// Ceres Solver - A fast non-linear least squares minimizer
// Copyright 2015 Google Inc. All rights reserved.
// http://ceres-solver.org/
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// Author: sameeragarwal@google.com (Sameer Agarwal)
#ifndef CERES_PUBLIC_CUBIC_INTERPOLATION_H_
#define CERES_PUBLIC_CUBIC_INTERPOLATION_H_
#include "ceres/internal/port.h"
#include "Eigen/Core"
#include "glog/logging.h"
namespace ceres {
// Given samples from a function sampled at four equally spaced points,
//
// p0 = f(-1)
// p1 = f(0)
// p2 = f(1)
// p3 = f(2)
//
// Evaluate the cubic Hermite spline (also known as the Catmull-Rom
// spline) at a point x that lies in the interval [0, 1].
//
// This is also the interpolation kernel (for the case of a = 0.5) as
// proposed by R. Keys, in:
//
// "Cubic convolution interpolation for digital image processing".
// IEEE Transactions on Acoustics, Speech, and Signal Processing
// 29 (6): 1153–1160.
//
// For more details see
//
// http://en.wikipedia.org/wiki/Cubic_Hermite_spline
// http://en.wikipedia.org/wiki/Bicubic_interpolation
//
// f if not NULL will contain the interpolated function values.
// dfdx if not NULL will contain the interpolated derivative values.
template <int kDataDimension>
void CubicHermiteSpline(const Eigen::Matrix<double, kDataDimension, 1>& p0,
const Eigen::Matrix<double, kDataDimension, 1>& p1,
const Eigen::Matrix<double, kDataDimension, 1>& p2,
const Eigen::Matrix<double, kDataDimension, 1>& p3,
const double x,
double* f,
double* dfdx) {
typedef Eigen::Matrix<double, kDataDimension, 1> VType;
const VType a = 0.5 * (-p0 + 3.0 * p1 - 3.0 * p2 + p3);
const VType b = 0.5 * (2.0 * p0 - 5.0 * p1 + 4.0 * p2 - p3);
const VType c = 0.5 * (-p0 + p2);
const VType d = p1;
// Use Horner's rule to evaluate the function value and its
// derivative.
// f = ax^3 + bx^2 + cx + d
if (f != NULL) {
Eigen::Map<VType>(f, kDataDimension) = d + x * (c + x * (b + x * a));
}
// dfdx = 3ax^2 + 2bx + c
if (dfdx != NULL) {
Eigen::Map<VType>(dfdx, kDataDimension) = c + x * (2.0 * b + 3.0 * a * x);
}
}
// Given as input an infinite one dimensional grid, which provides the
// following interface.
//
// class Grid {
// public:
// enum { DATA_DIMENSION = 2; };
// void GetValue(int n, double* f) const;
// };
//
// Here, GetValue gives the value of a function f (possibly vector
// valued) for any integer n.
//
// The enum DATA_DIMENSION indicates the dimensionality of the
// function being interpolated. For example if you are interpolating
// rotations in axis-angle format over time, then DATA_DIMENSION = 3.
//
// CubicInterpolator uses cubic Hermite splines to produce a smooth
// approximation to it that can be used to evaluate the f(x) and f'(x)
// at any point on the real number line.
//
// For more details on cubic interpolation see
//
// http://en.wikipedia.org/wiki/Cubic_Hermite_spline
//
// Example usage:
//
// const double data[] = {1.0, 2.0, 5.0, 6.0};
// Grid1D<double, 1> grid(x, 0, 4);
// CubicInterpolator<Grid1D<double, 1> > interpolator(grid);
// double f, dfdx;
// interpolator.Evaluator(1.5, &f, &dfdx);
template<typename Grid>
class CubicInterpolator {
public:
explicit CubicInterpolator(const Grid& grid)
: grid_(grid) {
// The + casts the enum into an int before doing the
// comparison. It is needed to prevent
// "-Wunnamed-type-template-args" related errors.
CHECK_GE(+Grid::DATA_DIMENSION, 1);
}
void Evaluate(double x, double* f, double* dfdx) const {
const int n = std::floor(x);
Eigen::Matrix<double, Grid::DATA_DIMENSION, 1> p0, p1, p2, p3;
grid_.GetValue(n - 1, p0.data());
grid_.GetValue(n, p1.data());
grid_.GetValue(n + 1, p2.data());
grid_.GetValue(n + 2, p3.data());
CubicHermiteSpline<Grid::DATA_DIMENSION>(p0, p1, p2, p3, x - n, f, dfdx);
}
// The following two Evaluate overloads are needed for interfacing
// with automatic differentiation. The first is for when a scalar
// evaluation is done, and the second one is for when Jets are used.
void Evaluate(const double& x, double* f) const {
Evaluate(x, f, NULL);
}
template<typename JetT> void Evaluate(const JetT& x, JetT* f) const {
double fx[Grid::DATA_DIMENSION], dfdx[Grid::DATA_DIMENSION];
Evaluate(x.a, fx, dfdx);
for (int i = 0; i < Grid::DATA_DIMENSION; ++i) {
f[i].a = fx[i];
f[i].v = dfdx[i] * x.v;
}
}
private:
const Grid& grid_;
};
// An object that implements an infinite one dimensional grid needed
// by the CubicInterpolator where the source of the function values is
// an array of type T on the interval
//
// [begin, ..., end - 1]
//
// Since the input array is finite and the grid is infinite, values
// outside this interval needs to be computed. Grid1D uses the value
// from the nearest edge.
//
// The function being provided can be vector valued, in which case
// kDataDimension > 1. The dimensional slices of the function maybe
// interleaved, or they maybe stacked, i.e, if the function has
// kDataDimension = 2, if kInterleaved = true, then it is stored as
//
// f01, f02, f11, f12 ....
//
// and if kInterleaved = false, then it is stored as
//
// f01, f11, .. fn1, f02, f12, .. , fn2
//
template <typename T,
int kDataDimension = 1,
bool kInterleaved = true>
struct Grid1D {
public:
enum { DATA_DIMENSION = kDataDimension };
Grid1D(const T* data, const int begin, const int end)
: data_(data), begin_(begin), end_(end), num_values_(end - begin) {
CHECK_LT(begin, end);
}
EIGEN_STRONG_INLINE void GetValue(const int n, double* f) const {
const int idx = std::min(std::max(begin_, n), end_ - 1) - begin_;
if (kInterleaved) {
for (int i = 0; i < kDataDimension; ++i) {
f[i] = static_cast<double>(data_[kDataDimension * idx + i]);
}
} else {
for (int i = 0; i < kDataDimension; ++i) {
f[i] = static_cast<double>(data_[i * num_values_ + idx]);
}
}
}
private:
const T* data_;
const int begin_;
const int end_;
const int num_values_;
};
// Given as input an infinite two dimensional grid like object, which
// provides the following interface:
//
// struct Grid {
// enum { DATA_DIMENSION = 1 };
// void GetValue(int row, int col, double* f) const;
// };
//
// Where, GetValue gives us the value of a function f (possibly vector
// valued) for any pairs of integers (row, col), and the enum
// DATA_DIMENSION indicates the dimensionality of the function being
// interpolated. For example if you are interpolating a color image
// with three channels (Red, Green & Blue), then DATA_DIMENSION = 3.
//
// BiCubicInterpolator uses the cubic convolution interpolation
// algorithm of R. Keys, to produce a smooth approximation to it that
// can be used to evaluate the f(r,c), df(r, c)/dr and df(r,c)/dc at
// any point in the real plane.
//
// For more details on the algorithm used here see:
//
// "Cubic convolution interpolation for digital image processing".
// Robert G. Keys, IEEE Trans. on Acoustics, Speech, and Signal
// Processing 29 (6): 1153–1160, 1981.
//
// http://en.wikipedia.org/wiki/Cubic_Hermite_spline
// http://en.wikipedia.org/wiki/Bicubic_interpolation
//
// Example usage:
//
// const double data[] = {1.0, 3.0, -1.0, 4.0,
// 3.6, 2.1, 4.2, 2.0,
// 2.0, 1.0, 3.1, 5.2};
// Grid2D<double, 1> grid(data, 3, 4);
// BiCubicInterpolator<Grid2D<double, 1> > interpolator(grid);
// double f, dfdr, dfdc;
// interpolator.Evaluate(1.2, 2.5, &f, &dfdr, &dfdc);
template<typename Grid>
class BiCubicInterpolator {
public:
explicit BiCubicInterpolator(const Grid& grid)
: grid_(grid) {
// The + casts the enum into an int before doing the
// comparison. It is needed to prevent
// "-Wunnamed-type-template-args" related errors.
CHECK_GE(+Grid::DATA_DIMENSION, 1);
}
// Evaluate the interpolated function value and/or its
// derivative. Returns false if r or c is out of bounds.
void Evaluate(double r, double c,
double* f, double* dfdr, double* dfdc) const {
// BiCubic interpolation requires 16 values around the point being
// evaluated. We will use pij, to indicate the elements of the
// 4x4 grid of values.
//
// col
// p00 p01 p02 p03
// row p10 p11 p12 p13
// p20 p21 p22 p23
// p30 p31 p32 p33
//
// The point (r,c) being evaluated is assumed to lie in the square
// defined by p11, p12, p22 and p21.
const int row = std::floor(r);
const int col = std::floor(c);
Eigen::Matrix<double, Grid::DATA_DIMENSION, 1> p0, p1, p2, p3;
// Interpolate along each of the four rows, evaluating the function
// value and the horizontal derivative in each row.
Eigen::Matrix<double, Grid::DATA_DIMENSION, 1> f0, f1, f2, f3;
Eigen::Matrix<double, Grid::DATA_DIMENSION, 1> df0dc, df1dc, df2dc, df3dc;
grid_.GetValue(row - 1, col - 1, p0.data());
grid_.GetValue(row - 1, col , p1.data());
grid_.GetValue(row - 1, col + 1, p2.data());
grid_.GetValue(row - 1, col + 2, p3.data());
CubicHermiteSpline<Grid::DATA_DIMENSION>(p0, p1, p2, p3, c - col,
f0.data(), df0dc.data());
grid_.GetValue(row, col - 1, p0.data());
grid_.GetValue(row, col , p1.data());
grid_.GetValue(row, col + 1, p2.data());
grid_.GetValue(row, col + 2, p3.data());
CubicHermiteSpline<Grid::DATA_DIMENSION>(p0, p1, p2, p3, c - col,
f1.data(), df1dc.data());
grid_.GetValue(row + 1, col - 1, p0.data());
grid_.GetValue(row + 1, col , p1.data());
grid_.GetValue(row + 1, col + 1, p2.data());
grid_.GetValue(row + 1, col + 2, p3.data());
CubicHermiteSpline<Grid::DATA_DIMENSION>(p0, p1, p2, p3, c - col,
f2.data(), df2dc.data());
grid_.GetValue(row + 2, col - 1, p0.data());
grid_.GetValue(row + 2, col , p1.data());
grid_.GetValue(row + 2, col + 1, p2.data());
grid_.GetValue(row + 2, col + 2, p3.data());
CubicHermiteSpline<Grid::DATA_DIMENSION>(p0, p1, p2, p3, c - col,
f3.data(), df3dc.data());
// Interpolate vertically the interpolated value from each row and
// compute the derivative along the columns.
CubicHermiteSpline<Grid::DATA_DIMENSION>(f0, f1, f2, f3, r - row, f, dfdr);
if (dfdc != NULL) {
// Interpolate vertically the derivative along the columns.
CubicHermiteSpline<Grid::DATA_DIMENSION>(df0dc, df1dc, df2dc, df3dc,
r - row, dfdc, NULL);
}
}
// The following two Evaluate overloads are needed for interfacing
// with automatic differentiation. The first is for when a scalar
// evaluation is done, and the second one is for when Jets are used.
void Evaluate(const double& r, const double& c, double* f) const {
Evaluate(r, c, f, NULL, NULL);
}
template<typename JetT> void Evaluate(const JetT& r,
const JetT& c,
JetT* f) const {
double frc[Grid::DATA_DIMENSION];
double dfdr[Grid::DATA_DIMENSION];
double dfdc[Grid::DATA_DIMENSION];
Evaluate(r.a, c.a, frc, dfdr, dfdc);
for (int i = 0; i < Grid::DATA_DIMENSION; ++i) {
f[i].a = frc[i];
f[i].v = dfdr[i] * r.v + dfdc[i] * c.v;
}
}
private:
const Grid& grid_;
};
// An object that implements an infinite two dimensional grid needed
// by the BiCubicInterpolator where the source of the function values
// is an grid of type T on the grid
//
// [(row_start, col_start), ..., (row_start, col_end - 1)]
// [ ... ]
// [(row_end - 1, col_start), ..., (row_end - 1, col_end - 1)]
//
// Since the input grid is finite and the grid is infinite, values
// outside this interval needs to be computed. Grid2D uses the value
// from the nearest edge.
//
// The function being provided can be vector valued, in which case
// kDataDimension > 1. The data maybe stored in row or column major
// format and the various dimensional slices of the function maybe
// interleaved, or they maybe stacked, i.e, if the function has
// kDataDimension = 2, is stored in row-major format and if
// kInterleaved = true, then it is stored as
//
// f001, f002, f011, f012, ...
//
// A commonly occuring example are color images (RGB) where the three
// channels are stored interleaved.
//
// If kInterleaved = false, then it is stored as
//
// f001, f011, ..., fnm1, f002, f012, ...
template <typename T,
int kDataDimension = 1,
bool kRowMajor = true,
bool kInterleaved = true>
struct Grid2D {
public:
enum { DATA_DIMENSION = kDataDimension };
Grid2D(const T* data,
const int row_begin, const int row_end,
const int col_begin, const int col_end)
: data_(data),
row_begin_(row_begin), row_end_(row_end),
col_begin_(col_begin), col_end_(col_end),
num_rows_(row_end - row_begin), num_cols_(col_end - col_begin),
num_values_(num_rows_ * num_cols_) {
CHECK_GE(kDataDimension, 1);
CHECK_LT(row_begin, row_end);
CHECK_LT(col_begin, col_end);
}
EIGEN_STRONG_INLINE void GetValue(const int r, const int c, double* f) const {
const int row_idx =
std::min(std::max(row_begin_, r), row_end_ - 1) - row_begin_;
const int col_idx =
std::min(std::max(col_begin_, c), col_end_ - 1) - col_begin_;
const int n =
(kRowMajor)
? num_cols_ * row_idx + col_idx
: num_rows_ * col_idx + row_idx;
if (kInterleaved) {
for (int i = 0; i < kDataDimension; ++i) {
f[i] = static_cast<double>(data_[kDataDimension * n + i]);
}
} else {
for (int i = 0; i < kDataDimension; ++i) {
f[i] = static_cast<double>(data_[i * num_values_ + n]);
}
}
}
private:
const T* data_;
const int row_begin_;
const int row_end_;
const int col_begin_;
const int col_end_;
const int num_rows_;
const int num_cols_;
const int num_values_;
};
} // namespace ceres
#endif // CERES_PUBLIC_CUBIC_INTERPOLATOR_H_