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DEFINITIONS
This source file includes following definitions.
- AcquireMagickMatrix
- GaussJordanElimination
- LeastSquaresAddTerms
- RelinquishMagickMatrix
/*
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% M M AAA TTTTT RRRR IIIII X X %
% MM MM A A T R R I X X %
% M M M AAAAA T RRRR I X %
% M M A A T R R I X X %
% M M A A T R R IIIII X X %
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% MagickCore Matrix Methods %
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% Software Design %
% John Cristy %
% August 2007 %
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% Copyright 1999-2011 ImageMagick Studio LLC, a non-profit organization %
% dedicated to making software imaging solutions freely available. %
% %
% You may not use this file except in compliance with the License. You may %
% obtain a copy of the License at %
% %
% http://www.imagemagick.org/script/license.php %
% %
% Unless required by applicable law or agreed to in writing, software %
% distributed under the License is distributed on an "AS IS" BASIS, %
% WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied. %
% See the License for the specific language governing permissions and %
% limitations under the License. %
% %
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*/
/*
Include declarations.
*/
#include "magick/studio.h"
#include "magick/matrix.h"
#include "magick/memory_.h"
#include "magick/utility.h"
/*
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% A c q u i r e M a g i c k M a t r i x %
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% AcquireMagickMatrix() allocates and returns a matrix in the form of an
% array of pointers to an array of doubles, with all values pre-set to zero.
%
% This used to generate the two dimentional matrix, and vectors required
% for the GaussJordanElimination() method below, solving some system of
% simultanious equations.
%
% The format of the AcquireMagickMatrix method is:
%
% double **AcquireMagickMatrix(const size_t number_rows,
% const size_t size)
%
% A description of each parameter follows:
%
% o number_rows: the number pointers for the array of pointers
% (first dimension).
%
% o size: the size of the array of doubles each pointer points to
% (second dimension).
%
*/
MagickExport double **AcquireMagickMatrix(const size_t number_rows,
const size_t size)
{
double
**matrix;
register ssize_t
i,
j;
matrix=(double **) AcquireQuantumMemory(number_rows,sizeof(*matrix));
if (matrix == (double **) NULL)
return((double **)NULL);
for (i=0; i < (ssize_t) number_rows; i++)
{
matrix[i]=(double *) AcquireQuantumMemory(size,sizeof(*matrix[i]));
if (matrix[i] == (double *) NULL)
{
for (j=0; j < i; j++)
matrix[j]=(double *) RelinquishMagickMemory(matrix[j]);
matrix=(double **) RelinquishMagickMemory(matrix);
return((double **) NULL);
}
for (j=0; j < (ssize_t) size; j++)
matrix[i][j]=0.0;
}
return(matrix);
}
/*
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% G a u s s J o r d a n E l i m i n a t i o n %
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% GaussJordanElimination() returns a matrix in reduced row echelon form,
% while simultaneously reducing and thus solving the augumented results
% matrix.
%
% See also http://en.wikipedia.org/wiki/Gauss-Jordan_elimination
%
% The format of the GaussJordanElimination method is:
%
% MagickBooleanType GaussJordanElimination(double **matrix,double **vectors,
% const size_t rank,const size_t number_vectors)
%
% A description of each parameter follows:
%
% o matrix: the matrix to be reduced, as an 'array of row pointers'.
%
% o vectors: the additional matrix argumenting the matrix for row reduction.
% Producing an 'array of column vectors'.
%
% o rank: The size of the matrix (both rows and columns).
% Also represents the number terms that need to be solved.
%
% o number_vectors: Number of vectors columns, argumenting the above matrix.
% Usally 1, but can be more for more complex equation solving.
%
% Note that the 'matrix' is given as a 'array of row pointers' of rank size.
% That is values can be assigned as matrix[row][column] where 'row' is
% typically the equation, and 'column' is the term of the equation.
% That is the matrix is in the form of a 'row first array'.
%
% However 'vectors' is a 'array of column pointers' which can have any number
% of columns, with each column array the same 'rank' size as 'matrix'.
%
% This allows for simpler handling of the results, especially is only one
% column 'vector' is all that is required to produce the desired solution.
%
% For example, the 'vectors' can consist of a pointer to a simple array of
% doubles. when only one set of simultanious equations is to be solved from
% the given set of coefficient weighted terms.
%
% double **matrix = AcquireMagickMatrix(8UL,8UL);
% double coefficents[8];
% ...
% GaussJordanElimination(matrix, &coefficents, 8UL, 1UL);
%
% However by specifing more 'columns' (as an 'array of vector columns',
% you can use this function to solve a set of 'separable' equations.
%
% For example a distortion function where u = U(x,y) v = V(x,y)
% And the functions U() and V() have separate coefficents, but are being
% generated from a common x,y->u,v data set.
%
% Another example is generation of a color gradient from a set of colors
% at specific coordients, such as a list x,y -> r,g,b,a
% (Reference to be added - Anthony)
%
% You can also use the 'vectors' to generate an inverse of the given 'matrix'
% though as a 'column first array' rather than a 'row first array'. For
% details see http://en.wikipedia.org/wiki/Gauss-Jordan_elimination
%
*/
MagickExport MagickBooleanType GaussJordanElimination(double **matrix,
double **vectors,const size_t rank,const size_t number_vectors)
{
#define GaussJordanSwap(x,y) \
{ \
if ((x) != (y)) \
{ \
(x)+=(y); \
(y)=(x)-(y); \
(x)=(x)-(y); \
} \
}
double
max,
scale;
register ssize_t
i,
j,
k;
ssize_t
column,
*columns,
*pivots,
row,
*rows;
columns=(ssize_t *) AcquireQuantumMemory(rank,sizeof(*columns));
rows=(ssize_t *) AcquireQuantumMemory(rank,sizeof(*rows));
pivots=(ssize_t *) AcquireQuantumMemory(rank,sizeof(*pivots));
if ((rows == (ssize_t *) NULL) || (columns == (ssize_t *) NULL) ||
(pivots == (ssize_t *) NULL))
{
if (pivots != (ssize_t *) NULL)
pivots=(ssize_t *) RelinquishMagickMemory(pivots);
if (columns != (ssize_t *) NULL)
columns=(ssize_t *) RelinquishMagickMemory(columns);
if (rows != (ssize_t *) NULL)
rows=(ssize_t *) RelinquishMagickMemory(rows);
return(MagickFalse);
}
(void) ResetMagickMemory(columns,0,rank*sizeof(*columns));
(void) ResetMagickMemory(rows,0,rank*sizeof(*rows));
(void) ResetMagickMemory(pivots,0,rank*sizeof(*pivots));
column=0;
row=0;
for (i=0; i < (ssize_t) rank; i++)
{
max=0.0;
for (j=0; j < (ssize_t) rank; j++)
if (pivots[j] != 1)
{
for (k=0; k < (ssize_t) rank; k++)
if (pivots[k] != 0)
{
if (pivots[k] > 1)
return(MagickFalse);
}
else
if (fabs(matrix[j][k]) >= max)
{
max=fabs(matrix[j][k]);
row=j;
column=k;
}
}
pivots[column]++;
if (row != column)
{
for (k=0; k < (ssize_t) rank; k++)
GaussJordanSwap(matrix[row][k],matrix[column][k]);
for (k=0; k < (ssize_t) number_vectors; k++)
GaussJordanSwap(vectors[k][row],vectors[k][column]);
}
rows[i]=row;
columns[i]=column;
if (matrix[column][column] == 0.0)
return(MagickFalse); /* sigularity */
scale=1.0/matrix[column][column];
matrix[column][column]=1.0;
for (j=0; j < (ssize_t) rank; j++)
matrix[column][j]*=scale;
for (j=0; j < (ssize_t) number_vectors; j++)
vectors[j][column]*=scale;
for (j=0; j < (ssize_t) rank; j++)
if (j != column)
{
scale=matrix[j][column];
matrix[j][column]=0.0;
for (k=0; k < (ssize_t) rank; k++)
matrix[j][k]-=scale*matrix[column][k];
for (k=0; k < (ssize_t) number_vectors; k++)
vectors[k][j]-=scale*vectors[k][column];
}
}
for (j=(ssize_t) rank-1; j >= 0; j--)
if (columns[j] != rows[j])
for (i=0; i < (ssize_t) rank; i++)
GaussJordanSwap(matrix[i][rows[j]],matrix[i][columns[j]]);
pivots=(ssize_t *) RelinquishMagickMemory(pivots);
rows=(ssize_t *) RelinquishMagickMemory(rows);
columns=(ssize_t *) RelinquishMagickMemory(columns);
return(MagickTrue);
}
/*
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% L e a s t S q u a r e s A d d T e r m s %
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% LeastSquaresAddTerms() adds one set of terms and associate results to the
% given matrix and vectors for solving using least-squares function fitting.
%
% The format of the AcquireMagickMatrix method is:
%
% void LeastSquaresAddTerms(double **matrix,double **vectors,
% const double *terms,const double *results,const size_t rank,
% const size_t number_vectors);
%
% A description of each parameter follows:
%
% o matrix: the square matrix to add given terms/results to.
%
% o vectors: the result vectors to add terms/results to.
%
% o terms: the pre-calculated terms (without the unknown coefficent
% weights) that forms the equation being added.
%
% o results: the result(s) that should be generated from the given terms
% weighted by the yet-to-be-solved coefficents.
%
% o rank: the rank or size of the dimentions of the square matrix.
% Also the length of vectors, and number of terms being added.
%
% o number_vectors: Number of result vectors, and number or results being
% added. Also represents the number of separable systems of equations
% that is being solved.
%
% Example of use...
%
% 2 dimentional Affine Equations (which are separable)
% c0*x + c2*y + c4*1 => u
% c1*x + c3*y + c5*1 => v
%
% double **matrix = AcquireMagickMatrix(3UL,3UL);
% double **vectors = AcquireMagickMatrix(2UL,3UL);
% double terms[3], results[2];
% ...
% for each given x,y -> u,v
% terms[0] = x;
% terms[1] = y;
% terms[2] = 1;
% results[0] = u;
% results[1] = v;
% LeastSquaresAddTerms(matrix,vectors,terms,results,3UL,2UL);
% ...
% if ( GaussJordanElimination(matrix,vectors,3UL,2UL) ) {
% c0 = vectors[0][0];
% c2 = vectors[0][1];
% c4 = vectors[0][2];
% c1 = vectors[1][0];
% c3 = vectors[1][1];
% c5 = vectors[1][2];
% }
% else
% printf("Matrix unsolvable\n);
% RelinquishMagickMatrix(matrix,3UL);
% RelinquishMagickMatrix(vectors,2UL);
%
*/
MagickExport void LeastSquaresAddTerms(double **matrix,double **vectors,
const double *terms,const double *results,const size_t rank,
const size_t number_vectors)
{
register ssize_t
i,
j;
for (j=0; j < (ssize_t) rank; j++)
{
for (i=0; i < (ssize_t) rank; i++)
matrix[i][j]+=terms[i]*terms[j];
for (i=0; i < (ssize_t) number_vectors; i++)
vectors[i][j]+=results[i]*terms[j];
}
return;
}
/*
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% R e l i n q u i s h M a g i c k M a t r i x %
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% RelinquishMagickMatrix() frees the previously acquired matrix (array of
% pointers to arrays of doubles).
%
% The format of the RelinquishMagickMatrix method is:
%
% double **RelinquishMagickMatrix(double **matrix,
% const size_t number_rows)
%
% A description of each parameter follows:
%
% o matrix: the matrix to relinquish
%
% o number_rows: the first dimension of the acquired matrix (number of
% pointers)
%
*/
MagickExport double **RelinquishMagickMatrix(double **matrix,
const size_t number_rows)
{
register ssize_t
i;
if (matrix == (double **) NULL )
return(matrix);
for (i=0; i < (ssize_t) number_rows; i++)
matrix[i]=(double *) RelinquishMagickMemory(matrix[i]);
matrix=(double **) RelinquishMagickMemory(matrix);
return(matrix);
}