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LSDRasterSpectral.cpp
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//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
//
// LSDRasterSpectral
// Land Surface Dynamics StatsTools
//
// An object for manipulating rasters developed for the University of Edinburgh
// Land Surface Dynamics group topographic toolbox. This is a derivative class
// from LSDRaster, for use specifically with spectral analysis.
//
// These tools have been seperated from the LSDRaster class mainly because
// they require the FFTW library and are therefore less portable than
// the standard LSDRaster object.
//
// Developed by:
// Simon M. Mudd
// Martin D. Hurst
// David T. Milodowski
// Stuart W.D. Grieve
// Declan A. Valters
// Fiona Clubb
//
// Copyright (C) 2013 Simon M. Mudd 2013
//
// Developer can be contacted by simon.m.mudd _at_ ed.ac.uk
//
// Simon Mudd
// University of Edinburgh
// School of GeoSciences
// Drummond Street
// Edinburgh, EH8 9XP
// Scotland
// United Kingdom
//
// This program is free software;
// you can redistribute it and/or modify it under the terms of the
// GNU General Public License as published by the Free Software Foundation;
// either version 2 of the License, or (at your option) any later version.
//
// This program is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY;
// without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
// See the GNU General Public License for more details.
//
// You should have received a copy of the
// GNU General Public License along with this program;
// if not, write to:
// Free Software Foundation, Inc.,
// 51 Franklin Street, Fifth Floor,
// Boston, MA 02110-1301
// USA
//
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
//
// LSDRasterSpectral.cpp
// cpp file for the LSDRasterSpectral object
// LSD stands for Land Surface Dynamics
// This object perform spectral analysis and is seperate from LSDRaster
// simply because it requires the FFTW package so this can be removed
// from compilation to retain portability
//
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
//
// This object is written by
// Simon M. Mudd, University of Edinburgh
// David T. Milodowski, University of Edinburgh
// Martin D. Hurst, British Geological Survey
// Fiona Clubb, University of Edinburgh
// Stuart Grieve, University of Edinburgh
//
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
//
// Version 0.0.1 02/04/2013
//
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
//-----------------------------------------------------------------
//DOCUMENTATION URL: http://www.geos.ed.ac.uk/~s0675405/LSD_Docs/
//-----------------------------------------------------------------
#include <iostream>
#include <fstream>
#include <iomanip>
#include <vector>
#include <string>
#include <math.h>
#include <string.h>
#include "TNT/tnt.h"
#include "TNT/jama_lu.h"
#include "TNT/jama_eig.h"
#include "LSDRaster.hpp"
#include "LSDRasterSpectral.hpp"
#include "LSDStatsTools.hpp"
#include "LSDIndexRaster.hpp"
#include "fftw-3.3.1/api/fftw3.h"
using namespace std;
using namespace TNT;
using namespace JAMA;
#ifndef LSDRasterSpectral_CPP
#define LSDRasterSpectral_CPP
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
// Assignment operator
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
LSDRasterSpectral& LSDRasterSpectral::operator=(const LSDRasterSpectral& rhs)
{
if (&rhs != this)
{
create(rhs.get_NRows(),rhs.get_NCols(),rhs.get_XMinimum(),rhs.get_YMinimum(),
rhs.get_DataResolution(),rhs.get_NoDataValue(),rhs.get_RasterData());
}
return *this;
}
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
// the create function. This is default and throws an error
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
void LSDRasterSpectral::create()
{
cout << "LSDRasterSpectral line 63 You need to initialize with a filename!" << endl;
exit(EXIT_FAILURE);
}
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
// create function that creates a LSDSpectralRaster with
// that is square, and that has dimensions NRow, NCols = 2^raster_order
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
void LSDRasterSpectral::create(int raster_order, float cellsize, float ndv)
{
// get the raster size
int raster_size = pow(2,raster_order);
cout << "You are making an LSDSpectralRaster with an size of: " << raster_size << endl;
// now set the raster properties
NRows = raster_size;
NCols = raster_size;
XMinimum = 0.0;
YMinimum = 0.0;
DataResolution = cellsize;
NoDataValue = ndv;
Ly = int(pow(2,ceil(log(NRows)/log(2))));
Lx = int(pow(2,ceil(log(NCols)/log(2))));
cout << "Created square raster with NRows = " << NRows << " and Ly: " << Ly << endl;
cout << "NCols: " << NCols << " and Lx: " << Lx << endl;
Array2D<float> test_data(NRows,NCols,ndv);
RasterData = test_data.copy();
}
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
// this creates a raster using an infile
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
void LSDRasterSpectral::create(string filename, string extension)
{
read_raster(filename,extension);
}
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
// this creates a raster filled with no data values
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
void LSDRasterSpectral::create(int nrows, int NCols, float xmin, float ymin,
float cellsize, float ndv, Array2D<float> data)
{
NRows = nrows;
NCols = NCols;
XMinimum = xmin;
YMinimum = ymin;
DataResolution = cellsize;
NoDataValue = ndv;
Ly = int(pow(2,ceil(log(NRows)/log(2))));
Lx = int(pow(2,ceil(log(NCols)/log(2))));
RasterData = data.copy();
if (RasterData.dim1() != NRows)
{
cout << "dimension of data is not the same as stated in NRows!" << endl;
exit(EXIT_FAILURE);
}
if (RasterData.dim2() != NCols)
{
cout << "dimension of data is not the same as stated in NRows!" << endl;
exit(EXIT_FAILURE);
}
}
// this function creates an LSDSpectralRaster from an LSDRaster
void LSDRasterSpectral::create(LSDRaster& An_LSDRaster)
{
NRows = An_LSDRaster.get_NRows();
NCols = An_LSDRaster.get_NCols();
XMinimum = An_LSDRaster.get_XMinimum();
YMinimum = An_LSDRaster.get_YMinimum();
DataResolution = An_LSDRaster.get_DataResolution();
NoDataValue = An_LSDRaster.get_NoDataValue();
Ly = int(pow(2,ceil(log(NRows)/log(2))));
Lx = int(pow(2,ceil(log(NCols)/log(2))));
RasterData = An_LSDRaster.get_RasterData();
}
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
// Fourier Helper functions
// These function don't really use the data stored within the object
// but are useful in Fourier analysis so are located here as member functions
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
// This function returns the frequencies in the direction of the rows
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
vector<float> LSDRasterSpectral::get_row_direction_frequencies_unshifted()
{
vector<float> freq_values(NRows);
// now get the frequencies based on the cellsize.
int lastrow = 0;
for(int row = 0; row<=(NRows-1)/2; row++)
{
freq_values[row] = float(row)*DataResolution/(float(NRows));
//cout << "Row: " << row << endl;
lastrow = row;
}
for(int row = (NRows)/2; row > 0; row--)
{
lastrow++;
freq_values[lastrow] = -float(row)*DataResolution/(float(NRows));
//cout << "Row is: " << lastrow << "and selection is: " << -row << endl;
}
return freq_values;
}
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
// This function returns the frequencies in the direction of the cols
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
vector<float> LSDRasterSpectral::get_col_direction_frequencies_unshifted()
{
vector<float> freq_values(NCols);
// now get the frequencies based on the cellsize.
int lastcol = 0;
for(int col = 0; col<=(NCols-1)/2; col++)
{
freq_values[col] = float(col)*DataResolution/(float(NCols));
//cout << "Col: " << col << endl;
lastcol = col;
}
for(int col = (NCols)/2; col > 0; col--)
{
lastcol++;
freq_values[lastcol] = -float(col)*DataResolution/(float(NCols));
//cout << "Col is: " << lastcol << "and selection is: " << -col << endl;
}
return freq_values;
}
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
//
// This function returns an array2d that contains entries scaled by 1/f^beta where
// beta is the fractal scaling
// This is used in generation of psuedo-fractal surfaces using the
// Fourier synthesis method of fractal generation
// (e.g., http://bringerp.free.fr/Files/Captain%20Blood/Saupe87d.pdf p.105
//
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
Array2D<float> LSDRasterSpectral::get_frequency_scaling_array(float beta)
{
Array2D<float> freq_scaling_array(NRows,NCols);
float radial_freq;
// get the frequencies
vector<float> row_freqs = get_row_direction_frequencies_unshifted();
vector<float> col_freqs = get_col_direction_frequencies_unshifted();
// get the frequency scaling
for(int row = 0; row<NRows; row++)
{
for (int col = 0; col<NCols; col++)
{
radial_freq = sqrt( row_freqs[row]*row_freqs[row]+col_freqs[col]*col_freqs[col]);
if (radial_freq == 0)
{
freq_scaling_array[row][col] = 0;
}
else
{
freq_scaling_array[row][col] = 1/pow(radial_freq,beta);
}
}
}
return freq_scaling_array;
}
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
//
// This creates a fractal surface using the spectral method
// The method works as follows:
// 1) Generate a random surface
// 2) Perform DFT on this random surface
// 3) Scale the tranform (both real and imaginary parts) by 1/f^beta
// 4) Perform the inverse DFT.
//
// This results in a pseudo fractal surface that can be used in comarison
// with real topography
//
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
void LSDRasterSpectral::generate_fractal_surface_spectral_method(float beta)
{
// first generate a random field
float range = 1.0;
rewrite_with_random_values(range);
// now get the frequency scaling
Array2D<float> freq_scaling = get_frequency_scaling_array(beta);
// now get the real and imaginary DFT arrays
// first set up the input and output arrays
Array2D<float> InputArray = RasterData.copy();
Array2D<float> OutputArrayReal = RasterData.copy();
Array2D<float> OutputArrayImaginary = RasterData.copy();
int transform_direction = -1; // this means it will be a forward transform
// perform the fourier annalysis
dfftw2D_fwd(InputArray, OutputArrayReal, OutputArrayImaginary,transform_direction);
cout << "Performed DFT! " << endl;
// now scale the DFT. We replace values in the output arrays
for(int row = 0; row<NRows; row++)
{
for (int col = 0; col<NCols; col++)
{
OutputArrayReal[row][col] = OutputArrayReal[row][col]*freq_scaling[row][col];
OutputArrayImaginary[row][col] = OutputArrayImaginary[row][col]*freq_scaling[row][col];
}
}
// now perform the inverse transform
// this overwrites the InputArray
transform_direction = 1;
dfftw2D_inv(OutputArrayReal, OutputArrayImaginary,
InputArray, transform_direction);
RasterData = InputArray;
}
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
//
// FFFFFF oOOo UU UU RRRR IIIIII EEEEEE RRRR
// FF oO Oo UU UU RR RR II EE RR RR
// FFFF OO OO UU UU RRRR II EEEE RRRR
// FF oO Oo UU UU RR RR II EE RR RR
// FF oOOo uUUu RR RR IIIIII EEEEEE RR RR
//
// AAAA NN NN AAAA LL YY YY sSSSs IIIIII sSSSs
// AA AA NNNN NN AA AA LL YY YY SS II SS
// AAAAAA NN NN NN AAAAAA LL YYYY sSSs II sSSs
// AA AA NN NNNN AA AA LL YY SS II SS
// AA AA NN NNN AA AA LLLLLL YY SSSSs IIIIII SSSSs
//
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
// FAST FOURIER TRANSFORM MODULE
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
// Computes both the forward and inverse fast fourier transforms of a 2D
// discrete dataset.
// FOR FORWARD TRANSFORM:
// - InputArray = zeta_padded (padded DEM)
// - transform_direction = -1
// - OutputArray = 2D spectrum
void LSDRasterSpectral::dfftw2D_fwd(Array2D<float>& InputArray, Array2D<float>& OutputArrayReal,
Array2D<float>& OutputArrayImaginary, int transform_direction)
{
// cout << Ly << " " << Lx << endl;
fftw_complex *input,*output;
fftw_plan plan;
// Declare one_dimensional contiguous arrays of dimension Ly*Lx
input = (fftw_complex*)fftw_malloc(sizeof(fftw_complex)*Ly*Lx);
output = (fftw_complex*)fftw_malloc(sizeof(fftw_complex)*Ly*Lx);
// SET UP PLAN
// -forwards, transform_direction==-1, -inverse, transform_direction==1
if (transform_direction==-1)
{
cout << " Running 2D discrete FORWARD fast fourier transform..." << endl;
plan = fftw_plan_dft_2d(Ly,Lx,input,output,transform_direction,FFTW_MEASURE);
}
else
{
cout << "\nFATAL ERROR: for the tranform direction\n\t -1 = FORWARD \n\t" << endl;
exit(EXIT_FAILURE);
}
// LOAD DATA INTO COMPLEX ARRAY FOR FFT IN ROW MAJOR ORDER
for (int i=0;i<Ly;++i)
{
for (int j=0;j<Lx;++j)
{
input[Lx*i+j][0] = InputArray[i][j];
}
}
// EXECUTE PLAN
fftw_execute(plan);
// RETRIEVE OUTPUT - since data is real, we only need to extract real part of
// the output.
for (int i=0;i<Ly;++i)
{
for (int j=0;j<Lx;++j)
{
OutputArrayReal[i][j] = output[Lx*i+j][0];
OutputArrayImaginary[i][j] = output[Lx*i+j][1];
}
}
// DEALLOCATE PLAN AND ARRAYS
fftw_destroy_plan(plan);
fftw_free(input);
fftw_free(output);
}
//------------------------------------------------------------------------------
// FOR INVERSE TRANSFORM:
// - InputArrays = Real and Imaginary components of 2D spectrum
// - transform_direction = 1
// - OutputArray = reconstructed DEM
void LSDRasterSpectral::dfftw2D_inv(Array2D<float>& InputArrayReal,
Array2D<float>& InputArrayImaginary,
Array2D<float>& OutputArray, int transform_direction)
{
fftw_complex *input,*output;
fftw_plan plan;
// Declare one_dimensional contiguous arrays of dimension Ly*Lx
input = (fftw_complex*)fftw_malloc(sizeof(fftw_complex)*Ly*Lx);
output = (fftw_complex*)fftw_malloc(sizeof(fftw_complex)*Ly*Lx);
// SET UP PLAN
// -forwards => transform_direction==-1, -inverse => transform_direction==1
if (transform_direction==1)
{
cout << " Running 2D discrete INVERSE fast fourier transform..." << endl;
plan = fftw_plan_dft_2d(Ly,Lx,input,output,transform_direction,FFTW_MEASURE);
}
else
{
cout << "\nFATAL ERROR: for the tranform direction\n\t 1 = INVERSE \n\t" << endl;
exit(EXIT_FAILURE);
}
// LOAD DATA INTO COMPLEX ARRAY FOR FFT IN ROW MAJOR ORDER
for (int i=0;i<Ly;++i)
{
for (int j=0;j<Lx;++j)
{
input[Lx*i+j][0] = InputArrayReal[i][j];
input[Lx*i+j][1] = InputArrayImaginary[i][j];
}
}
// EXECUTE PLAN
fftw_execute(plan);
// RETRIEVE OUTPUT ARRAY
for (int i=0;i<Ly;++i)
{
for (int j=0;j<Lx;++j)
{
OutputArray[i][j] = output[Lx*i+j][0];
}
}
// DEALLOCATE PLAN AND ARRAYS
fftw_destroy_plan(plan);
fftw_free(input);
fftw_free(output);
}
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
// DETREND DATA MODULE
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
// FIT PLANE BY LEAST SQUARES REGRESSION AND USE COEFFICIENTS TO DETERMINE
// LOCAL SLOPE ax + by + c = z
// Have N simultaneous linear equations, and N unknowns.
// => b = Ax, where x is a 1xN array containing the coefficients we need for
// surface fitting.
// A is constructed using different combinations of x and y, thus we only need
// to compute this once, since the window size does not change.
// For 1st order surface fitting, there are 3 coefficients, therefore A is a
// 3x3 matrix
// Module kicks out detrended array, and an array with the trend plane
void LSDRasterSpectral::detrend2D(Array2D<float>& zeta, Array2D<float>& zeta_detrend, Array2D<float>& trend_plane)
{
cout << " Detrending the DEM by fitting a planar surface..." << endl;
Array2D<float> A(3,3,0.0);
Array1D<float> bb(3,0.0);
Array1D<float> coeffs(3);
for (int i=0; i<NRows; ++i)
{
for (int j=0; j<NCols; ++j)
{
if(zeta[i][j] != NoDataValue)
{
float x = j;
float y = i;
// Generate matrix A
A[0][0] += pow(x,2);
A[0][1] += x*y;
A[0][2] += x;
A[1][0] += y*x;
A[1][1] += pow(y,2);
A[1][2] += y;
A[2][0] += x;
A[2][1] += y;
A[2][2] += 1;
// Generate vector bb
bb[0] += zeta[i][j]*x;
bb[1] += zeta[i][j]*y;
bb[2] += zeta[i][j];
}
}
}
// Solve matrix equations using LU decomposition using the TNT JAMA package:
// A.coefs = b, where coefs is the coefficients vector.
LU<float> sol_A(A); // Create LU object
coeffs = sol_A.solve(bb);
float a_plane = coeffs[0];
float b_plane = coeffs[1];
float c_plane = coeffs[2];
// Create detrended surface
for (int i=0; i<NRows; ++i)
{
for (int j=0; j<NCols; ++j)
{
float x = j;
float y = i;
trend_plane[i][j] = a_plane*x + b_plane*y + c_plane;
if(zeta[i][j] != NoDataValue)
{
zeta_detrend[i][j] = zeta[i][j] - trend_plane[i][j];
}
else // Set NoDataValues as 0 on detrended surface
{
zeta_detrend[i][j] = 0;
}
}
}
}
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
// HANN WINDOW MODULE
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
// Use 2D elliptical Hann (raised cosine) window on data matrix, to reduce
// spectral leakage and retain good frequency resolution.
// Return windowed data, the Hann window and also the summed square of the
// weighting coefficients, WSS.
// Another option would be to use a 2D Welch window, but functionality is very
// similar.
void LSDRasterSpectral::window_data_Hann2D(Array2D<float>& zeta_detrend, Array2D<float>& zeta_Hann2D, Array2D<float>& Hann2D)
{
float PI = 3.14159265;
cout << " Windowing DEM using an elliptical 2D Hann window..." << endl;
float ny = NRows;
float nx = NCols;
// Get matrix coordinates of centroid of matrix
float a = (nx-1)/2;
float b = (ny-1)/2;
// Set up data window
Array2D<float> r_prime_matrix(NRows,NCols,0.0);
Array2D<float> id(NRows,NCols,0.0);
Array2D<float> theta_matrix(NRows,NCols,0.0);
float r; // radial polar coordinate
float theta; // angular polar coordinate
float rprime;
float HannCoefficient = 0;
for(int i = 0; i < NRows; ++i)
{
for(int j = 0; j < NCols; ++j)
{
float x = j;
float y = i;
if(x == a)
{
theta = (PI/2);
}
else
{
theta = atan2((y - b),(x - a));
}
r = sqrt((y - b)*(y - b) + (x - a)*(x - a)); // distance from centre to this point
rprime = sqrt((a*a)*(b*b)/(b*b*(cos(theta)*cos(theta)) + a*a*(sin(theta)*sin(theta)))); // distance from centre to edge of ellipse for this particular theta
if(r < rprime)
{
HannCoefficient = 0.5 * (1 + cos(PI * r/rprime));
Hann2D[i][j] = HannCoefficient;
WSS += HannCoefficient*HannCoefficient;
zeta_Hann2D[i][j] = zeta_detrend[i][j] * HannCoefficient;
id[i][j]=1;
}
}
}
}
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
// SHIFT ORIGIN OF SPECTRUM IN FOURIER DOMAIN
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
// "The output of most algorithms that compute the DFT must be rearranged to
// place the zero wavenumber element near the center of the array. Provided Nx
// and Ny are even, dividing the output array into four equal quadrants and
// exchanging the nonadjacent quadrants will place the zero wavenumber element
// at the position (Nx/2, Ny/2) in the new array." (Perron et al., 2008)
void LSDRasterSpectral::shift_spectrum(Array2D<float>& spectrum_real, Array2D<float>& spectrum_imaginary, Array2D<float>& spectrum_real_shift, Array2D<float>& spectrum_imaginary_shift)
{
int QuadrantRows = Ly/2;
int QuadrantCols = Lx/2;
for(int i = 0; i < QuadrantRows; ++i)
{
for(int j = 0; j< QuadrantCols; ++j)
{
spectrum_real_shift[i][j] = spectrum_real[i+QuadrantRows][j+QuadrantCols]; // top left to bottom right
spectrum_real_shift[i+QuadrantRows][j] = spectrum_real[i][j+QuadrantCols]; // bottom right to top left
spectrum_real_shift[i][j+QuadrantCols] = spectrum_real[i+QuadrantRows][j]; // top right to bottom left
spectrum_real_shift[i+QuadrantRows][j+QuadrantCols] = spectrum_real[i][j]; // bottom right to top left
spectrum_imaginary_shift[i][j] = spectrum_imaginary[i+QuadrantRows][j+QuadrantCols]; // etc...
spectrum_imaginary_shift[i+QuadrantRows][j] = spectrum_imaginary[i][j+QuadrantCols];
spectrum_imaginary_shift[i][j+QuadrantCols] = spectrum_imaginary[i+QuadrantRows][j];
spectrum_imaginary_shift[i+QuadrantRows][j+QuadrantCols] = spectrum_imaginary[i][j];
}
}
}
//------------------------------------------------------------------------------
// DE-SHIFT ORIGIN OF SPECTRUM
// Inverse process of above to return filtered spectrum to original format
// required for the inverse fourier transform algorithm.
void LSDRasterSpectral::shift_spectrum_inv(Array2D<float>& FilteredSpectrumReal, Array2D<float>& FilteredSpectrumImaginary, Array2D<float>& FilteredSpectrumReal_deshift, Array2D<float>& FilteredSpectrumImaginary_deshift)
{
int QuadrantRows = Ly/2;
int QuadrantCols = Lx/2;
for(int i = 0; i < QuadrantRows; ++i)
{
for(int j = 0; j< QuadrantCols; ++j)
{
FilteredSpectrumReal_deshift[i+QuadrantRows][j+QuadrantCols] = FilteredSpectrumReal[i][j];
FilteredSpectrumReal_deshift[i][j+QuadrantCols] = FilteredSpectrumReal[i+QuadrantRows][j];
FilteredSpectrumReal_deshift[i+QuadrantRows][j] = FilteredSpectrumReal[i][j+QuadrantCols];
FilteredSpectrumReal_deshift[i][j] = FilteredSpectrumReal[i+QuadrantRows][j+QuadrantCols];
FilteredSpectrumImaginary_deshift[i+QuadrantRows][j+QuadrantCols] = FilteredSpectrumImaginary[i][j];
FilteredSpectrumImaginary_deshift[i][j+QuadrantCols] = FilteredSpectrumImaginary[i+QuadrantRows][j];
FilteredSpectrumImaginary_deshift[i+QuadrantRows][j] = FilteredSpectrumImaginary[i][j+QuadrantCols];
FilteredSpectrumImaginary_deshift[i][j] = FilteredSpectrumImaginary[i+QuadrantRows][j+QuadrantCols];
}
}
}
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
// CALCULATE THE DFT PERIODOGRAM
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
// Multiply fourier analysis output by complex conjugate and normalise.
// Note that for complex number z=x+iy, z*=x-iy, z.z* = x^2 + y^2
// Returns 2D PSD as only output
void LSDRasterSpectral::calculate_2D_PSD(Array2D<float>& spectrum_real_shift, Array2D<float>& spectrum_imaginary_shift)
{
Array2D<float> temp(Ly,Lx,0.0);
P_DFT = temp.copy();
for (int i=0; i<Ly; ++i)
{
for (int j=0; j<Lx; ++j)
{
P_DFT[i][j] = (pow(spectrum_real_shift[i][j],2) + pow(spectrum_imaginary_shift[i][j],2))/(Ly*Lx*WSS);
}
}
}
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
// GET RADIAL POWER SPECTRUM
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
// Collapse 2D PSD into a radial PSD
void LSDRasterSpectral::calculate_radial_PSD()
{
// CALCULATE FREQUENCY INCREMENTS - for generation of power spectrum
// Frequency goes from zero to 1/(2*resolution), the Nyquist frequency in
// NRows_padded/2 increments.
float dfx = 1/(DataResolution*Lx);
float dfy = 1/(DataResolution*Ly);
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
vector<float> RadialFrequencyRaw(Ly*(Lx/2+1),0.0); // This is the distance from the origin in Frequency space. Note that half of spectrum is redundant, since the fourier transform of a real dataset is symmetric, with a degeneracy of two.
vector<float> RadialPSDRaw(Ly*(Lx/2+1),0.0);
float NyquistFreq = 1/(2*DataResolution);
float RadialFreq;
int count = 0;
for (int i=0; i < Ly; ++i)
{
for (int j=0; j < (Lx/2+1); ++j)
{
float x = j;
float y = i;
RadialFreq = sqrt((y - (Ly/2))*(y - (Ly/2))*dfy*dfy + (x - (Lx/2))*(x - (Lx/2))*dfx*dfx); // distance from centre to this point. Converting position in frequency into an absolute frequency
if (RadialFreq <= NyquistFreq) // Ignore radial frequencies greater than the Nyquist frequency as these are aliased
{
RadialFrequencyRaw[count] = RadialFreq;
RadialPSDRaw[count] = 2*P_DFT[i][j]; // Due to degeneracy
++count;
}
}
}
// Sort radial frequency
vector<size_t> index_map;
matlab_float_sort(RadialFrequencyRaw,RadialFrequencyRaw,index_map);
// Reorder amplitudes to match sorted frequencies
matlab_float_reorder(RadialPSDRaw,index_map,RadialPSDRaw);
// Get number of discrete radial frequencies
int n_freqs = 0;
for (int i=0; i<(Ly*(Lx/2+1)); ++i)
{
if (RadialFrequencyRaw[i] != RadialFrequencyRaw[i+1])
{
++n_freqs;
}
}
// Convert to spatially averaged spectrum
cout << " Converting to radially averaged PSD..." << endl;
vector<float> RadialFrequency_grouped(n_freqs,0.0); // This is the distance from the origin in Frequency space
vector<float> RadialPSD_average(n_freqs,0.0); // This will ultimately contain the radially averaged PSD
int n_occurences = 0; // This will keep track of the number of occurences of each radial frequency
int pointer = 0;
for (int i=0; i<(Ly*(Lx/2+1)); ++i)
{
RadialFrequency_grouped[pointer] = RadialFrequencyRaw[i];
RadialPSD_average[pointer] += RadialPSDRaw[i];
++n_occurences;
if (RadialFrequencyRaw[i] != RadialFrequencyRaw[i+1])
{
RadialPSD_average[pointer] = RadialPSD_average[pointer]/n_occurences;
// increment pointer and reset n_occurences
++pointer;
n_occurences = 0;
}
}
RadiallyAveragedPSD = RadialPSD_average;
RadialFrequency = RadialFrequency_grouped;
// // Copy across to output vectors
// RadialFrequency_output = RadialFrequency_grouped;
// RadialPSD_output = RadialPSD_average;
}
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
// MAIN FUNCTIONS USING SPECTRAL ANALYSIS
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
// COMPUTE DISCRETE FAST FOURIER TRANSFORM OF A REAL, 2-DIMENSIONAL DATASET.
// Computes the 2D and radial power spectra of a 2D array.
// Input arguement is the width of the logarithmically spaced bins. For
// topography, suggest this is 0.1
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
void LSDRasterSpectral::fftw2D_spectral_analysis(char* file_id, float LogBinWidth)
{
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
// DETREND DATA
// FIT PLANE BY LEAST SQUARES REGRESSION AND USE COEFFICIENTS TO DETERMINE
// LOCAL SLOPE ax + by + c = z
Array2D<float> zeta_detrend(NRows,NCols);
Array2D<float> trend_plane(NRows,NCols);
detrend2D(RasterData, zeta_detrend, trend_plane);
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
// USE ELLIPTICAL 2D HANN (raised cosine) WINDOW ON ZETA MATRIX.
// RETURN WINDOWED DATA AND ALSO THE SUMMED SQUARE OF THE WEIGHTING
// COEFFICIENTS.
Array2D<float> Hann2D(NRows,NCols,0.0);
Array2D<float> zeta_Hann2D(NRows,NCols,0.0);
WSS = 0; // summed square of weighting coefficients
window_data_Hann2D(zeta_detrend, zeta_Hann2D, Hann2D);
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
// 2D DISCRETE FAST FOURIER TRANSFORM
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
// PAD DATA WITH ZEROS TO A POWER OF TWO (facilitates FFT)
// Ly = int(pow(2,ceil(log(NRows)/log(2))));
// Lx = int(pow(2,ceil(log(NCols)/log(2))));
Array2D<float> zeta_padded(Ly,Lx);
for (int i=0;i<Ly;++i)
{
for (int j=0;j<Lx;++j)
{
if (i<NRows && j<NCols)
{
zeta_padded[i][j] = zeta_Hann2D[i][j];
}
else
{
zeta_padded[i][j]=0;
}
}
}
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
// DO 2D FORWARD FAST FOURIER TRANSFORM
int transform_direction = -1;
Array2D<float> SpectrumReal(Ly,Lx);
Array2D<float> SpectrumImaginary(Ly,Lx);
dfftw2D_fwd(zeta_padded, SpectrumReal, SpectrumImaginary, transform_direction);
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
// REARRANGE SPECTRUM SO THAT ORIGIN IS AT THE CENTRE
Array2D<float> SpectrumReal_shift(Ly,Lx);
Array2D<float> SpectrumImaginary_shift(Ly,Lx);
shift_spectrum(SpectrumReal, SpectrumImaginary, SpectrumReal_shift, SpectrumImaginary_shift);
// CALCULATE THE DFT PERIODOGRAM
// Multiply output by complex conjugate and normalise.
// Note that for complex number z=x+iy, z*=x-iy, z.z* = x^2 + y^2
calculate_2D_PSD(SpectrumReal_shift, SpectrumImaginary_shift);
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
// GET RADIAL POWER SPECTRUM
// For forward transform, return the spectral power of the topography both
// in a 2D array, and also as a one dimensional array of radial frequency
calculate_radial_PSD();
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
// BIN POWER SPECTRUM INTO LOGARITMICALLY SPACED BINS OF RADIAL FREQUENCY TO
// GET MODEL "SIGNAL" FOR WIENER FILTER
cout << " Binning radial PSD into logarithmically spaced bins..." << endl;
// Initiate output vectors
vector<float> Bin_MeanRadialFreq;
vector<float> Bin_RadialPSD;
vector<float> BinMidpoints;
vector<float> StandardDeviationRadialFreq;
vector<float> StandardDeviationRadialPSD;
// Execute log binning
log_bin_data(RadialFrequency, RadiallyAveragedPSD, LogBinWidth, Bin_MeanRadialFreq, Bin_RadialPSD, BinMidpoints,
StandardDeviationRadialFreq, StandardDeviationRadialPSD, int(NoDataValue));
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
// WRITE OUTPUTS TO FILE
int len = strlen(file_id);
cout << " Writing output files..." << endl;
// 2D PSD
char *PSD_file = new char[len+6];
strcpy(PSD_file,file_id);
strcat(PSD_file,"_P_DFT");
//cout << "Line 601 LSDRSpectral, Lx: " << Lx << " Ly "<< Ly << "SRshift, dim1,2: " << SpectrumReal_shift.dim1() << " " << SpectrumReal_shift.dim2() << endl;
LSDRaster PowerSpectrum(Ly,Lx,-(Lx/2),(Lx/2-1),DataResolution,NoDataValue,P_DFT);
PowerSpectrum.write_raster(PSD_file,"flt");
//----------------------------------------------------------------------------
ofstream ofs;
//----------------------------------------------------------------------------
// Radially averaged PSD
char *RadialPSD_file = new char[len+14];
strcpy(RadialPSD_file,file_id);
strcat(RadialPSD_file,"_RadialPSD.txt");
ofs.open(RadialPSD_file);
if( ofs.fail() )
{
cout << "\nFATAL ERROR: unable to write to " << RadialPSD_file << endl;
exit(EXIT_FAILURE);
}
ofs << "Freq Wavelength PSD Model_PSD Model_noise\n";
for(int i=0; i < int(RadialFrequency.size()); ++i)
{
ofs << RadialFrequency[i] << " " << 1/RadialFrequency[i] << " " << RadiallyAveragedPSD[i] << " \n";
}
ofs.close();
//----------------------------------------------------------------------------
// Binned averaged PSD
char *RadialPSD_binned_file = new char[len+21];
strcpy(RadialPSD_binned_file,file_id);
strcat(RadialPSD_binned_file,"_RadialPSD_binned.txt");
ofs.open(RadialPSD_binned_file);
if( ofs.fail() )
{
cout << "\nFATAL ERROR: unable to write to " << RadialPSD_binned_file << endl;
exit(EXIT_FAILURE);
}
ofs << "Freq Wavelength PSD Sigma Model Noise\n";
for(int i=0; i < int(Bin_MeanRadialFreq.size()); ++i)
{
ofs << Bin_MeanRadialFreq[i] << " " << 1/Bin_MeanRadialFreq[i] << " " << Bin_RadialPSD[i] << " " << StandardDeviationRadialPSD[i] << " \n";
}
ofs.close();
//----------------------------------------------------------------------------
cout << " DONE!" << endl;
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
}
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
//
// sSSSs PPPPP EEEEEE CCCC TTTTTT RRRR AAAA LL
// SS PP PP EE CC TT RR RR AA AA LL
// sSSs PPPPP EEEE CC TT RRRR AAAAAA LL
// SS PP EE CC TT RR RR AA AA LL
// SSSSs PP EEEEEE CCCC TT RR RR AA AA LLLLLL
//
// FFFFFF IIIIII LL TTTTTT EEEEEE RRRR sSSSs
// FF II LL TT EE RR RR SS
// FFFF II LL TT EEEE RRRR sSSs
// FF II LL TT EE RR RR SS
// FF IIIIII LLLLLL TT EEEEEE RR RR SSSSs
//
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
// FILTER WEIGHTS
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
// BANDPASS FILTER
// Filter array to band between frequency bands f1 and f2. The bandpass filter
// is a gaussian filter centred at (f1+f2)/2 and with a SD of |f2-f1|/6.
void LSDRasterSpectral::bandpass_filter(Array2D<float>& RawSpectrumReal, Array2D<float>& RawSpectrumImaginary,
Array2D<float>& FilteredSpectrumReal, Array2D<float>& FilteredSpectrumImaginary,
float f1, float f2)
{
// CALCULATE FREQUENCY INCREMENTS - for generation of power spectrum
// Frequency goes from zero to 1/(2*resolution), the Nyquist frequency in
// NRows_padded/2 increments.
float dfx = 1/(DataResolution*Lx);
float dfy = 1/(DataResolution*Ly);
//=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-
cout << " Gaussian bandpass filter between f1 = " << f1 << " and f2 = " << f2 << endl;
float f; // radial frequency
float weight; // Filter weight
float sigma = sqrt((f2-f1)*(f2-f1))/6; // Standard Deviation of Gaussian filter
for (int i=0; i < Ly; ++i)
{
for (int j=0; j < Lx; ++j)
{
float x = j;
float y = i;
// Converting position in frequency space into an absolute frequency
f = sqrt((y - (Ly/2))*(y - (Ly/2))*dfy*dfy + (x - (Lx/2))*(x - (Lx/2))*dfx*dfx);
weight = exp(-(f - 0.5*(f1 + f2))*(f - 0.5*(f1 + f2))/(2*sigma*sigma));
FilteredSpectrumReal[i][j] = weight*RawSpectrumReal[i][j];
FilteredSpectrumImaginary[i][j] = weight*RawSpectrumImaginary[i][j];
}
}
}
//------------------------------------------------------------------------------
// LOWPASS FILTER
// Filter array to retain frequencies below f1. The filter edge is a radial
// gaussian function with a SD of |f2-f1|/3.
void LSDRasterSpectral::lowpass_filter(Array2D<float>& RawSpectrumReal, Array2D<float>& RawSpectrumImaginary,
Array2D<float>& FilteredSpectrumReal, Array2D<float>& FilteredSpectrumImaginary,
float f1, float f2)
{
// CALCULATE FREQUENCY INCREMENTS - for generation of power spectrum
// Frequency goes from zero to 1/(2*resolution), the Nyquist frequency in
// NRows_padded/2 increments.
float dfx = 1/(DataResolution*Lx);
float dfy = 1/(DataResolution*Ly);
//----------------------------------------------------------------------------
cout << " Lowpass filter with edges controlled by radial Gaussian function between f1 = " << f1 << " and f2 = " << f2 << endl;
float f; // radial frequency
float weight; // Filter weight
float sigma; // Standard Deviation of Gaussian edge
for (int i=0; i < Ly; ++i)
{