mirror of
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1201 lines
30 KiB
C++
1201 lines
30 KiB
C++
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// This code is in the public domain -- Ignacio Castaño <castano@gmail.com>
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#include "fitting.h"
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#include "vector.inl"
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#include "plane.inl"
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#include "matrix.inl"
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#include "nvcore/array.inl"
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#include "nvcore/utils.h" // max, swap
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using namespace nv;
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// @@ Move to EigenSolver.h
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// @@ We should be able to do something cheaper...
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static Vector3 estimatePrincipalComponent(const float * __restrict matrix)
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{
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const Vector3 row0(matrix[0], matrix[1], matrix[2]);
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const Vector3 row1(matrix[1], matrix[3], matrix[4]);
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const Vector3 row2(matrix[2], matrix[4], matrix[5]);
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float r0 = lengthSquared(row0);
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float r1 = lengthSquared(row1);
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float r2 = lengthSquared(row2);
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if (r0 > r1 && r0 > r2) return row0;
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if (r1 > r2) return row1;
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return row2;
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}
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static inline Vector3 firstEigenVector_PowerMethod(const float *__restrict matrix)
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{
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if (matrix[0] == 0 && matrix[3] == 0 && matrix[5] == 0)
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{
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return Vector3(0.0f);
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}
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Vector3 v = estimatePrincipalComponent(matrix);
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const int NUM = 8;
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for (int i = 0; i < NUM; i++)
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{
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float x = v.x * matrix[0] + v.y * matrix[1] + v.z * matrix[2];
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float y = v.x * matrix[1] + v.y * matrix[3] + v.z * matrix[4];
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float z = v.x * matrix[2] + v.y * matrix[4] + v.z * matrix[5];
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float norm = max(max(x, y), z);
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v = Vector3(x, y, z) / norm;
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}
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return v;
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}
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Vector3 nv::Fit::computeCentroid(int n, const Vector3 *__restrict points)
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{
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Vector3 centroid(0.0f);
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for (int i = 0; i < n; i++)
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{
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centroid += points[i];
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}
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centroid /= float(n);
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return centroid;
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}
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Vector3 nv::Fit::computeCentroid(int n, const Vector3 *__restrict points, const float *__restrict weights, Vector3::Arg metric)
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{
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Vector3 centroid(0.0f);
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float total = 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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centroid /= total;
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return centroid;
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}
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Vector4 nv::Fit::computeCentroid(int n, const Vector4 *__restrict points)
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{
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Vector4 centroid(0.0f);
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for (int i = 0; i < n; i++)
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{
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centroid += points[i];
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}
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centroid /= float(n);
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return centroid;
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}
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Vector4 nv::Fit::computeCentroid(int n, const Vector4 *__restrict points, const float *__restrict weights, Vector4::Arg metric)
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{
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Vector4 centroid(0.0f);
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float total = 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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centroid /= total;
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return centroid;
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}
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Vector3 nv::Fit::computeCovariance(int n, const Vector3 *__restrict points, float *__restrict covariance)
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{
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// compute the centroid
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Vector3 centroid = computeCentroid(n, points);
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// compute covariance matrix
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for (int i = 0; i < 6; i++)
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{
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covariance[i] = 0.0f;
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}
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for (int i = 0; i < n; i++)
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{
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Vector3 v = points[i] - centroid;
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covariance[0] += v.x * v.x;
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covariance[1] += v.x * v.y;
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covariance[2] += v.x * v.z;
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covariance[3] += v.y * v.y;
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covariance[4] += v.y * v.z;
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covariance[5] += v.z * v.z;
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}
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return centroid;
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}
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Vector3 nv::Fit::computeCovariance(int n, const Vector3 *__restrict points, const float *__restrict weights, Vector3::Arg metric, float *__restrict covariance)
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{
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// compute the centroid
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Vector3 centroid = computeCentroid(n, points, weights, metric);
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// compute covariance matrix
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for (int i = 0; i < 6; i++)
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{
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covariance[i] = 0.0f;
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}
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for (int i = 0; i < n; i++)
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{
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Vector3 a = (points[i] - centroid) * metric;
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Vector3 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 centroid;
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}
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Vector4 nv::Fit::computeCovariance(int n, const Vector4 *__restrict points, float *__restrict covariance)
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{
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// compute the centroid
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Vector4 centroid = computeCentroid(n, points);
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// compute covariance matrix
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for (int i = 0; i < 10; i++)
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{
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covariance[i] = 0.0f;
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}
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for (int i = 0; i < n; i++)
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{
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Vector4 v = points[i] - centroid;
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covariance[0] += v.x * v.x;
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covariance[1] += v.x * v.y;
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covariance[2] += v.x * v.z;
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covariance[3] += v.x * v.w;
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covariance[4] += v.y * v.y;
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covariance[5] += v.y * v.z;
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covariance[6] += v.y * v.w;
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covariance[7] += v.z * v.z;
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covariance[8] += v.z * v.w;
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covariance[9] += v.w * v.w;
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}
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return centroid;
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}
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Vector4 nv::Fit::computeCovariance(int n, const Vector4 *__restrict points, const float *__restrict weights, Vector4::Arg metric, float *__restrict covariance)
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{
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// compute the centroid
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Vector4 centroid = computeCentroid(n, points, weights, metric);
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// compute covariance matrix
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for (int i = 0; i < 10; i++)
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{
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covariance[i] = 0.0f;
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}
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for (int i = 0; i < n; i++)
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{
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Vector4 a = (points[i] - centroid) * metric;
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Vector4 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.x * b.w;
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covariance[4] += a.y * b.y;
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covariance[5] += a.y * b.z;
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covariance[6] += a.y * b.w;
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covariance[7] += a.z * b.z;
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covariance[8] += a.z * b.w;
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covariance[9] += a.w * b.w;
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}
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return centroid;
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}
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Vector3 nv::Fit::computePrincipalComponent_PowerMethod(int n, const Vector3 *__restrict points)
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{
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float matrix[6];
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computeCovariance(n, points, matrix);
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return firstEigenVector_PowerMethod(matrix);
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}
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Vector3 nv::Fit::computePrincipalComponent_PowerMethod(int n, const Vector3 *__restrict points, const float *__restrict weights, Vector3::Arg metric)
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{
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float matrix[6];
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computeCovariance(n, points, weights, metric, matrix);
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return firstEigenVector_PowerMethod(matrix);
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}
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static inline Vector3 firstEigenVector_EigenSolver3(const float *__restrict matrix)
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{
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if (matrix[0] == 0 && matrix[3] == 0 && matrix[5] == 0)
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{
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return Vector3(0.0f);
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}
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float eigenValues[3];
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Vector3 eigenVectors[3];
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if (!nv::Fit::eigenSolveSymmetric3(matrix, eigenValues, eigenVectors))
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{
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return Vector3(0.0f);
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}
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return eigenVectors[0];
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}
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Vector3 nv::Fit::computePrincipalComponent_EigenSolver(int n, const Vector3 *__restrict points)
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{
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float matrix[6];
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computeCovariance(n, points, matrix);
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return firstEigenVector_EigenSolver3(matrix);
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}
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Vector3 nv::Fit::computePrincipalComponent_EigenSolver(int n, const Vector3 *__restrict points, const float *__restrict weights, Vector3::Arg metric)
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{
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float matrix[6];
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computeCovariance(n, points, weights, metric, matrix);
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return firstEigenVector_EigenSolver3(matrix);
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}
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static inline Vector4 firstEigenVector_EigenSolver4(const float *__restrict matrix)
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{
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if (matrix[0] == 0 && matrix[4] == 0 && matrix[7] == 0&& matrix[9] == 0)
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{
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return Vector4(0.0f);
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}
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float eigenValues[4];
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Vector4 eigenVectors[4];
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if (!nv::Fit::eigenSolveSymmetric4(matrix, eigenValues, eigenVectors))
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{
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return Vector4(0.0f);
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}
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return eigenVectors[0];
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}
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Vector4 nv::Fit::computePrincipalComponent_EigenSolver(int n, const Vector4 *__restrict points)
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{
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float matrix[10];
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computeCovariance(n, points, matrix);
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return firstEigenVector_EigenSolver4(matrix);
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}
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Vector4 nv::Fit::computePrincipalComponent_EigenSolver(int n, const Vector4 *__restrict points, const float *__restrict weights, Vector4::Arg metric)
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{
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float matrix[10];
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computeCovariance(n, points, weights, metric, matrix);
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return firstEigenVector_EigenSolver4(matrix);
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}
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void ArvoSVD(int rows, int cols, float * Q, float * diag, float * R);
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Vector3 nv::Fit::computePrincipalComponent_SVD(int n, const Vector3 *__restrict points)
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{
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// Store the points in an n x n matrix
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Array<float> Q; Q.resize(n*n, 0.0f);
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for (int i = 0; i < n; ++i)
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{
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Q[i*n+0] = points[i].x;
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Q[i*n+1] = points[i].y;
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Q[i*n+2] = points[i].z;
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}
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// Alloc space for the SVD outputs
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Array<float> diag; diag.resize(n, 0.0f);
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Array<float> R; R.resize(n*n, 0.0f);
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ArvoSVD(n, n, &Q[0], &diag[0], &R[0]);
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// Get the principal component
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return Vector3(R[0], R[1], R[2]);
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}
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Vector4 nv::Fit::computePrincipalComponent_SVD(int n, const Vector4 *__restrict points)
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{
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// Store the points in an n x n matrix
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Array<float> Q; Q.resize(n*n, 0.0f);
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for (int i = 0; i < n; ++i)
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{
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Q[i*n+0] = points[i].x;
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Q[i*n+1] = points[i].y;
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Q[i*n+2] = points[i].z;
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Q[i*n+3] = points[i].w;
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}
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// Alloc space for the SVD outputs
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Array<float> diag; diag.resize(n, 0.0f);
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Array<float> R; R.resize(n*n, 0.0f);
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ArvoSVD(n, n, &Q[0], &diag[0], &R[0]);
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// Get the principal component
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return Vector4(R[0], R[1], R[2], R[3]);
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}
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Plane nv::Fit::bestPlane(int n, const Vector3 *__restrict points)
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{
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// compute the centroid and covariance
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float matrix[6];
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Vector3 centroid = computeCovariance(n, points, matrix);
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if (matrix[0] == 0 && matrix[3] == 0 && matrix[5] == 0)
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{
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// If no plane defined, then return a horizontal plane.
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return Plane(Vector3(0, 0, 1), centroid);
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}
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float eigenValues[3];
|
||
|
|
Vector3 eigenVectors[3];
|
||
|
|
if (!eigenSolveSymmetric3(matrix, eigenValues, eigenVectors)) {
|
||
|
|
// If no plane defined, then return a horizontal plane.
|
||
|
|
return Plane(Vector3(0, 0, 1), centroid);
|
||
|
|
}
|
||
|
|
|
||
|
|
return Plane(eigenVectors[2], centroid);
|
||
|
|
}
|
||
|
|
|
||
|
|
bool nv::Fit::isPlanar(int n, const Vector3 * points, float epsilon/*=NV_EPSILON*/)
|
||
|
|
{
|
||
|
|
// compute the centroid and covariance
|
||
|
|
float matrix[6];
|
||
|
|
computeCovariance(n, points, matrix);
|
||
|
|
|
||
|
|
float eigenValues[3];
|
||
|
|
Vector3 eigenVectors[3];
|
||
|
|
if (!eigenSolveSymmetric3(matrix, eigenValues, eigenVectors)) {
|
||
|
|
return false;
|
||
|
|
}
|
||
|
|
|
||
|
|
return eigenValues[2] < epsilon;
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
// Tridiagonal solver from Charles Bloom.
|
||
|
|
// Householder transforms followed by QL decomposition.
|
||
|
|
// Seems to be based on the code from Numerical Recipes in C.
|
||
|
|
|
||
|
|
static void EigenSolver3_Tridiagonal(float mat[3][3], float * diag, float * subd);
|
||
|
|
static bool EigenSolver3_QLAlgorithm(float mat[3][3], float * diag, float * subd);
|
||
|
|
|
||
|
|
bool nv::Fit::eigenSolveSymmetric3(const float matrix[6], float eigenValues[3], Vector3 eigenVectors[3])
|
||
|
|
{
|
||
|
|
nvDebugCheck(matrix != NULL && eigenValues != NULL && eigenVectors != NULL);
|
||
|
|
|
||
|
|
float subd[3];
|
||
|
|
float diag[3];
|
||
|
|
float work[3][3];
|
||
|
|
|
||
|
|
work[0][0] = matrix[0];
|
||
|
|
work[0][1] = work[1][0] = matrix[1];
|
||
|
|
work[0][2] = work[2][0] = matrix[2];
|
||
|
|
work[1][1] = matrix[3];
|
||
|
|
work[1][2] = work[2][1] = matrix[4];
|
||
|
|
work[2][2] = matrix[5];
|
||
|
|
|
||
|
|
EigenSolver3_Tridiagonal(work, diag, subd);
|
||
|
|
if (!EigenSolver3_QLAlgorithm(work, diag, subd))
|
||
|
|
{
|
||
|
|
for (int i = 0; i < 3; i++) {
|
||
|
|
eigenValues[i] = 0;
|
||
|
|
eigenVectors[i] = Vector3(0);
|
||
|
|
}
|
||
|
|
return false;
|
||
|
|
}
|
||
|
|
|
||
|
|
for (int i = 0; i < 3; i++) {
|
||
|
|
eigenValues[i] = (float)diag[i];
|
||
|
|
}
|
||
|
|
|
||
|
|
// eigenvectors are the columns; make them the rows :
|
||
|
|
|
||
|
|
for (int i=0; i < 3; i++)
|
||
|
|
{
|
||
|
|
for (int j = 0; j < 3; j++)
|
||
|
|
{
|
||
|
|
eigenVectors[j].component[i] = (float) work[i][j];
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
// shuffle to sort by singular value :
|
||
|
|
if (eigenValues[2] > eigenValues[0] && eigenValues[2] > eigenValues[1])
|
||
|
|
{
|
||
|
|
swap(eigenValues[0], eigenValues[2]);
|
||
|
|
swap(eigenVectors[0], eigenVectors[2]);
|
||
|
|
}
|
||
|
|
if (eigenValues[1] > eigenValues[0])
|
||
|
|
{
|
||
|
|
swap(eigenValues[0], eigenValues[1]);
|
||
|
|
swap(eigenVectors[0], eigenVectors[1]);
|
||
|
|
}
|
||
|
|
if (eigenValues[2] > eigenValues[1])
|
||
|
|
{
|
||
|
|
swap(eigenValues[1], eigenValues[2]);
|
||
|
|
swap(eigenVectors[1], eigenVectors[2]);
|
||
|
|
}
|
||
|
|
|
||
|
|
nvDebugCheck(eigenValues[0] >= eigenValues[1] && eigenValues[0] >= eigenValues[2]);
|
||
|
|
nvDebugCheck(eigenValues[1] >= eigenValues[2]);
|
||
|
|
|
||
|
|
return true;
|
||
|
|
}
|
||
|
|
|
||
|
|
static void EigenSolver3_Tridiagonal(float mat[3][3], float * diag, float * subd)
|
||
|
|
{
|
||
|
|
// Householder reduction T = Q^t M Q
|
||
|
|
// Input:
|
||
|
|
// mat, symmetric 3x3 matrix M
|
||
|
|
// Output:
|
||
|
|
// mat, orthogonal matrix Q
|
||
|
|
// diag, diagonal entries of T
|
||
|
|
// subd, subdiagonal entries of T (T is symmetric)
|
||
|
|
const float epsilon = 1e-08f;
|
||
|
|
|
||
|
|
float a = mat[0][0];
|
||
|
|
float b = mat[0][1];
|
||
|
|
float c = mat[0][2];
|
||
|
|
float d = mat[1][1];
|
||
|
|
float e = mat[1][2];
|
||
|
|
float f = mat[2][2];
|
||
|
|
|
||
|
|
diag[0] = a;
|
||
|
|
subd[2] = 0.f;
|
||
|
|
if (fabsf(c) >= epsilon)
|
||
|
|
{
|
||
|
|
const float ell = sqrtf(b*b+c*c);
|
||
|
|
b /= ell;
|
||
|
|
c /= ell;
|
||
|
|
const float q = 2*b*e+c*(f-d);
|
||
|
|
diag[1] = d+c*q;
|
||
|
|
diag[2] = f-c*q;
|
||
|
|
subd[0] = ell;
|
||
|
|
subd[1] = e-b*q;
|
||
|
|
mat[0][0] = 1; mat[0][1] = 0; mat[0][2] = 0;
|
||
|
|
mat[1][0] = 0; mat[1][1] = b; mat[1][2] = c;
|
||
|
|
mat[2][0] = 0; mat[2][1] = c; mat[2][2] = -b;
|
||
|
|
}
|
||
|
|
else
|
||
|
|
{
|
||
|
|
diag[1] = d;
|
||
|
|
diag[2] = f;
|
||
|
|
subd[0] = b;
|
||
|
|
subd[1] = e;
|
||
|
|
mat[0][0] = 1; mat[0][1] = 0; mat[0][2] = 0;
|
||
|
|
mat[1][0] = 0; mat[1][1] = 1; mat[1][2] = 0;
|
||
|
|
mat[2][0] = 0; mat[2][1] = 0; mat[2][2] = 1;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
static bool EigenSolver3_QLAlgorithm(float mat[3][3], float * diag, float * subd)
|
||
|
|
{
|
||
|
|
// QL iteration with implicit shifting to reduce matrix from tridiagonal
|
||
|
|
// to diagonal
|
||
|
|
const int maxiter = 32;
|
||
|
|
|
||
|
|
for (int ell = 0; ell < 3; ell++)
|
||
|
|
{
|
||
|
|
int iter;
|
||
|
|
for (iter = 0; iter < maxiter; iter++)
|
||
|
|
{
|
||
|
|
int m;
|
||
|
|
for (m = ell; m <= 1; m++)
|
||
|
|
{
|
||
|
|
float dd = fabsf(diag[m]) + fabsf(diag[m+1]);
|
||
|
|
if ( fabsf(subd[m]) + dd == dd )
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
if ( m == ell )
|
||
|
|
break;
|
||
|
|
|
||
|
|
float g = (diag[ell+1]-diag[ell])/(2*subd[ell]);
|
||
|
|
float r = sqrtf(g*g+1);
|
||
|
|
if ( g < 0 )
|
||
|
|
g = diag[m]-diag[ell]+subd[ell]/(g-r);
|
||
|
|
else
|
||
|
|
g = diag[m]-diag[ell]+subd[ell]/(g+r);
|
||
|
|
float s = 1, c = 1, p = 0;
|
||
|
|
for (int i = m-1; i >= ell; i--)
|
||
|
|
{
|
||
|
|
float f = s*subd[i], b = c*subd[i];
|
||
|
|
if ( fabsf(f) >= fabsf(g) )
|
||
|
|
{
|
||
|
|
c = g/f;
|
||
|
|
r = sqrtf(c*c+1);
|
||
|
|
subd[i+1] = f*r;
|
||
|
|
c *= (s = 1/r);
|
||
|
|
}
|
||
|
|
else
|
||
|
|
{
|
||
|
|
s = f/g;
|
||
|
|
r = sqrtf(s*s+1);
|
||
|
|
subd[i+1] = g*r;
|
||
|
|
s *= (c = 1/r);
|
||
|
|
}
|
||
|
|
g = diag[i+1]-p;
|
||
|
|
r = (diag[i]-g)*s+2*b*c;
|
||
|
|
p = s*r;
|
||
|
|
diag[i+1] = g+p;
|
||
|
|
g = c*r-b;
|
||
|
|
|
||
|
|
for (int k = 0; k < 3; k++)
|
||
|
|
{
|
||
|
|
f = mat[k][i+1];
|
||
|
|
mat[k][i+1] = s*mat[k][i]+c*f;
|
||
|
|
mat[k][i] = c*mat[k][i]-s*f;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
diag[ell] -= p;
|
||
|
|
subd[ell] = g;
|
||
|
|
subd[m] = 0;
|
||
|
|
}
|
||
|
|
|
||
|
|
if ( iter == maxiter )
|
||
|
|
// should not get here under normal circumstances
|
||
|
|
return false;
|
||
|
|
}
|
||
|
|
|
||
|
|
return true;
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
// Tridiagonal solver for 4x4 symmetric matrices.
|
||
|
|
|
||
|
|
static void EigenSolver4_Tridiagonal(float mat[4][4], float * diag, float * subd);
|
||
|
|
static bool EigenSolver4_QLAlgorithm(float mat[4][4], float * diag, float * subd);
|
||
|
|
|
||
|
|
bool nv::Fit::eigenSolveSymmetric4(const float matrix[10], float eigenValues[4], Vector4 eigenVectors[4])
|
||
|
|
{
|
||
|
|
nvDebugCheck(matrix != NULL && eigenValues != NULL && eigenVectors != NULL);
|
||
|
|
|
||
|
|
float subd[4];
|
||
|
|
float diag[4];
|
||
|
|
float work[4][4];
|
||
|
|
|
||
|
|
work[0][0] = matrix[0];
|
||
|
|
work[0][1] = work[1][0] = matrix[1];
|
||
|
|
work[0][2] = work[2][0] = matrix[2];
|
||
|
|
work[0][3] = work[3][0] = matrix[3];
|
||
|
|
work[1][1] = matrix[4];
|
||
|
|
work[1][2] = work[2][1] = matrix[5];
|
||
|
|
work[1][3] = work[3][1] = matrix[6];
|
||
|
|
work[2][2] = matrix[7];
|
||
|
|
work[2][3] = work[3][2] = matrix[8];
|
||
|
|
work[3][3] = matrix[9];
|
||
|
|
|
||
|
|
EigenSolver4_Tridiagonal(work, diag, subd);
|
||
|
|
if (!EigenSolver4_QLAlgorithm(work, diag, subd))
|
||
|
|
{
|
||
|
|
for (int i = 0; i < 4; i++) {
|
||
|
|
eigenValues[i] = 0;
|
||
|
|
eigenVectors[i] = Vector4(0);
|
||
|
|
}
|
||
|
|
return false;
|
||
|
|
}
|
||
|
|
|
||
|
|
for (int i = 0; i < 4; i++) {
|
||
|
|
eigenValues[i] = (float)diag[i];
|
||
|
|
}
|
||
|
|
|
||
|
|
// eigenvectors are the columns; make them the rows
|
||
|
|
|
||
|
|
for (int i = 0; i < 4; i++)
|
||
|
|
{
|
||
|
|
for (int j = 0; j < 4; j++)
|
||
|
|
{
|
||
|
|
eigenVectors[j].component[i] = (float) work[i][j];
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
// sort by singular value
|
||
|
|
|
||
|
|
for (int i = 0; i < 3; ++i)
|
||
|
|
{
|
||
|
|
for (int j = i+1; j < 4; ++j)
|
||
|
|
{
|
||
|
|
if (eigenValues[j] > eigenValues[i])
|
||
|
|
{
|
||
|
|
swap(eigenValues[i], eigenValues[j]);
|
||
|
|
swap(eigenVectors[i], eigenVectors[j]);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
nvDebugCheck(eigenValues[0] >= eigenValues[1] && eigenValues[0] >= eigenValues[2] && eigenValues[0] >= eigenValues[3]);
|
||
|
|
nvDebugCheck(eigenValues[1] >= eigenValues[2] && eigenValues[1] >= eigenValues[3]);
|
||
|
|
nvDebugCheck(eigenValues[2] >= eigenValues[2]);
|
||
|
|
|
||
|
|
return true;
|
||
|
|
}
|
||
|
|
|
||
|
|
inline float signNonzero(float x)
|
||
|
|
{
|
||
|
|
return (x >= 0.0f) ? 1.0f : -1.0f;
|
||
|
|
}
|
||
|
|
|
||
|
|
static void EigenSolver4_Tridiagonal(float mat[4][4], float * diag, float * subd)
|
||
|
|
{
|
||
|
|
// Householder reduction T = Q^t M Q
|
||
|
|
// Input:
|
||
|
|
// mat, symmetric 3x3 matrix M
|
||
|
|
// Output:
|
||
|
|
// mat, orthogonal matrix Q
|
||
|
|
// diag, diagonal entries of T
|
||
|
|
// subd, subdiagonal entries of T (T is symmetric)
|
||
|
|
|
||
|
|
static const int n = 4;
|
||
|
|
|
||
|
|
// Set epsilon relative to size of elements in matrix
|
||
|
|
static const float relEpsilon = 1e-6f;
|
||
|
|
float maxElement = FLT_MAX;
|
||
|
|
for (int i = 0; i < n; ++i)
|
||
|
|
for (int j = 0; j < n; ++j)
|
||
|
|
maxElement = max(maxElement, fabsf(mat[i][j]));
|
||
|
|
float epsilon = relEpsilon * maxElement;
|
||
|
|
|
||
|
|
// Iterative algorithm, works for any size of matrix but might be slower than
|
||
|
|
// a closed-form solution for symmetric 4x4 matrices. Based on this article:
|
||
|
|
// http://en.wikipedia.org/wiki/Householder_transformation#Tridiagonalization
|
||
|
|
|
||
|
|
Matrix A, Q(identity);
|
||
|
|
memcpy(&A, mat, sizeof(float)*n*n);
|
||
|
|
|
||
|
|
// We proceed from left to right, making the off-tridiagonal entries zero in
|
||
|
|
// one column of the matrix at a time.
|
||
|
|
for (int k = 0; k < n - 2; ++k)
|
||
|
|
{
|
||
|
|
float sum = 0.0f;
|
||
|
|
for (int j = k+1; j < n; ++j)
|
||
|
|
sum += A(j,k)*A(j,k);
|
||
|
|
float alpha = -signNonzero(A(k+1,k)) * sqrtf(sum);
|
||
|
|
float r = sqrtf(0.5f * (alpha*alpha - A(k+1,k)*alpha));
|
||
|
|
|
||
|
|
// If r is zero, skip this column - already in tridiagonal form
|
||
|
|
if (fabsf(r) < epsilon)
|
||
|
|
continue;
|
||
|
|
|
||
|
|
float v[n] = {};
|
||
|
|
v[k+1] = 0.5f * (A(k+1,k) - alpha) / r;
|
||
|
|
for (int j = k+2; j < n; ++j)
|
||
|
|
v[j] = 0.5f * A(j,k) / r;
|
||
|
|
|
||
|
|
Matrix P(identity);
|
||
|
|
for (int i = 0; i < n; ++i)
|
||
|
|
for (int j = 0; j < n; ++j)
|
||
|
|
P(i,j) -= 2.0f * v[i] * v[j];
|
||
|
|
|
||
|
|
A = mul(mul(P, A), P);
|
||
|
|
Q = mul(Q, P);
|
||
|
|
}
|
||
|
|
|
||
|
|
nvDebugCheck(fabsf(A(2,0)) < epsilon);
|
||
|
|
nvDebugCheck(fabsf(A(0,2)) < epsilon);
|
||
|
|
nvDebugCheck(fabsf(A(3,0)) < epsilon);
|
||
|
|
nvDebugCheck(fabsf(A(0,3)) < epsilon);
|
||
|
|
nvDebugCheck(fabsf(A(3,1)) < epsilon);
|
||
|
|
nvDebugCheck(fabsf(A(1,3)) < epsilon);
|
||
|
|
|
||
|
|
for (int i = 0; i < n; ++i)
|
||
|
|
diag[i] = A(i,i);
|
||
|
|
for (int i = 0; i < n - 1; ++i)
|
||
|
|
subd[i] = A(i+1,i);
|
||
|
|
subd[n-1] = 0.0f;
|
||
|
|
|
||
|
|
memcpy(mat, &Q, sizeof(float)*n*n);
|
||
|
|
}
|
||
|
|
|
||
|
|
static bool EigenSolver4_QLAlgorithm(float mat[4][4], float * diag, float * subd)
|
||
|
|
{
|
||
|
|
// QL iteration with implicit shifting to reduce matrix from tridiagonal
|
||
|
|
// to diagonal
|
||
|
|
const int maxiter = 32;
|
||
|
|
|
||
|
|
for (int ell = 0; ell < 4; ell++)
|
||
|
|
{
|
||
|
|
int iter;
|
||
|
|
for (iter = 0; iter < maxiter; iter++)
|
||
|
|
{
|
||
|
|
int m;
|
||
|
|
for (m = ell; m < 3; m++)
|
||
|
|
{
|
||
|
|
float dd = fabsf(diag[m]) + fabsf(diag[m+1]);
|
||
|
|
if ( fabsf(subd[m]) + dd == dd )
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
if ( m == ell )
|
||
|
|
break;
|
||
|
|
|
||
|
|
float g = (diag[ell+1]-diag[ell])/(2*subd[ell]);
|
||
|
|
float r = sqrtf(g*g+1);
|
||
|
|
if ( g < 0 )
|
||
|
|
g = diag[m]-diag[ell]+subd[ell]/(g-r);
|
||
|
|
else
|
||
|
|
g = diag[m]-diag[ell]+subd[ell]/(g+r);
|
||
|
|
float s = 1, c = 1, p = 0;
|
||
|
|
for (int i = m-1; i >= ell; i--)
|
||
|
|
{
|
||
|
|
float f = s*subd[i], b = c*subd[i];
|
||
|
|
if ( fabsf(f) >= fabsf(g) )
|
||
|
|
{
|
||
|
|
c = g/f;
|
||
|
|
r = sqrtf(c*c+1);
|
||
|
|
subd[i+1] = f*r;
|
||
|
|
c *= (s = 1/r);
|
||
|
|
}
|
||
|
|
else
|
||
|
|
{
|
||
|
|
s = f/g;
|
||
|
|
r = sqrtf(s*s+1);
|
||
|
|
subd[i+1] = g*r;
|
||
|
|
s *= (c = 1/r);
|
||
|
|
}
|
||
|
|
g = diag[i+1]-p;
|
||
|
|
r = (diag[i]-g)*s+2*b*c;
|
||
|
|
p = s*r;
|
||
|
|
diag[i+1] = g+p;
|
||
|
|
g = c*r-b;
|
||
|
|
|
||
|
|
for (int k = 0; k < 4; k++)
|
||
|
|
{
|
||
|
|
f = mat[k][i+1];
|
||
|
|
mat[k][i+1] = s*mat[k][i]+c*f;
|
||
|
|
mat[k][i] = c*mat[k][i]-s*f;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
diag[ell] -= p;
|
||
|
|
subd[ell] = g;
|
||
|
|
subd[m] = 0;
|
||
|
|
}
|
||
|
|
|
||
|
|
if ( iter == maxiter )
|
||
|
|
// should not get here under normal circumstances
|
||
|
|
return false;
|
||
|
|
}
|
||
|
|
|
||
|
|
return true;
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
int nv::Fit::compute4Means(int n, const Vector3 *__restrict points, const float *__restrict weights, Vector3::Arg metric, Vector3 *__restrict cluster)
|
||
|
|
{
|
||
|
|
// Compute principal component.
|
||
|
|
float matrix[6];
|
||
|
|
Vector3 centroid = computeCovariance(n, points, weights, metric, matrix);
|
||
|
|
Vector3 principal = firstEigenVector_PowerMethod(matrix);
|
||
|
|
|
||
|
|
// Pick initial solution.
|
||
|
|
int mini, maxi;
|
||
|
|
mini = maxi = 0;
|
||
|
|
|
||
|
|
float mindps, maxdps;
|
||
|
|
mindps = maxdps = dot(points[0] - centroid, principal);
|
||
|
|
|
||
|
|
for (int i = 1; i < n; ++i)
|
||
|
|
{
|
||
|
|
float dps = dot(points[i] - centroid, principal);
|
||
|
|
|
||
|
|
if (dps < mindps) {
|
||
|
|
mindps = dps;
|
||
|
|
mini = i;
|
||
|
|
}
|
||
|
|
else {
|
||
|
|
maxdps = dps;
|
||
|
|
maxi = i;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
cluster[0] = centroid + mindps * principal;
|
||
|
|
cluster[1] = centroid + maxdps * principal;
|
||
|
|
cluster[2] = (2.0f * cluster[0] + cluster[1]) / 3.0f;
|
||
|
|
cluster[3] = (2.0f * cluster[1] + cluster[0]) / 3.0f;
|
||
|
|
|
||
|
|
// Now we have to iteratively refine the clusters.
|
||
|
|
while (true)
|
||
|
|
{
|
||
|
|
Vector3 newCluster[4] = { Vector3(0.0f), Vector3(0.0f), Vector3(0.0f), Vector3(0.0f) };
|
||
|
|
float total[4] = {0, 0, 0, 0};
|
||
|
|
|
||
|
|
for (int i = 0; i < n; ++i)
|
||
|
|
{
|
||
|
|
// Find nearest cluster.
|
||
|
|
int nearest = 0;
|
||
|
|
float mindist = FLT_MAX;
|
||
|
|
for (int j = 0; j < 4; j++)
|
||
|
|
{
|
||
|
|
float dist = lengthSquared((cluster[j] - points[i]) * metric);
|
||
|
|
if (dist < mindist)
|
||
|
|
{
|
||
|
|
mindist = dist;
|
||
|
|
nearest = j;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
newCluster[nearest] += weights[i] * points[i];
|
||
|
|
total[nearest] += weights[i];
|
||
|
|
}
|
||
|
|
|
||
|
|
for (int j = 0; j < 4; j++)
|
||
|
|
{
|
||
|
|
if (total[j] != 0)
|
||
|
|
newCluster[j] /= total[j];
|
||
|
|
}
|
||
|
|
|
||
|
|
if (equal(cluster[0], newCluster[0]) && equal(cluster[1], newCluster[1]) &&
|
||
|
|
equal(cluster[2], newCluster[2]) && equal(cluster[3], newCluster[3]))
|
||
|
|
{
|
||
|
|
return (total[0] != 0) + (total[1] != 0) + (total[2] != 0) + (total[3] != 0);
|
||
|
|
}
|
||
|
|
|
||
|
|
cluster[0] = newCluster[0];
|
||
|
|
cluster[1] = newCluster[1];
|
||
|
|
cluster[2] = newCluster[2];
|
||
|
|
cluster[3] = newCluster[3];
|
||
|
|
|
||
|
|
// Sort clusters by weight.
|
||
|
|
for (int i = 0; i < 4; i++)
|
||
|
|
{
|
||
|
|
for (int j = i; j > 0 && total[j] > total[j - 1]; j--)
|
||
|
|
{
|
||
|
|
swap( total[j], total[j - 1] );
|
||
|
|
swap( cluster[j], cluster[j - 1] );
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
|
||
|
|
// Adaptation of James Arvo's SVD code, as found in ZOH.
|
||
|
|
|
||
|
|
inline float Sqr(float x) { return x*x; }
|
||
|
|
|
||
|
|
inline float svd_pythag( float a, float b )
|
||
|
|
{
|
||
|
|
float at = fabsf(a);
|
||
|
|
float bt = fabsf(b);
|
||
|
|
if( at > bt )
|
||
|
|
return at * sqrtf( 1.0f + Sqr( bt / at ) );
|
||
|
|
else if( bt > 0.0f )
|
||
|
|
return bt * sqrtf( 1.0f + Sqr( at / bt ) );
|
||
|
|
else return 0.0f;
|
||
|
|
}
|
||
|
|
|
||
|
|
inline float SameSign( float a, float b )
|
||
|
|
{
|
||
|
|
float t;
|
||
|
|
if( b >= 0.0f ) t = fabsf( a );
|
||
|
|
else t = -fabsf( a );
|
||
|
|
return t;
|
||
|
|
}
|
||
|
|
|
||
|
|
void ArvoSVD(int rows, int cols, float * Q, float * diag, float * R)
|
||
|
|
{
|
||
|
|
static const int MaxIterations = 30;
|
||
|
|
|
||
|
|
int i, j, k, l, p, q, iter;
|
||
|
|
float c, f, h, s, x, y, z;
|
||
|
|
float norm = 0.0f;
|
||
|
|
float g = 0.0f;
|
||
|
|
float scale = 0.0f;
|
||
|
|
|
||
|
|
Array<float> temp; temp.resize(cols, 0.0f);
|
||
|
|
|
||
|
|
for( i = 0; i < cols; i++ )
|
||
|
|
{
|
||
|
|
temp[i] = scale * g;
|
||
|
|
scale = 0.0f;
|
||
|
|
g = 0.0f;
|
||
|
|
s = 0.0f;
|
||
|
|
l = i + 1;
|
||
|
|
|
||
|
|
if( i < rows )
|
||
|
|
{
|
||
|
|
for( k = i; k < rows; k++ ) scale += fabsf( Q[k*cols+i] );
|
||
|
|
if( scale != 0.0f )
|
||
|
|
{
|
||
|
|
for( k = i; k < rows; k++ )
|
||
|
|
{
|
||
|
|
Q[k*cols+i] /= scale;
|
||
|
|
s += Sqr( Q[k*cols+i] );
|
||
|
|
}
|
||
|
|
f = Q[i*cols+i];
|
||
|
|
g = -SameSign( sqrtf(s), f );
|
||
|
|
h = f * g - s;
|
||
|
|
Q[i*cols+i] = f - g;
|
||
|
|
if( i != cols - 1 )
|
||
|
|
{
|
||
|
|
for( j = l; j < cols; j++ )
|
||
|
|
{
|
||
|
|
s = 0.0f;
|
||
|
|
for( k = i; k < rows; k++ ) s += Q[k*cols+i] * Q[k*cols+j];
|
||
|
|
f = s / h;
|
||
|
|
for( k = i; k < rows; k++ ) Q[k*cols+j] += f * Q[k*cols+i];
|
||
|
|
}
|
||
|
|
}
|
||
|
|
for( k = i; k < rows; k++ ) Q[k*cols+i] *= scale;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
diag[i] = scale * g;
|
||
|
|
g = 0.0f;
|
||
|
|
s = 0.0f;
|
||
|
|
scale = 0.0f;
|
||
|
|
|
||
|
|
if( i < rows && i != cols - 1 )
|
||
|
|
{
|
||
|
|
for( k = l; k < cols; k++ ) scale += fabsf( Q[i*cols+k] );
|
||
|
|
if( scale != 0.0f )
|
||
|
|
{
|
||
|
|
for( k = l; k < cols; k++ )
|
||
|
|
{
|
||
|
|
Q[i*cols+k] /= scale;
|
||
|
|
s += Sqr( Q[i*cols+k] );
|
||
|
|
}
|
||
|
|
f = Q[i*cols+l];
|
||
|
|
g = -SameSign( sqrtf(s), f );
|
||
|
|
h = f * g - s;
|
||
|
|
Q[i*cols+l] = f - g;
|
||
|
|
for( k = l; k < cols; k++ ) temp[k] = Q[i*cols+k] / h;
|
||
|
|
if( i != rows - 1 )
|
||
|
|
{
|
||
|
|
for( j = l; j < rows; j++ )
|
||
|
|
{
|
||
|
|
s = 0.0f;
|
||
|
|
for( k = l; k < cols; k++ ) s += Q[j*cols+k] * Q[i*cols+k];
|
||
|
|
for( k = l; k < cols; k++ ) Q[j*cols+k] += s * temp[k];
|
||
|
|
}
|
||
|
|
}
|
||
|
|
for( k = l; k < cols; k++ ) Q[i*cols+k] *= scale;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
norm = max( norm, fabsf( diag[i] ) + fabsf( temp[i] ) );
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
for( i = cols - 1; i >= 0; i-- )
|
||
|
|
{
|
||
|
|
if( i < cols - 1 )
|
||
|
|
{
|
||
|
|
if( g != 0.0f )
|
||
|
|
{
|
||
|
|
for( j = l; j < cols; j++ ) R[i*cols+j] = ( Q[i*cols+j] / Q[i*cols+l] ) / g;
|
||
|
|
for( j = l; j < cols; j++ )
|
||
|
|
{
|
||
|
|
s = 0.0f;
|
||
|
|
for( k = l; k < cols; k++ ) s += Q[i*cols+k] * R[j*cols+k];
|
||
|
|
for( k = l; k < cols; k++ ) R[j*cols+k] += s * R[i*cols+k];
|
||
|
|
}
|
||
|
|
}
|
||
|
|
for( j = l; j < cols; j++ )
|
||
|
|
{
|
||
|
|
R[i*cols+j] = 0.0f;
|
||
|
|
R[j*cols+i] = 0.0f;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
R[i*cols+i] = 1.0f;
|
||
|
|
g = temp[i];
|
||
|
|
l = i;
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
for( i = cols - 1; i >= 0; i-- )
|
||
|
|
{
|
||
|
|
l = i + 1;
|
||
|
|
g = diag[i];
|
||
|
|
if( i < cols - 1 ) for( j = l; j < cols; j++ ) Q[i*cols+j] = 0.0f;
|
||
|
|
if( g != 0.0f )
|
||
|
|
{
|
||
|
|
g = 1.0f / g;
|
||
|
|
if( i != cols - 1 )
|
||
|
|
{
|
||
|
|
for( j = l; j < cols; j++ )
|
||
|
|
{
|
||
|
|
s = 0.0f;
|
||
|
|
for( k = l; k < rows; k++ ) s += Q[k*cols+i] * Q[k*cols+j];
|
||
|
|
f = ( s / Q[i*cols+i] ) * g;
|
||
|
|
for( k = i; k < rows; k++ ) Q[k*cols+j] += f * Q[k*cols+i];
|
||
|
|
}
|
||
|
|
}
|
||
|
|
for( j = i; j < rows; j++ ) Q[j*cols+i] *= g;
|
||
|
|
}
|
||
|
|
else
|
||
|
|
{
|
||
|
|
for( j = i; j < rows; j++ ) Q[j*cols+i] = 0.0f;
|
||
|
|
}
|
||
|
|
Q[i*cols+i] += 1.0f;
|
||
|
|
}
|
||
|
|
|
||
|
|
|
||
|
|
for( k = cols - 1; k >= 0; k-- )
|
||
|
|
{
|
||
|
|
for( iter = 1; iter <= MaxIterations; iter++ )
|
||
|
|
{
|
||
|
|
int jump;
|
||
|
|
|
||
|
|
for( l = k; l >= 0; l-- )
|
||
|
|
{
|
||
|
|
q = l - 1;
|
||
|
|
if( fabsf( temp[l] ) + norm == norm ) { jump = 1; break; }
|
||
|
|
if( fabsf( diag[q] ) + norm == norm ) { jump = 0; break; }
|
||
|
|
}
|
||
|
|
|
||
|
|
if( !jump )
|
||
|
|
{
|
||
|
|
c = 0.0f;
|
||
|
|
s = 1.0f;
|
||
|
|
for( i = l; i <= k; i++ )
|
||
|
|
{
|
||
|
|
f = s * temp[i];
|
||
|
|
temp[i] *= c;
|
||
|
|
if( fabsf( f ) + norm == norm ) break;
|
||
|
|
g = diag[i];
|
||
|
|
h = svd_pythag( f, g );
|
||
|
|
diag[i] = h;
|
||
|
|
h = 1.0f / h;
|
||
|
|
c = g * h;
|
||
|
|
s = -f * h;
|
||
|
|
for( j = 0; j < rows; j++ )
|
||
|
|
{
|
||
|
|
y = Q[j*cols+q];
|
||
|
|
z = Q[j*cols+i];
|
||
|
|
Q[j*cols+q] = y * c + z * s;
|
||
|
|
Q[j*cols+i] = z * c - y * s;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
z = diag[k];
|
||
|
|
if( l == k )
|
||
|
|
{
|
||
|
|
if( z < 0.0f )
|
||
|
|
{
|
||
|
|
diag[k] = -z;
|
||
|
|
for( j = 0; j < cols; j++ ) R[k*cols+j] *= -1.0f;
|
||
|
|
}
|
||
|
|
break;
|
||
|
|
}
|
||
|
|
if( iter >= MaxIterations ) return;
|
||
|
|
x = diag[l];
|
||
|
|
q = k - 1;
|
||
|
|
y = diag[q];
|
||
|
|
g = temp[q];
|
||
|
|
h = temp[k];
|
||
|
|
f = ( ( y - z ) * ( y + z ) + ( g - h ) * ( g + h ) ) / ( 2.0f * h * y );
|
||
|
|
g = svd_pythag( f, 1.0f );
|
||
|
|
f = ( ( x - z ) * ( x + z ) + h * ( ( y / ( f + SameSign( g, f ) ) ) - h ) ) / x;
|
||
|
|
c = 1.0f;
|
||
|
|
s = 1.0f;
|
||
|
|
for( j = l; j <= q; j++ )
|
||
|
|
{
|
||
|
|
i = j + 1;
|
||
|
|
g = temp[i];
|
||
|
|
y = diag[i];
|
||
|
|
h = s * g;
|
||
|
|
g = c * g;
|
||
|
|
z = svd_pythag( f, h );
|
||
|
|
temp[j] = z;
|
||
|
|
c = f / z;
|
||
|
|
s = h / z;
|
||
|
|
f = x * c + g * s;
|
||
|
|
g = g * c - x * s;
|
||
|
|
h = y * s;
|
||
|
|
y = y * c;
|
||
|
|
for( p = 0; p < cols; p++ )
|
||
|
|
{
|
||
|
|
x = R[j*cols+p];
|
||
|
|
z = R[i*cols+p];
|
||
|
|
R[j*cols+p] = x * c + z * s;
|
||
|
|
R[i*cols+p] = z * c - x * s;
|
||
|
|
}
|
||
|
|
z = svd_pythag( f, h );
|
||
|
|
diag[j] = z;
|
||
|
|
if( z != 0.0f )
|
||
|
|
{
|
||
|
|
z = 1.0f / z;
|
||
|
|
c = f * z;
|
||
|
|
s = h * z;
|
||
|
|
}
|
||
|
|
f = c * g + s * y;
|
||
|
|
x = c * y - s * g;
|
||
|
|
for( p = 0; p < rows; p++ )
|
||
|
|
{
|
||
|
|
y = Q[p*cols+j];
|
||
|
|
z = Q[p*cols+i];
|
||
|
|
Q[p*cols+j] = y * c + z * s;
|
||
|
|
Q[p*cols+i] = z * c - y * s;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
temp[l] = 0.0f;
|
||
|
|
temp[k] = f;
|
||
|
|
diag[k] = x;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
|
||
|
|
// Sort the singular values into descending order.
|
||
|
|
|
||
|
|
for( i = 0; i < cols - 1; i++ )
|
||
|
|
{
|
||
|
|
float biggest = diag[i]; // Biggest singular value so far.
|
||
|
|
int bindex = i; // The row/col it occurred in.
|
||
|
|
for( j = i + 1; j < cols; j++ )
|
||
|
|
{
|
||
|
|
if( diag[j] > biggest )
|
||
|
|
{
|
||
|
|
biggest = diag[j];
|
||
|
|
bindex = j;
|
||
|
|
}
|
||
|
|
}
|
||
|
|
if( bindex != i ) // Need to swap rows and columns.
|
||
|
|
{
|
||
|
|
// Swap columns in Q.
|
||
|
|
for (int j = 0; j < rows; ++j)
|
||
|
|
swap(Q[j*cols+i], Q[j*cols+bindex]);
|
||
|
|
|
||
|
|
// Swap rows in R.
|
||
|
|
for (int j = 0; j < rows; ++j)
|
||
|
|
swap(R[i*cols+j], R[bindex*cols+j]);
|
||
|
|
|
||
|
|
// Swap elements in diag.
|
||
|
|
swap(diag[i], diag[bindex]);
|
||
|
|
}
|
||
|
|
}
|
||
|
|
}
|