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