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Print version ISSN 1679-4508
Einstein (São Paulo) vol.10 no.2 São Paulo Apr./June 2012
Rafael do Espírito Santo
Instituto do Cérebro - InCe, Hospital Israelita Albert Einstein HIAE, São Paulo (SP), Brazil
OBJECTIVE: To describe the use of a statistical tool (Principal Component Analysis PCA) for the recognition of patterns and compression, applying these concepts to digital images used in Medicine.
METHODS: The description of Principal Component Analysis is made by means of the explanation of eigenvalues and eigenvectors of a matrix. This concept is presented on a digital image collected in the clinical routine of a hospital, based on the functional aspects of a matrix. The analysis of potential for recovery of the original image was made in terms of the rate of compression obtained.
RESULTS: The compressed medical images maintain the principal characteristics until approximately one-fourth of their original size, highlighting the use of Principal Component Analysis as a tool for image compression. Secondarily, the parameter obtained may reflect the complexity and potentially, the texture of the original image.
CONCLUSION: The quantity of principal components used in the compression influences the recovery of the original image from the final (compacted) image.
Keywords: Principal component analysis; Eigenvalues; Eigenvectors; Image compressing; Patters; Dimensionality reduction
Principal Components Analysis (PCA)(1) is a mathematical formulation used in the reduction of data dimensions(2). Thus, the PCA technique allows the identification of standards in data and their expression in such a way that their similarities and differences are emphasized. Once patterns are found, they can be compressed, i.e., their dimensions can be reduced without much loss of information. In summary, the PCA formulation may be used as a digital image compression algorithm with a low level of loss.
In the PCA approach, the information contained in a set of data is stored in a computational structure with reduced dimensions based on the integral projection of the data set onto a subspace generated by a system of orthogonal axes(3). The optimal system of axes may be obtained using the Singular Values Decomposition (SVD) method(4). The reduced dimension computational structure is selected so that relevant data characteristics are identified with little loss of information(3). Such a reduction is advantageous in several instances: for image compression, data representation, calculation reduction necessary in subsequent processing, etc.
Use of the PCA technique in data dimension reduction is justified by the easy representation of multidimensional data, using the information contained in the data covariance matrix, principles of linear algebra(3) and basic statistics. The studies carried out by Mashal et al.(5) adopted the PCA formulation in the selections of images from a multimedia database. According to Smith(6), PCA is an authentic image compression algorithm with minimal loss of information.
The relevance of this work is in the performance evaluation of the PCA formulation in compressing digital images from the measurement of the degree of compression and the degree of information loss that the PCA introduces into the compressed images in discarding some principal components.
This article has the purpose of describing the PCA of a population of data and the possibility of applying it to the compression of digital images. The application of the technique in pattern recognition is also emphasized.
Admitting digital processing, a continuous (analogical) datum is converted into a matrix of simple elements (pixels) that assume discrete values (gray levels), that is:
In which the values of x and y (x, y) are the coordinates of the pixels in the image, and f(x,y) is the corresponding level of gray(7).
COVARIANCE OF AN IMAGE
The covariance matrix of an image is given by:
A PCA may be characterized from the data of p variables for n individuals, as is indicated on table 1.
By definition(1), the first principal component is the linear combination of variables X1X2;...;Xp, that is,
The second principal component
The third principal component,
and so forth. If there are p variables, then there are at most p principal components, always calculated according to expressions similar to expressions (3) or (4) or (5).
The results of a PCA, that is, the principal components Zp are obtained from an analysis that consists in finding the eigenvalues(3-6) of a sample covariance matrix(8). The covariance matrix is symmetrical and has the form:
in which the elements cjj, positioned along the primary diagonal, are the variances of Xi (var(Xi)) and the cij's of the secondary diagonal represent the covariance between the variables Xi Xj (cov (Xi, Xj)).
The eigenvalues of matrix C are the variances of the principal components. There are p eigenvalues. They are always numbers greater than or equal to zero, represented by the symbol λ. Negative λ's are not allowed in a covariance matrix(6). Assuming that the eigenvalues are ordered as λ1>λ2>...λp> 0, then λ1 corresponds to the first principal component (expression 1), and λi to the i-th principal component, or:
As was mentioned, var(Zi) = λi and the constants ai1, ai2,..., aip are the elements of the corresponding eigenvector, graduated so that(6)
The fact that cii is the variance of Xi and that λi is the variance of Zi implies that the sum of variances of the principal components is equal to the sum of variances of the original variances(6). Thus, in a way, the principal components contain all the variation of the original data(5,6).
The steps normally followed in a PCA of a digital image can now be established:
Step 1: In the computational model of a digital image, in expression 1, the variables X1, X2,...,Xp are the columns of the image. The PCA is begun by coding (correcting) the image to that its columns have zero means and unitary variances. This is common, in order to avoid one or the other of the columns having undue influence on the principal components(6):
Step 2: The covariance matrix C is calculated using expression 6, implemented computationally, that is:
Step 3: The eigenvalues λ1,λ2,...,λp and the corresponding eigenvectors a1, a2,..., ap. are calculated.
Step 4: The value of a vector of characteristics is obtained, a matrix with vectors containing the list of eigenvectors (matrix columns) of the covariance matrix(6).
Step 5: The final data are obtained, that is, a matrix with all the eigenvectors (components) of the covariance matrix.
Step 6: The original image is obtained from the final data without compression using the expression
Step 7: Any components that explain only a small portion of the variation in data for the effect of image compression are discarded. The eliminations have the effect of reducing the quantity of eigenvectors of the characteristics vectors and can produce final data with a smaller dimension. The use of expression 13 in these conditions allow the recovery of the original image with compression.
According to Castro(9,10), low-loss compression afforded by the present method may be expressed in terms of the compression factor of (ρ) and of the mean squared error (MSE) committed in the approximation of A (original image) by Ã (image obtained from the disposal of some of the components). The compression factor is defined by:
And the MSE committed in the approximation of A by Ã is:
This section shows examples of compression of digitalized images using the PCA formulation. Various situations are presented as examples.
Example 1: Recovering a TIFF image with 512x512 pixels with all the components (512) of image covariance matrix (without compression, i.e., steps 1 to 6).
Example 2: Recovery of a TIFF image with 512x512 pixels with 112 principal components of the covariance matrix of the image (with compression, that is, steps from 1 to 5 to 7).
Example 3: Recovery of an image with 32 principal components of the image covariance matrix (with compression).
Example 4: Recovery of an image with 12 principal components of the covariance matrix of the image (with compression).
Examples 1 to 4 show the effects of the reduction in number of principal components (elevation of the image compression rate) in the increased loss of information. This application may bring great savings in storage of medical images. However, the level of information preserved depends on the parameters (compression rate), and should be modulated by the user's interest. The higher the compression rate (the fewer principal components are used in the characteristics vector) the more degraded the quality of the image recovered (examples 3 and 4).
In certain applications, such as brain function images, the central principle is the variation of the resonance signal over time. In these conditions, the spatial information may be maintained in a reference file, making it possible to compress subsequent images with no loss. On the other hand, it is still necessary to evaluate the pertinence of the application of high compression rates when an assessment of structures of reduced dimensions relative to the size of the voxels is needed.
Furthermore, the observation of the results from the application of the PCA technique in medical images may be considered a complexity measure. In other words, images with dense texture patterns tend to produce different results with the use of the technique described. Nevertheless, this hypothesis was not tested in this project; it only points to the line of investigation, in which the results may certify and quantify this possibility.
New secondary applications (based on the results here described) may encompass various conditions in the medical routine. These applications benefit from the procedures described herein. In this way, the comprehension of the principles here presented is important for the better use of medical applications based on these foundations.
The quantity of principal components used in compression influences the recovery of the original image from the compacted image. This tool allows significant savings of storage space, which can be critical in clinical applications and in processing large volumes of data. As a secondary property, these components also have the potential of reflecting the complexity of the image, enabling their correlation with the texture of the image.
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Rafael do Espírito Santo
Avenida Morumbi, 627/701 Morumbi
Zip code: 05651-901 São Paulo (SP), Brazil
Phone: (55 11) 2151-1366
Fax: (55 11) 2151-0273
Received on: Sep 5, 2011
Accepted on: Jun 13, 2012
Conflict of interest: none.
Study carried out at Instituto do Cérebro InCe, Hospital Israelita Albert Einstein HIAE, São Paulo (SP), Brazil.