Classification of Crystallographic Data Using Canonical Correlation Analysis

Hindawi Publishing Corporation EURASIP Journal on Advances in Signal Processing Volume 2007, Article ID 19260, 8 pages doi:10.1155/2007/19260 Research Article Classification of Crystallographic Data Using Canonical Correlation Analysis M. Ladisa,1 A. Lamura,2 and T. Laudadio2 1 Istituto 2 Istituto di Cristallografia (IC), CNR, Via Amendola 122/O, 70126 Bari, Italy Applicazioni Calcolo (IAC), CNR, Via Amendola 122/D, 70126 Bari, Italy Received 28 September 2006; Revised 10 January 2007; Accepted 4 March 2007 Recommended by Sabine Van Huffel A reliable and automatic method is applied to crystallographic data for tissue typing. The technique is based on canonical correlation analysis, a statistical method which makes use of the spectral-spatial information characterizing X-ray diffraction data measured from bone samples with implanted tissues. The performance has been compared with a standard crystallographic technique in terms of accuracy and automation. The proposed approach is able to provide reliable tissue classification with a direct tissue visualization without requiring any user interaction. Copyright © 2007 M. Ladisa et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. INTRODUCTION One of the main goals of tissue engineering is the reconstruction of highly damaged bony segments. To this aim, it is possible to exploit the patient’s own cells, which are isolated, expanded in vitro, loaded onto a bioceramic scaffold, and, finally, reimplanted into the lesion site. Generally, bone marrow stromal cells (BMSC) are adopted, as described in [1]. In this respect it would be important to characterize the structure of the engineered bone and to evaluate whether the BMSC extracellular matrix deposition on a bioceramic scaffold repeats the morphogenesis of the natural bone development. In addition, it is also interesting to look into the interaction between the newly deposited bone and the scaffold in order to recuperate damaged tissues. This is due to the fact that the spatial organization of the new bone and the bonebiomaterial integration is regulated by the chemistry and the geometry of the scaffold used to place BMSC in the lesion site [1–3]. In this context the standard crystallographic approach to detect the different tissues is based on a quantitative analysis performed by the Rietveld technique [4, 5]. This method allows to determine the relative amounts of different tissue components but it is rather sophisticated and computationally demanding. The aim of this paper is to propose a new technique based on a statistical method called canonical correlation analysis (CCA) [6]. This method is the multivariate variant of the ordinary correlation analysis (OCA) and has already been successfully applied to several applications in biomedical signal processing [7, 8]. Here, CCA is applied to X-ray diffraction data in order to construct a nosologic image [9] of the bone sample in which all the detected tissues are visualized. The goal is achieved by combining the spectral-spatial information provided by the X-ray diffraction patterns and a signal subspace that models the spectrum of a characteristic tissue type. Such images can be easily interpreted by crystallographers. The paper is organized as follows. In Section 2, we present the mathematical aspects of the CCA method. Then the application of CCA to crystallographic data is reported in Section 3. In Section 4, the numerical results are described and discussed and, finally, we draw our conclusions. 2. CCA CCA is a statistical technique developed by Hotelling in 1936 in order to assess the relationship between two sets of variables [6]. It is a multichannel generalization of OCA, which quantifies the relationship between two random variables x and y by means of the so-called correlation coefficient Cov[x, y] , ρ=  V [x]V [y] (1) where Cov and V stand for covariance and variance, respectively. The correlation coefficient is a scalar with value 2 EURASIP Journal on Advances in Signal Processing between −1 and 1 that measures the degree of linear dependence between x and y. For zero-mean variables, (1) is replaced by E[xy] ρ =     , E x2 E y 2 (2) where E stands for expected value. Canonical correlation analysis can be applied to multichannel signal processing as follows: consider two zero-mean multivariate random vectors x = [x1 (t), . . . , xm (t)]T and y = [y1 (t), . . . , yn (t)]T , with t = 1, . . . , N, where the superscript T denotes the transpose. The following linear combinations of the components in x and y are defined, which, respectively, represent two new scalar random variables X and Y : X = wx1 x1 + · · · + wxm xm = wxT x, Y = w y1 y1 + · · · + w yn yn = wTy y. (3) CCA computes the linear combination coefficients wx = [wx1 , . . . , wxm ]T and w y = [w y1 , . . . , w yn ]T , called regression weights, so that the correlation between the new variables X and Y is maximum. The solution wx = w y = 0 is not allowed and the new variables X and Y are called canonical variates. Several implementations of CCA are available in the literature. However, as shown in [7], the most reliable and fastest implementation is based on the interpretation of CCA in terms of principal angles between linear subspaces [6, 10]. For further details the reader is referred to [7] and references therein. Here, an outline of the aforementioned implementation is provided for the sake of clarity. 2.1. Algorithm CCA (CCA by computing principal angles) Given the zero-mean multivariate random vectors x = [x1 (t), . . . , xm (t)] and y = [y1 (t), . . . , yn (t)], with t = 1, . . . , N.  and Y  , defined as follows: Step 1. Consider the matrices X ⎡ ⎢  =⎢ X ⎣ ⎡ ⎤ x1 (1) · · · xm (1) .. .. ⎥ ⎥ . . ⎦, x1 (N) · · · xm (N) ⎢  =⎢ Y ⎣ ⎤ y1 (1) · · · yn (1) .. .. ⎥ ⎥ . . ⎦. y1 (N) · · · yn (N) (4)  and Y : Step 2. Compute the QR decompositions [11] of X  = QX RX , X  = QY RY , Y (5) where QX and QY are orthogonal matrices and RX and RY are upper triangular matrices. Step 3. Compute the SVD [11] of QTX QY : QTX QY = USVT , (6) where S is a diagonal matrix and U and V are orthogonal matrices. The cosines of the principal angles are given by the diagonal elements of S. Figure 1: X-ray diffraction patterns of the investigated bone sample. Step 4. Set the canonical correlation coefficients equal to the diagonal elements of the matrix S and compute the corresponding regression weights as wX = RX−1 U and wY = RY− 1 V. The computation of the principal angles yields the most robust implementation of CCA, since it is able to provide re and Y  are singular. liable results even when the matrices X 3. CCA APPLIED TO CRYSTALLOGRAPHIC DATA During the data acquisition procedure, a number of microscopic X-ray diffraction images (XRDI) displaying the spatial v (...truncated)