# class cv::PCA

## Overview

Principal Component Analysis. More…

#include <core.hpp>

class PCA
{
public:
// enums

enum Flags;

// fields

Mat eigenvalues;
Mat eigenvectors;
Mat mean;

// construction

PCA();

PCA(
InputArray data,
InputArray mean,
int flags,
int maxComponents = 0
);

PCA(
InputArray data,
InputArray mean,
int flags,
double retainedVariance
);

// methods

Mat
backProject(InputArray vec) const;

void
backProject(
InputArray vec,
OutputArray result
) const;

PCA&
operator()(
InputArray data,
InputArray mean,
int flags,
int maxComponents = 0
);

PCA&
operator()(
InputArray data,
InputArray mean,
int flags,
double retainedVariance
);

Mat
project(InputArray vec) const;

void
project(
InputArray vec,
OutputArray result
) const;

void

void
write(FileStorage& fs) const;
};


## Detailed Documentation

Principal Component Analysis.

The class is used to calculate a special basis for a set of vectors. The basis will consist of eigenvectors of the covariance matrix calculated from the input set of vectors. The class PCA can also transform vectors to/from the new coordinate space defined by the basis. Usually, in this new coordinate system, each vector from the original set (and any linear combination of such vectors) can be quite accurately approximated by taking its first few components, corresponding to the eigenvectors of the largest eigenvalues of the covariance matrix. Geometrically it means that you calculate a projection of the vector to a subspace formed by a few eigenvectors corresponding to the dominant eigenvalues of the covariance matrix. And usually such a projection is very close to the original vector. So, you can represent the original vector from a high-dimensional space with a much shorter vector consisting of the projected vector’s coordinates in the subspace. Such a transformation is also known as Karhunen-Loeve Transform, or KLT. See http://en.wikipedia.org/wiki/Principal_component_analysis

The sample below is the function that takes two matrices. The first function stores a set of vectors (a row per vector) that is used to calculate PCA. The second function stores another “test” set of vectors (a row per vector). First, these vectors are compressed with PCA, then reconstructed back, and then the reconstruction error norm is computed and printed for each vector. :

using namespace cv;

PCA compressPCA(const Mat& pcaset, int maxComponents,
const Mat& testset, Mat& compressed)
{
PCA pca(pcaset, // pass the data
Mat(), // we do not have a pre-computed mean vector,
// so let the PCA engine to compute it
PCA::DATA_AS_ROW, // indicate that the vectors
// are stored as matrix rows
// (use PCA::DATA_AS_COL if the vectors are
// the matrix columns)
maxComponents // specify, how many principal components to retain
);
// if there is no test data, just return the computed basis, ready-to-use
if( !testset.data )
return pca;
CV_Assert( testset.cols == pcaset.cols );

compressed.create(testset.rows, maxComponents, testset.type());

Mat reconstructed;
for( int i = 0; i < testset.rows; i++ )
{
Mat vec = testset.row(i), coeffs = compressed.row(i), reconstructed;
// compress the vector, the result will be stored
// in the i-th row of the output matrix
pca.project(vec, coeffs);
// and then reconstruct it
pca.backProject(coeffs, reconstructed);
// and measure the error
printf("%d. diff = %g\n", i, norm(vec, reconstructed, NORM_L2));
}
return pca;
}


### Fields

Mat eigenvalues


eigenvalues of the covariation matrix

Mat eigenvectors


eigenvectors of the covariation matrix

Mat mean


mean value subtracted before the projection and added after the back projection

### Construction

PCA()


default constructor

The default constructor initializes an empty PCA structure. The other constructors initialize the structure and call PCA::operator()().

PCA(
InputArray data,
InputArray mean,
int flags,
int maxComponents = 0
)


This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

Parameters:

 data input samples stored as matrix rows or matrix columns. mean optional mean value; if the matrix is empty (noArray()), the mean is computed from the data. flags operation flags; currently the parameter is only used to specify the data layout (PCA::Flags) maxComponents maximum number of components that PCA should retain; by default, all the components are retained.
PCA(
InputArray data,
InputArray mean,
int flags,
double retainedVariance
)


This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

Parameters:

 data input samples stored as matrix rows or matrix columns. mean optional mean value; if the matrix is empty (noArray()), the mean is computed from the data. flags operation flags; currently the parameter is only used to specify the data layout (PCA::Flags) retainedVariance Percentage of variance that PCA should retain. Using this parameter will let the PCA decided how many components to retain but it will always keep at least 2.

### Methods

Mat
backProject(InputArray vec) const


Reconstructs vectors from their PC projections.

The methods are inverse operations to PCA::project. They take PC coordinates of projected vectors and reconstruct the original vectors. Unless all the principal components have been retained, the reconstructed vectors are different from the originals. But typically, the difference is small if the number of components is large enough (but still much smaller than the original vector dimensionality). As a result, PCA is used.

Parameters:

 vec coordinates of the vectors in the principal component subspace, the layout and size are the same as of PCA::project output vectors.
void
backProject(
InputArray vec,
OutputArray result
) const


This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

Parameters:

 vec coordinates of the vectors in the principal component subspace, the layout and size are the same as of PCA::project output vectors. result reconstructed vectors; the layout and size are the same as of PCA::project input vectors.
PCA&
operator()(
InputArray data,
InputArray mean,
int flags,
int maxComponents = 0
)


performs PCA

The operator performs PCA of the supplied dataset. It is safe to reuse the same PCA structure for multiple datasets. That is, if the structure has been previously used with another dataset, the existing internal data is reclaimed and the new eigenvalues, eigenvectors and mean are allocated and computed.

The computed eigenvalues are sorted from the largest to the smallest and the corresponding eigenvectors are stored as eigenvectors rows.

Parameters:

 data input samples stored as the matrix rows or as the matrix columns. mean optional mean value; if the matrix is empty (noArray()), the mean is computed from the data. flags operation flags; currently the parameter is only used to specify the data layout. (Flags) maxComponents maximum number of components that PCA should retain; by default, all the components are retained.
PCA&
operator()(
InputArray data,
InputArray mean,
int flags,
double retainedVariance
)


This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

Parameters:

 data input samples stored as the matrix rows or as the matrix columns. mean optional mean value; if the matrix is empty (noArray()), the mean is computed from the data. flags operation flags; currently the parameter is only used to specify the data layout. (PCA::Flags) retainedVariance Percentage of variance that PCA should retain. Using this parameter will let the PCA decided how many components to retain but it will always keep at least 2.
Mat
project(InputArray vec) const


Projects vector(s) to the principal component subspace.

The methods project one or more vectors to the principal component subspace, where each vector projection is represented by coefficients in the principal component basis. The first form of the method returns the matrix that the second form writes to the result. So the first form can be used as a part of expression while the second form can be more efficient in a processing loop.

Parameters:

 vec input vector(s); must have the same dimensionality and the same layout as the input data used at PCA phase, that is, if DATA_AS_ROW are specified, then vec.cols==data.cols (vector dimensionality) and vec.rows is the number of vectors to project, and the same is true for the PCA::DATA_AS_COL case.
void
project(
InputArray vec,
OutputArray result
) const


This is an overloaded member function, provided for convenience. It differs from the above function only in what argument(s) it accepts.

Parameters:

 vec input vector(s); must have the same dimensionality and the same layout as the input data used at PCA phase, that is, if DATA_AS_ROW are specified, then vec.cols==data.cols (vector dimensionality) and vec.rows is the number of vectors to project, and the same is true for the PCA::DATA_AS_COL case. result output vectors; in case of PCA::DATA_AS_COL, the output matrix has as many columns as the number of input vectors, this means that result.cols==vec.cols and the number of rows match the number of principal components (for example, maxComponents parameter passed to the constructor).
void

void