/* ----------------------------------------------------------------------------- Copyright (c) 2006 Simon Brown si@sjbrown.co.uk Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions: The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software. THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. -------------------------------------------------------------------------- */ /*! @file The symmetric eigensystem solver algorithm is from http://www.geometrictools.com/Documentation/EigenSymmetric3x3.pdf */ #include "maths.h" #include "simd.h" #include <cfloat> namespace squish { Sym3x3 ComputeWeightedCovariance( int n, Vec3 const* points, float const* weights ) { // compute the centroid float total = 0.0f; Vec3 centroid( 0.0f ); for( int i = 0; i < n; ++i ) { total += weights[i]; centroid += weights[i]*points[i]; } if( total > FLT_EPSILON ) centroid /= total; // accumulate the covariance matrix Sym3x3 covariance( 0.0f ); for( int i = 0; i < n; ++i ) { Vec3 a = points[i] - centroid; Vec3 b = weights[i]*a; covariance[0] += a.X()*b.X(); covariance[1] += a.X()*b.Y(); covariance[2] += a.X()*b.Z(); covariance[3] += a.Y()*b.Y(); covariance[4] += a.Y()*b.Z(); covariance[5] += a.Z()*b.Z(); } // return it return covariance; } #if 0 static Vec3 GetMultiplicity1Evector( Sym3x3 const& matrix, float evalue ) { // compute M Sym3x3 m; m[0] = matrix[0] - evalue; m[1] = matrix[1]; m[2] = matrix[2]; m[3] = matrix[3] - evalue; m[4] = matrix[4]; m[5] = matrix[5] - evalue; // compute U Sym3x3 u; u[0] = m[3]*m[5] - m[4]*m[4]; u[1] = m[2]*m[4] - m[1]*m[5]; u[2] = m[1]*m[4] - m[2]*m[3]; u[3] = m[0]*m[5] - m[2]*m[2]; u[4] = m[1]*m[2] - m[4]*m[0]; u[5] = m[0]*m[3] - m[1]*m[1]; // find the largest component float mc = std::fabs( u[0] ); int mi = 0; for( int i = 1; i < 6; ++i ) { float c = std::fabs( u[i] ); if( c > mc ) { mc = c; mi = i; } } // pick the column with this component switch( mi ) { case 0: return Vec3( u[0], u[1], u[2] ); case 1: case 3: return Vec3( u[1], u[3], u[4] ); default: return Vec3( u[2], u[4], u[5] ); } } static Vec3 GetMultiplicity2Evector( Sym3x3 const& matrix, float evalue ) { // compute M Sym3x3 m; m[0] = matrix[0] - evalue; m[1] = matrix[1]; m[2] = matrix[2]; m[3] = matrix[3] - evalue; m[4] = matrix[4]; m[5] = matrix[5] - evalue; // find the largest component float mc = std::fabs( m[0] ); int mi = 0; for( int i = 1; i < 6; ++i ) { float c = std::fabs( m[i] ); if( c > mc ) { mc = c; mi = i; } } // pick the first eigenvector based on this index switch( mi ) { case 0: case 1: return Vec3( -m[1], m[0], 0.0f ); case 2: return Vec3( m[2], 0.0f, -m[0] ); case 3: case 4: return Vec3( 0.0f, -m[4], m[3] ); default: return Vec3( 0.0f, -m[5], m[4] ); } } Vec3 ComputePrincipleComponent( Sym3x3 const& matrix ) { // compute the cubic coefficients float c0 = matrix[0]*matrix[3]*matrix[5] + 2.0f*matrix[1]*matrix[2]*matrix[4] - matrix[0]*matrix[4]*matrix[4] - matrix[3]*matrix[2]*matrix[2] - matrix[5]*matrix[1]*matrix[1]; float c1 = matrix[0]*matrix[3] + matrix[0]*matrix[5] + matrix[3]*matrix[5] - matrix[1]*matrix[1] - matrix[2]*matrix[2] - matrix[4]*matrix[4]; float c2 = matrix[0] + matrix[3] + matrix[5]; // compute the quadratic coefficients float a = c1 - ( 1.0f/3.0f )*c2*c2; float b = ( -2.0f/27.0f )*c2*c2*c2 + ( 1.0f/3.0f )*c1*c2 - c0; // compute the root count check float Q = 0.25f*b*b + ( 1.0f/27.0f )*a*a*a; // test the multiplicity if( FLT_EPSILON < Q ) { // only one root, which implies we have a multiple of the identity return Vec3( 1.0f ); } else if( Q < -FLT_EPSILON ) { // three distinct roots float theta = std::atan2( std::sqrt( -Q ), -0.5f*b ); float rho = std::sqrt( 0.25f*b*b - Q ); float rt = std::pow( rho, 1.0f/3.0f ); float ct = std::cos( theta/3.0f ); float st = std::sin( theta/3.0f ); float l1 = ( 1.0f/3.0f )*c2 + 2.0f*rt*ct; float l2 = ( 1.0f/3.0f )*c2 - rt*( ct + ( float )sqrt( 3.0f )*st ); float l3 = ( 1.0f/3.0f )*c2 - rt*( ct - ( float )sqrt( 3.0f )*st ); // pick the larger if( std::fabs( l2 ) > std::fabs( l1 ) ) l1 = l2; if( std::fabs( l3 ) > std::fabs( l1 ) ) l1 = l3; // get the eigenvector return GetMultiplicity1Evector( matrix, l1 ); } else // if( -FLT_EPSILON <= Q && Q <= FLT_EPSILON ) { // two roots float rt; if( b < 0.0f ) rt = -std::pow( -0.5f*b, 1.0f/3.0f ); else rt = std::pow( 0.5f*b, 1.0f/3.0f ); float l1 = ( 1.0f/3.0f )*c2 + rt; // repeated float l2 = ( 1.0f/3.0f )*c2 - 2.0f*rt; // get the eigenvector if( std::fabs( l1 ) > std::fabs( l2 ) ) return GetMultiplicity2Evector( matrix, l1 ); else return GetMultiplicity1Evector( matrix, l2 ); } } #else #define POWER_ITERATION_COUNT 8 Vec3 ComputePrincipleComponent( Sym3x3 const& matrix ) { Vec4 const row0( matrix[0], matrix[1], matrix[2], 0.0f ); Vec4 const row1( matrix[1], matrix[3], matrix[4], 0.0f ); Vec4 const row2( matrix[2], matrix[4], matrix[5], 0.0f ); Vec4 v = VEC4_CONST( 1.0f ); for( int i = 0; i < POWER_ITERATION_COUNT; ++i ) { // matrix multiply Vec4 w = row0*v.SplatX(); w = MultiplyAdd(row1, v.SplatY(), w); w = MultiplyAdd(row2, v.SplatZ(), w); // get max component from xyz in all channels Vec4 a = Max(w.SplatX(), Max(w.SplatY(), w.SplatZ())); // divide through and advance v = w*Reciprocal(a); } return v.GetVec3(); } #endif } // namespace squish