INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume 11, 2017 Recursive Adaptive Color KLT for image sequences Roumen K. Kountchev and Roumiana A. Kountcheva Abstract - The method Recursive color space transform for a RDgDb, HSV or HSL, CMY, PhotoYCC, Lab, Luv, etc. sequence of correlated RGB images is based on the algorithm [2,3,8,12,13], is the maximum decorrelation of the transformed Adaptive Color KLT (AC-KLT). The non-recursive AC-KLT for a color components L1L2L3. The main disadvantage of the color sequence of color images is not efficient enough in this case, because KLТ is that it has higher computational complexity than the each image should be processed individually. Depending on the kind deterministic transforms. To partially overcome the problem, of the images in the processed sequence (for example, color video, approximately optimal color KLT with fixed coefficients is multi-view, medical sequences, etc.), the pixels of same position, but from two neighbor images, have significant similarity which in most presented in [8]. Another approach for KLT complexity cases is over 90 %. In such case, strong possibility exists for high reduction is given in [14], where it is used for recursive coding reduction of the information redundancy of the color information. To of image blocks of small size. In this case, the full transform is this end, here is offered the method Recursive AC-KLT (RAC-KLT) executed for the first block only, and for the next blocks are whose basic idea is to execute the AC-KLT for the first image in the calculated the values of the difference parameters in respect to sequence, and for the next images to calculate the values of the the previous. This approach was further developed in [15] for difference parameters only, which are needed for the successful local adaptive color transform of blocks and was aimed at the restoration of the color information. The efficiency of the new image compression. In each block of the image the coefficients method grows up together with the mutual correlation between the of the color transform matrix are determined from the consecutive images in the sequence. The evaluation of the computational complexity of the new algorithm confirms its previously compressed four neighbor blocks by using the advantage towards the basic AC-KLT. In the paper are also given weighted sums of the RGB pixel values, which makes the some experimental results from the comparison between color transform block-specific. The efficiency of the block coding is transforms YCrCb, AC-KLT and RAC-KLT which show the higher related to the spatial inter-block correlation, which depends on efficiency of the new approach. Here are also commented the the image contents and in many cases could be not too high. possible application areas for efficient representation of correlated For a large part of the existing image sequences the image sequences. correlation between pixels of same spatial position in two neighbor images is much higher than the inter-block Keywords—Adaptive Color Space Transform, Karhunen-Loeve correlation inside the image. To enhance the efficiency of the Transform, Recursive Adaptive Color KLT, Sequence of Correlated Color Images. RGB components processing, for each pixel from a given image in the group here is offered to execute recursive I. INTRODUCTION calculation of the KLT transform matrix of size 3×3. This approach is the basics of the method called Recursive Adaptive The processing and analysis of sequences of correlated color images are recently the object of significant number of Color KLT (RAC-KLT), presented here. It is based on the scientific investigations [1-4] aimed at the reduction of the adaptive RGB space transform for a single image, called color information redundancy, noise filtration, segmentation Adaptive Color KLT (AC-KLT) [16], which has lower and objects recognition based on their color, tracing of moving computational complexity than the famous KLT algorithms objects in video sequences, analysis of the color characteristics based on iterative computational methods [5,6]. Together with in selected areas in a group of images, etc. One of the most the fast development of the contemporary computer efficient transforms of the initial colors space (RGB) is the technologies the color space transform based on the AC-KLT Karhunen-Loeve Transform (KLТ) [5-10] also known as becomes suitable for more application areas, where the Hotelling Transform (НТ), or Principal Component Analysis processing time satisfies the real-time implementation (PCA). Its most important advantage towards the well-known requirements. deterministic color transforms YCrCb (the most commonly The paper is arranged as follows: In Section II is given the used color space for compression), YUV, YIQ, RCT, YCoCg, short description of AC-KLT for color images; in Sections III and IV are correspondingly presented the RAC-KLT method Prof. D. Sc. PhD. R. K. Kountchev is with the Technical University of and algorithm; in Section V the computational complexity of Sofia, Department of Radio Communications and Video Technologies, Bul. RAC-KLT is compared with that of AC-KLT; in Section VI are Kl. Ohridsky 8, Sofia 1000, Bulgaria (corresponding author, phone: shown some experimental results; in Section VII are +899924284) e-mail: rkountch@ tu-sofia.bg). Dr. R. A. Kountcheva is with TK Engineering, Sofia, Bulgaria (e-mail: commented a part of the possible applications of the new [email protected]). algorithm, and Section VIII contains the conclusions. ISSN: 1998-4464 454 INTERNATIONAL JOURNAL OF CIRCUITS, SYSTEMS AND SIGNAL PROCESSING Volume 11, 2017 3 2 det| ki j− λδ i j |=λ + a λ + b λ + c = 0, (7) II. ADAPTIVE COLOR KLT FOR THE IMAGE RGB COMPONENTS 1, i= j, where for δji = is got: Here follows the brief mathematical description of the 0, i≠ j. RGB color space transform of the color image by using the a = −(k1 +k 2 + k 3 ), AC-KLT, in accordance with [15]. The transform has two 2 2 2 basic forms: Arithmetic and Trigonometric. b= k1 k 2 + k 1 k 3 + k 2 k 3 −(k 4 +k5 + k6), (8) 2 2 2 A. Arithmetic form of AC-KLT c= k1 k 6 + k2 k 5 + k3 k 4 −(k1k 2 k 3 + 2k 4 k 5 k 6), Direct AC-KLT for the R,G,B components of the image of The matrix [KC] is symmetrical and its eigenvalues are S pixels and of size m×n=S: always real numbers. They can be defined using the Cardano L[= Φ](C − m) or L= [ Φ ]X for s=1,2,..,S. (1) relations for the “casus irreducibilis” (or the so-called s s s s “trigonometric solution”): T Here Cs= [R s ,G s ,B s ] is the color vector which represents λ1 =2 p /3cos()ϕ /3 − (a /3) , the processed pixel s from the image of S pixels; (9) T λ3,2 =−2 p /3cos[(ϕ π ) /3] − (a /3) . Ls= [L 1s ,L 2s ,L 3s ] - the transformed color vector of the pixel s; For these eigenvalues are satisfied the relations: T λ1 ≥ λ 2 ≥ λ 3 ≥ 0 and λ1 + λ 2 + λ 3 =k 1 + k 1 + k 1 . m= [E(R s ),E(Gs),E(B s )] - the mean color vector, S In (9) are used the following quantities: 3 where x= E(x)s = (1/S)∑ xs is the averaging operator for xs in = − + s= 1 q 2(a/3) (ab)/3 c, 2 respect to s; p= -(a /3) + b < 0, (10) T = − = 3 Xs C s m [x1s ,x 2s ,x 3s ] - the modified color vector s, ϕ = − arccos [ q / 2 (| )3/|p ]. for which E(Xs )= 0; The solution of the systems of equations (6) defines the T [][,,]Φ = Φ Φ Φ - the transposed matrix of size 3×3 elements Φik of the matrix []Φ when i,k=1,2,3, and the rows 1 2 3 used for the adaptive color KLT; of the matrix comprise the eigen vectors Φ . k Φ =[,,] Φ Φ Φ T when k=1,2,3 are the eigen vectors of k 1k 2k 3k Φi1 =A ii /P , Φ i2 =B ii /P , Φ i3 = D ii /P for i =1,2,3, (11) the color covariance matrix defined on the basis of the where T modified vectors Xs= [x 1s ,x 2s ,x 3s ] : Ai=(k 3 − λ i 5(k)[k 2−λ i) − k 4 k 6 ], (12) S T 1 T Bi= (k 3 −λ i)[k 6 (k 1 −λ i ) − k 4 k 5 ], [Kc ]= E(Xs X s ) = ∑XX.s s (2) S− 1 s= 1 D= k [2k k− k (k − λ)] −k2 (k − λ ), The elements k of this matrix are symmetric in respect to i 6 4 5 6 1 i 5 2 i ij (13) its main diagonal, and it is represented as: 2 2 2 PABD.i= i +i + i k11 k 12 k 13 k1 k 4 k 5 Then from (11) it follows that the direct AC-KLT could be = = [Kc ] k21 k 22 k 23 k4 k 2 k 6 . (3) represented as given below: L A /P B /P D /P R E(R ) k31 k 32 k 33 k5 k 6 k 3 s1 1 1 1 1 1 1 s s = × − In the relation above are used the substitutions: k1=k11, L s2 A2 /P 2 B 2 /P 2 D 2 /P 2 Gs E(Gs ). (14) k2=k22, k3=k33, k4=k12=k21, k5=k13=k31, k6=k23=k32. The L s3 A3 /P 3 B 3 /P 3 D 3 /P 3 Bs E(Bs ) elements k1 ÷ k6 of the covariance matrix are calculated from the relations: The inverse AC-KLT is defined by the relations: 2 2 2 2 2 2 = ΦT + = Φ T k1= E(R s ) − (R), k2 = E(Gs ) − (G), k3 = E(B)-(B); s (4) Cs [ ] Ls m or X[]L.s s (15) k4= E(Rs G s ) − (R)(G), From (14) and (15) it follows that the inverse AC-KLT could be represented as: k5= E(Rs B s ) − (R)(B), (5) R s A1 /P 1 A 2 /P 2 A 3 /P 3 L s1 E(Rs ) k6= E(Gs B s ) − (G)(B).