, v12); + condNegSwap(c, b21, b22); condNegSwap(c, v21, v22); + condNegSwap(c, b31, b32); condNegSwap(c, v31, v32); + condSwap(c, &rho1, &rho2); + c = rho1 < rho3; + condNegSwap(c, b11, b13); condNegSwap(c, v11, v13); + condNegSwap(c, b21, b23); condNegSwap(c, v21, v23); + condNegSwap(c, b31, b33); condNegSwap(c, v31, v33); + condSwap(c, &rho1, &rho3); + c = rho2 < rho3; + condNegSwap(c, b12, b13); condNegSwap(c, v12, v13); + condNegSwap(c, b22, b23); condNegSwap(c, v22, v23); + condNegSwap(c, b32, b33); condNegSwap(c, v32, v33); +} + + +void QRGivensQuaternion(float a1, float a2, float *ch, float *sh) +{ + // a1 = pivot point on diagonal + // a2 = lower triangular entry we want to annihilate + float epsilon = (float)EPSILON; + float rho = accurateSqrt(a1*a1 + a2*a2); + + *sh = rho > epsilon ? a2 : 0; + *ch = fabsf(a1) + fmaxf(rho, epsilon); + bool b = a1 < 0; + condSwap(b, sh, ch); + float w = rsqrt((*ch)*(*ch) + (*sh)*(*sh)); + *ch *= w; + *sh *= w; +} + + +inline void QRDecomposition(// matrix that we want to decompose + float b11, float b12, float b13, + float b21, float b22, float b23, + float b31, float b32, float b33, + // output Q + float *q11, float *q12, float *q13, + float *q21, float *q22, float *q23, + float *q31, float *q32, float *q33, + // output R + float *r11, float *r12, float *r13, + float *r21, float *r22, float *r23, + float *r31, float *r32, float *r33) +{ + float ch1, sh1, ch2, sh2, ch3, sh3; + float a, b; + + // first givens rotation (ch,0,0,sh) + QRGivensQuaternion(b11, b21, &ch1, &sh1); + a = 1 - 2 * sh1*sh1; + b = 2 * ch1*sh1; + // apply B = Q' * B + *r11 = a*b11 + b*b21; *r12 = a*b12 + b*b22; *r13 = a*b13 + b*b23; + *r21 = -b*b11 + a*b21; *r22 = -b*b12 + a*b22; *r23 = -b*b13 + a*b23; + *r31 = b31; *r32 = b32; *r33 = b33; + + // second givens rotation (ch,0,-sh,0) + QRGivensQuaternion(*r11, *r31, &ch2, &sh2); + a = 1 - 2 * sh2*sh2; + b = 2 * ch2*sh2; + // apply B = Q' * B; + b11 = a*(*r11) + b*(*r31); b12 = a*(*r12) + b*(*r32); b13 = a*(*r13) + b*(*r33); + b21 = *r21; b22 = *r22; b23 = *r23; + b31 = -b*(*r11) + a*(*r31); b32 = -b*(*r12) + a*(*r32); b33 = -b*(*r13) + a*(*r33); + + // third givens rotation (ch,sh,0,0) + QRGivensQuaternion(b22, b32, &ch3, &sh3); + a = 1 - 2 * sh3*sh3; + b = 2 * ch3*sh3; + // R is now set to desired value + *r11 = b11; *r12 = b12; *r13 = b13; + *r21 = a*b21 + b*b31; *r22 = a*b22 + b*b32; *r23 = a*b23 + b*b33; + *r31 = -b*b21 + a*b31; *r32 = -b*b22 + a*b32; *r33 = -b*b23 + a*b33; + + // construct the cumulative rotation Q=Q1 * Q2 * Q3 + // the number of floating point operations for three quaternion multiplications + // is more or less comparable to the explicit form of the joined matrix. + // certainly more memory-efficient! + float sh12 = sh1*sh1; + float sh22 = sh2*sh2; + float sh32 = sh3*sh3; + + *q11 = (-1 + 2 * sh12)*(-1 + 2 * sh22); + *q12 = 4 * ch2*ch3*(-1 + 2 * sh12)*sh2*sh3 + 2 * ch1*sh1*(-1 + 2 * sh32); + *q13 = 4 * ch1*ch3*sh1*sh3 - 2 * ch2*(-1 + 2 * sh12)*sh2*(-1 + 2 * sh32); + + *q21 = 2 * ch1*sh1*(1 - 2 * sh22); + *q22 = -8 * ch1*ch2*ch3*sh1*sh2*sh3 + (-1 + 2 * sh12)*(-1 + 2 * sh32); + *q23 = -2 * ch3*sh3 + 4 * sh1*(ch3*sh1*sh3 + ch1*ch2*sh2*(-1 + 2 * sh32)); + + *q31 = 2 * ch2*sh2; + *q32 = 2 * ch3*(1 - 2 * sh22)*sh3; + *q33 = (-1 + 2 * sh22)*(-1 + 2 * sh32); +} + +void svd(// input A + float a11, float a12, float a13, + float a21, float a22, float a23, + float a31, float a32, float a33, + // output U + float *u11, float *u12, float *u13, + float *u21, float *u22, float *u23, + float *u31, float *u32, float *u33, + // output S + float *s11, float *s12, float *s13, + float *s21, float *s22, float *s23, + float *s31, float *s32, float *s33, + // output V + float *v11, float *v12, float *v13, + float *v21, float *v22, float *v23, + float *v31, float *v32, float *v33) +{ + // normal equations matrix + float ATA11, ATA12, ATA13; + float ATA21, ATA22, ATA23; + float ATA31, ATA32, ATA33; + + multAtB(a11, a12, a13, a21, a22, a23, a31, a32, a33, + a11, a12, a13, a21, a22, a23, a31, a32, a33, + &ATA11, &ATA12, &ATA13, &ATA21, &ATA22, &ATA23, &ATA31, &ATA32, &ATA33); + + // symmetric eigenalysis + float qV[4]; + jacobiEigenanlysis(&ATA11, &ATA21, &ATA22, &ATA31, &ATA32, &ATA33, qV); + quatToMat3(qV, v11, v12, v13, v21, v22, v23, v31, v32, v33); + + float b11, b12, b13; + float b21, b22, b23; + float b31, b32, b33; + multAB(a11, a12, a13, a21, a22, a23, a31, a32, a33, + *v11, *v12, *v13, *v21, *v22, *v23, *v31, *v32, *v33, + &b11, &b12, &b13, &b21, &b22, &b23, &b31, &b32, &b33); + + // sort singular values and find V + sortSingularValues(&b11, &b12, &b13, &b21, &b22, &b23, &b31, &b32, &b33, + v11, v12, v13, v21, v22, v23, v31, v32, v33); + + // QR decomposition + QRDecomposition(b11, b12, b13, b21, b22, b23, b31, b32, b33, + u11, u12, u13, u21, u22, u23, u31, u32, u33, + s11, s12, s13, s21, s22, s23, s31, s32, s33 + ); +} + +/// polar decomposition can be reconstructed trivially from SVD result +// A = UP +void pd(float a11, float a12, float a13, + float a21, float a22, float a23, + float a31, float a32, float a33, + // output U + float *u11, float *u12, float *u13, + float *u21, float *u22, float *u23, + float *u31, float *u32, float *u33, + // output P + float *p11, float *p12, float *p13, + float *p21, float *p22, float *p23, + float *p31, float *p32, float *p33) +{ + float w11, w12, w13, w21, w22, w23, w31, w32, w33; + float s11, s12, s13, s21, s22, s23, s31, s32, s33; + float v11, v12, v13, v21, v22, v23, v31, v32, v33; + + svd(a11, a12, a13, a21, a22, a23, a31, a32, a33, + &w11, &w12, &w13, &w21, &w22, &w23, &w31, &w32, &w33, + &s11, &s12, &s13, &s21, &s22, &s23, &s31, &s32, &s33, + &v11, &v12, &v13, &v21, &v22, &v23, &v31, &v32, &v33); + + // P = VSV' + float t11, t12, t13, t21, t22, t23, t31, t32, t33; + multAB(v11, v12, v13, v21, v22, v23, v31, v32, v33, + s11, s12, s13, s21, s22, s23, s31, s32, s33, + &t11, &t12, &t13, &t21, &t22, &t23, &t31, &t32, &t33); + + multAB(t11, t12, t13, t21, t22, t23, t31, t32, t33, + v11, v21, v31, v12, v22, v32, v13, v23, v33, + p11, p12, p13, p21, p22, p23, p31, p32, p33); + + // U = WV' + multAB(w11, w12, w13, w21, w22, w23, w31, w32, w33, + v11, v21, v31, v12, v22, v32, v13, v23, v33, + u11, u12, u13, u21, u22, u23, u31, u32, u33); +} + +#endif \ No newline at end of file -- cgit v1.2.3