Image Quality Measurement Besides Distortion Type Classifying Ahmad MAHMOUDI-AZNAVEH, Azadeh MANSOURI, Farah TORKAMANI-AZAR  , and Mohammad ESLAMI Faculty of Electrical and Computer Engineering, University of Shahid Beheshti, G.C. Tehran, Iran (Received July 7, 2008; Accepted October 15, 2008) To identify the distortion type and quantify the quality of images, a new method is presented based on a comparison among the structural properties as well as consideration of the luminance characteristics of the two compared images. To fulfill this aim, the mathematical concept of the singular value decomposition (SVD) theorem has been applied. The difference vector of the reflection coefficients of the disturbed and the original image on the right singular vector matrix of the original image are considered. Many tests were conducted to evaluate the performance, using a widespread subjective study involving 779 images from the Live Image Quality Assessment Database, Release 2005. The results showed a greatly improved performance along with the ability to distinguish distortion type of images. # 2009 The Optical Society of Japan Keywords: image processing, image quality, matrix algebra, matrix decomposition, singular value decomposition 1. Introduction Evaluating the quality of digital image/video with respect to the purpose of the visual information applications is one of the challenging problems in the field of image and video processing. Subjective assessments, which are evaluated by humans, are widely used, however, careful subjective methods are experimentally difficult and time consuming. 1) Some researchers have tried to modify the existing quanti- tative measures to accommodate the factor of human visual perception while these methods work with just pixel values. 2–11) Two important quantitative measures were introduced: the universal image quality index (UQI), 3) and its improved form called the structural similarity index (SSIM), 4) as in eq. (1). SS-SSIM ¼ ð2 x  y þ C 1 Þð2 xy þ C 2 Þ ð 2 x  2 y þ C 1 Þð 2 x  2 y þ C 2 Þ ; ð1Þ where x and y are the original and disturbed images, and  x ,  x , and  xy are the mean of x, the variance of x, and the covariance of x and y, respectively. In fact,  x and  x can be viewed as estimates of the luminance and contrast of x, and  xy measures the tendency of x and y to vary together, thus can be considered as an indication of structural similarity. Adding the two small constants C 1 and C 2 prevents the drawback of a case with unstable measurements. The methods use a sliding window approach to calculate the SSIM index within the local window. To evaluate the overall image quality, a mean SSIM index of the whole distorted image is used. Wang et al. generalized the SS-SSIM idea in a multi-scale image and proposed the Multi-Scale-Structural Similarity Index: MS-SSIM, 5) which could produced a better result than before. However, the requirement of processing in different scales is considered a serious disadvantage. In 2006, Eskicioglu et al. used the singular value decom- position theorem (SVD) to measure the quality of images. This algorithm was based on the differences between singular values of the original and distorted images in small blocks with the size of m  m as M SVD. 9) Another method, which is given in ref. 11, used the fundamentals of Webber’s law and JND parameter. 12) The ratio of the first and second singular value of each block to the mean brightness value is used to show the rate of distortion. This algorithm could identify the type of distortion as well. In two recent SVD based methods only the singular values were used to quantify the image quality. Although there was some improvement, the SVs include mainly the luminance factor of the image and do not consider the structural information. Indeed, the singular vectors of each image describe the structural characteristics. To evaluate the perceptual quality of images, not only is the brightness important, but also the relation between the structural components has an important role in measuring the quality. In this paper, we used the singular vector matrices to improve the performance of quality measurement. In the following sections, the mathematical foundations of the proposed algorithm are explained. In §3, the simulation results are explained for validation purposes. 2. The Proposed Method SVD is a method for identifying and ordering the dimensions along which data points exhibit the most variation. SVD is based on a theorem from linear algebra which says that a rectangular matrix A can be broken down into the product of three matrices, an orthogonal matrix U,a diagonal matrix S, and the transpose of an orthogonal matrix V . The theorem is usually presented as A mn ¼ U mm S mn V T nn ; ð2Þ where U T U ¼ I and V T V ¼ I (I is identity matrix). Also the columns of U ¼fU 1 ; U 2 ; ... ; U m g are orthonormal eigen- vectors of AA T , the columns of V ¼fV 1 ; V 2 ; ... ; V n g are orthonormal eigenvectors of A T A, and S is a diagonal matrix containing singular values, s i , which are the square roots of eigenvalues from U or V in descending order. To show the pivotal effect of the eigenvectors in image formation, the  E-mail address: f-torkamani@sbu.ac.ir OPTICAL REVIEW Vol. 16, No. 1 (2009) 30–34 30