International Journal of Signal Processing Roumen Kountchev, Roumiana Kountcheva http://iaras.org/iaras/journals/ijsp

Color Space Transform for Correlated Images Based on the Recursive Adaptive KLT

ROUMEN KOUNTCHEV ROUMIANA KOUNTCHEVA Radio Communications and Video Technologies Technical University of Sofia TK Engineering Boul. Kl. Ohridsy 8, Sofia 1000 Drujba2, Bl. 404/2, Sofia 1582 BULGARIA BULGARIA [email protected] http://www.tu-sofia.bg [email protected]

Abstract: - In this work is presented a method for RGB color space transform for a group of correlated images, based on the recursive calculation of the algorithm Adaptive Color KLT (AC-KLT), developed earlier by the authors. However, the use of the algorithm AC-KLT for a group of color images is not sufficiently efficient, because each image should be processed individually. Depending on the image kind (for example, video- or multi-view sequences, medical image sequences, etc.) the pixels of same spatial position can have very high similarity, which in some cases is more than 90%. In such case, there is a high possibility to reduce significantly the information redundancy of the color information for the processed group of images. For this, here is offered the method Recursive AC-KLT, whose basic idea is the AC-KLT to be calculated for the first image in the group only, and for all other images to calculate the values of the difference parameters, needed to restore the color information of the image. The efficiency of the presented approach depends on the mutual correlation between images in the group. The algorithm efficiency is evaluated, and the obtained results confirm its advantage over the basic AC-KLT algorithm. The new method could be used for processing groups of correlated images in the systems for compression and processing of visual information, computer vision, pattern recognition, digital watermarking, etc.

Key-Words: Karhunen-Loève Transform, Adaptive Color Space Transform, Recursive Adaptive Color KLT, Group of Correlated Color Images

1 Introduction color KLТ the higher computational complexity than that of the deterministic transforms. However, The processing of sequences of color images became together with the development of the contemporary recently an object of many scientific investigations computer technologies, it becomes easily overcome, [1-4], aimed at the reduction of the color information which defines the color KLТ actuality. redundancy, noise filtration, color-based In [12] are given some applications of the KLT for segmentation and recognition of the objects in the recursive image block coding. In this case, the image, tracing of moving objects in video sequences, transform is calculated for the first image block, and color analysis of selected objects in groups of for the next blocks are calculated the difference images, etc. One of the most efficient transforms of parameters only. The efficiency of this approach is the primary image space RGB in mean square sense strongly dependable on the spatial inter-block is the Karhunen-Loève Transform (KLТ) [5-9], also correlation, which in some cases (depending on the known as the Hotelling Transform (НТ) or the image contents) is not high enough. As it is known, Principal Component Analysis (PCA). Its most for many kinds of images obtained from various important advantage in respect of the famous sources, the correlation between pixels of same determined color transforms of the kind YCrCb, spatial position in two neighbor images is much YUV, YIQ, YCoCg, RCT, HSV (or HSL), CMY, higher than the corresponding inter-block PhotoYCC, Lab, Luv, etc. [2,3,10,11], is the correlation. Depending on the image kind (for maximum decorrelation of the transformed color example, color video or multi-view sequences, components L1L2L3. The main disadvantage of the medical sequences, etc.) the similarity between

ISSN: 2367-8984 72 Volume 2, 2017 International Journal of Signal Processing Roumen Kountchev, Roumiana Kountcheva http://iaras.org/iaras/journals/ijsp

neighbor image couples could be very high (above 2.1 Arithmetic representation of AC-KLT 90%). In such case, there is a possibility for The direct AC-KLT of the color components R,G,B significant reduction of the information redundancy for an image of S pixels and of size m×n=S, is: for the processed group of images.     To enhance the processing efficiency here is s [L ]( s−Φ= )mC or s Φ= X][L s , for s=1,2,..,S. (1)  offered the method, called Recursive Adaptive Color = T KLT (RAC-KLT). In accordance with the new Here ssss ]B,G,R[C is the color vector which approach, on the RGB components of each pixel is represents the pixel s in the image of S pixels;  T applied recursive KLT with transform matrix of size = s3s2s1s ]L,L,L[L - the transformed color vector 3×3. The basis of the algorithm is the Adaptive RGB  = T space transform algorithm for a single image, based of the pixel s; R(E[m ), G(E ), B(E sss )] - the on the KLT, and called Adaptive Color KLT (AC- S == x)S/1()x(Ex KLT) [13]. It has lower computational complexity mean color vector, where s ∑ s is =1s than the KLT algorithms, which use iterative the mean operator (for x in respect to s); calculation methods [5,6]. The basic advantage of s    T AC-KLT towards the famous deterministic color ss =−= s3s2s1 ]x,x,x[mCX - the modified color transforms is the maximum decorrelation of the  T  vector s, for which = ;0)X(E ΦΦΦ=Φ ],,[][ transformed three components, in result of which the s 321 color energy concentration is maximum in the first - the transposed matrix of size 3×3 used for the  T one. However, the use of AC-KLT for a group of adaptive color KLT; Φ ΦΦΦ= k3k2k1k ],,[ for color images is not efficient enough, because each k=1,2,3 - the eigen vectors of the color covariance image should be processed individually, and the matrix ]K[ , defined on the basis of the modified computational complexity of AC-KLT is higher than c  T that of the deterministic color transforms. These vectors = s3s2s1s :]x,x,x[X disadvantages are overcome by RAC-KLT. S  T 1  T The paper is arranged as follows. In Section 2 is c ss )XX(E]K[ == ∑ ss .XX (2) given the short presentation of the algorithm AC- −1S =1s KLT for a single color image; in Sections 3 and 4 The elements kij of this matrix are symmetric in are presented the method and the corresponding respect to its main diagonal, and it is represented as: algorithm for RAC-KLT for a group of correlated images; in Section 5 is evaluated the efficiency of  kkk 131211   kkk 541  RAC-KLT, compared to AC-KLT, in Section 6 are =   =   c ]K[  kkk 232221   kkk 624 . (3) commented the possible applications of the new  kkk   kkk  algorithm, and section 7 contains the Conclusions.  333231   365  In the relation above are used the substitutions: k1=k11, k2=k22, k3=k33, k4=k12=k21, k5=k13=k31, 2 Adaptive Color KLT for the Image k6=k23=k32. The elements k1 ÷ k6 of the covariance RGB Components matrix are calculated from the relations: 2 2 2 2 22 The use of the Karhunen-Loève Transform for s1 ,)R()R(Ek s2 ,)G()G(Ek =−=−= s3 ;)B(-)B(Ek (4) processing of the image primary color components results in their decorrelation, which ensures the ss4 −= R()GR(Ek )(G), ss5 −= R()BR(Ek )(B), enhancement of such operations as: compression, ss6 −= G()BG(Ek )(B). (5) color-based segmentation, etc.  The basic problem is the high computational The eigenvectors Φk of the matrix [KC] are the complexity of KLT. The method Adaptive Color solution of the set of equations: KLT offers precise, and together with this, non- 3   2 2 iterative calculation of the covariance matrix ]K[ Φλ=Φ kkkC and k ∑ ki =Φ=Φ 1 for k=1, 2, 3.(6) eigenvectors and has lower computational =1i complexity. The detailed presentation of the method The eigenvalues ,, λλλ of the matrix [KC] are is given in earlier publication of the authors [13]. 321 the solution of its characteristic equation: The AC-KLT has two basic forms, given below. 23 det − λδ jiji =+λ+λ+λ= ,0cba|k| (7)

ISSN: 2367-8984 73 Volume 2, 2017 International Journal of Signal Processing Roumen Kountchev, Roumiana Kountcheva http://iaras.org/iaras/journals/ijsp

  ,1 for i= j, L is maximum, the second ( L ) is much smaller, where: =δ  s s2 ji ,0 for i≠ j,  and the last ( L s3 ) is close to zero. 2 2 2 a (k kk 321 ), b kkkkkk (k4323121 5++−++=++−= kk 6),

2 2 kkkkkkc 2 (k +−++= .)kkk2kk (8) 61 52 43 654321 2.2. Trigonometric representation of AC-KLT

Since the matrix [KC] is symmetrical, its The matrix Φ][ could be represented through the eigenvalues are always real numbers. They can be Euler rotation angles (α,β,γ), as follows: defined using the Cardano relations for the “casus  ΦΦΦ 131211  irreducibilis” (“trigonometric solution”):   ][ =Φ ΦΦΦ = =λ 3/p2 cos()−ϕ )3/a(3/ ,  232221  (17) 1  ΦΦΦ  (9)  333231  −=λ −πϕ 3,2 3/p2 cos[(  )3/a(]3/) . 1([ )] 2 ([ )][ 3 ( )] ,,([ γβαΦ=γΦβΦαΦ= )], For these eigenvalues are satisfied the relations: where the angles α,β,γ define the position of the

321 ≥λ≥λ≥λ 0 and ++=λ+λ+λ 111321 .kkk transform coordinate axes L,L,L 321 in respect to In Eqs. (9) are used the following quantities: the original color RGB space; 2q (a/ 3 ()3 ab 3/) c, p -(a 2 <+=+−= ,0b)3/ cos α−α 0sin    (10) ([ )]=αΦ sin cosαα 0 ; arccos −=ϕ 2/q[ (| )3/|p 3 ]. 1    100  The solution of the systems of equations (6)  cosβ sin0 β defines the elements Φik of the matrix [Φ], when   ([ )]=βΦ 010 ; (18) i,k=1,2,3, and the rows of the matrix comprise the 2     0sin cosββ−  eigen vectors Φk. The corresponding elements are: cos γ−γ 0sin  ,P/A ,P/B =Φ=Φ=Φ P/D for i=1,2,3, (11) ii3iii2iii1i =γΦ  γγ  3 ([ )] sin cos 0. A (k λ (k)[k λ −−−= kk) ], 64i25i3i  100  k(B )[ −λ−λ−= kk)k(k 54i16i3i ], (12) From Eqs. (17) and (18) it follows, that the elements 2 of the matrix Φ][ are defined by the relations: k(kkk2[kD )] 5i16546i k(k λ−−λ−−= i2 ), coscoscos γα−γβα=Φ ;sinsin 2 2++= 2 .DBAP (13) 11 ii i i coscos sinsin cosγα−γβα−=Φ ; (19) Then from Eq. (11) it follows, that the direct AC- 12 βα=Φ KLT could be presented as: 13 cos ;sin sin coscoscos γα+γβα=Φ ;sin L   P/DP/BP/A   R   )R(E  21 s1 111111  s s          22 sin cos sin coscos γα+γβα−=Φ ; (20) L = P/DP/BP/A × G − )G(E . (14)  s2   222222   s   s  βα=Φ ;sinsin          23 L s3   P/DP/BP/A 333333   Bs   s )B(E  31 sin cosγβ−=Φ ; 32 γβ=Φ ;sinsin 33 cosβ=Φ . (21) The inverse AC-KLT is defined by the relations: The matrix of the inverse AC-KLT in this case is  T   T  defined as: s s+Φ= mL][C , or s Φ= L][X s , (15) − T1 1([][][ )] 2 ([ )] 3 ([ α−Φβ−Φγ−Φ=Φ=Φ )]. (22) From Eqs. (14) and (15) it follows that the inverse −1 Hence, to define Φ][ are needed the Euler angles AC-KLT could be represented as: α, β and γ only. These angles are calculated using R s   P/AP/AP/A 332211  L s1   s )R(E  the matrix elements:         G = P/BP/BP/B × L + )G(E . (16) 2 2 2  s   332211   s2   s  =α arcsin(Φ 1 Φ =− +BAPPD(arcsin) ); (23)         23 33 3232 3 Bs   P/DP/DP/D 332211  L s3   s )B(E  arccos()()33 =Φ=β arccos 33 ;P/D (24) As a result of the direct AC-KLT, the energy of the first component ( L ) of the transformed vector 2 2 2 s1 arccos( 32 33 =Φ−Φ=γ 33 + 3 .)BABarccos()1 (25)

ISSN: 2367-8984 74 Volume 2, 2017 International Journal of Signal Processing Roumen Kountchev, Roumiana Kountcheva http://iaras.org/iaras/journals/ijsp

The elements of the matrix Φ][ −1 are restored by The RAC-KLT processing for one image in the group is based on the following operations: using the angles α, β, γ. As a result, to the decoder are transferred the values of angles α, β, γ only, • Recursive calculation of the color covariance matrix [K ] of the processed image with sequential instead of all 9 elements of the matrix Φ][ −1 , i.e. the c,t number t, based on [Kc,t-1], calculated for the number of needed coefficients is 3 times smaller. preceding image, with number (t-1):

∆+= −1t,ct,c K[]K[]K[ t,c ], (26) 3 Recursive Adaptive Color KLT for a where Group of Color Images  − − − − − −1t,5t,51t,4t,41t,1t,1 )kk()kk()kk(    =∆ − − − The method, called Recursive Adaptive Color t,c ]K[  − − −1t,6t,61t,2t,21t,4t,4 )kk()kk()kk( . (27)   KLT (RAC-KLT), is aimed at the processing of a  − − − − − −1t,3t,31t,6t,61t,5t,5 )kk()kk()kk(  sequence of correlated images, separated into It is supposed here that in the related equations all groups with fixed, or adaptively changing length. In quantities with index (t-1) are already calculated for the first case, the fixed length is defined on the basis the previous image with number (t-1) and are stored of the averaged range of the mutual correlation in the dynamic buffer memory. calculated for the images in the group. In the second  case, the length is set in accordance with the value • Recursive calculation of the eigen vectors Φ t,k  of the mutual correlation between the couples of ∆+= ΦΦΦ , for k = 3,2,1 , (28) neighbor images in the group, which should be − t,k1t,kt,k where each kth difference eigen vector higher than a pre-selected threshold. The method  RAC-KLT is applied for the images from each −=∆ ΦΦΦ −1t,kt,kt,k is the solution of the system of group, got after its separation into sub-groups with linear equations for k=1,2,3: fixed, or adaptive changing length (number of   3 images in the group).  2 2 c, ]K[ Φ ∆λ∆=∆∆ Φ t,kt,kt,kt , k =Φ∆ ∑ ∆Φ ki = .1 (29) =1i

The difference eigen values λ∆ t,k of the matrix

∆ t,c ]K[ are defined by relations, similar to Eqs. (9), (10), and the matrix Φ ][ - by relations, similar to TV Frame 1 TV Frame 2 t Eqs. (19)-(21). Each of the participating quantities should be assigned the index t.

• Recursive calculation of the angles ,, γβα ttt for the processed image t:

TV Frame 3 TV Frame 4 − α∆+α=α t1tt , − β∆+β=β t1tt , − γ∆+γ=γ t1tt , (30) where the difference angles are defined similarly to Eqs. (23) - (25) as follows: 2 2 αt arcsin t,3t,2t,3t,2 ∆+∆∆∆∆=∆ BAPPD( 3 t, ),

TV Frame 5 TV Frame 6 t arccos()∆∆=β∆ t,3t,3 ,P/D (31) 2 2 t arccos( t,3 t,3 ∆+∆∆=γ∆ t,3 .)BAB • Calculation of the direct RAC-KLT for the vectors    −= mCX tt,st,s in accordance with the relations: TV Frame 7 TV Frame 8   s ,,([L γβαΦ= )]X t,sttt , for s=1,2,..,S. (32) Fig. 1. A sequence of 8 TV frames from video camera with 25 fps The matrix ,,([ γβαΦ ttt )] is defined by the Examples for groups of correlated images of relations from Table 1, where in the last line the different kind (sequence of TV frames) with fixed mutual correlation between the corresponding two length (N=8) are shown on Fig. 1. sequential color images with sequential numbers (t) and (t-1) is close to 100 %. This permits the

ISSN: 2367-8984 75 Volume 2, 2017 International Journal of Signal Processing Roumen Kountchev, Roumiana Kountcheva http://iaras.org/iaras/journals/ijsp

calculations needed for AC-KLT to be reduced = − R()GR(Ek )(G000,s0,s0,4 ), many times, because the matrix Φ ][ should not be t = − R()BR(Ek )(B ), (36) calculated (instead, the already calculated matrix 000,s0.s0,5 = − G()BG(Ek )(B000,s0,s0,6 ). Φ −1t ][ is used). Table 1 Step 2. Calculation of the parameters, defined by the elements of the matrix [Kc,0]: t ,,([][ γβαΦ=Φ ttt )] Condition a (k ++−= kk 0,30,20,10 ), (37) ([ )][ ( )] ([ γΦβΦαΦ t3t2t1 )] ttt ≠γ∆≠β∆≠α∆ 0,0,0 kkkkkkb (k 2 2 ++−++= kk 2 ), (38) ([ − )][ ( )] ([ γΦβΦαΦ t3t21t1 )] ,0,0 ttt ≠γ∆≠β∆≈α∆ 0 0,40,30,20,30,10,20,10 0,5 0,6 2 2 2 ([ )][ ( − )] ([ γΦβΦαΦ t31t2t1 )] ttt ≠γ∆≈β∆≠α∆ 0,0,0 0,60,10 0,50,2 kkkkkkc 0,40,3 −++= (k + 0,60,50,40,30,20,1 .)kkk2kk 3 3 ([ )][ ( )] ([ γΦβΦαΦ − )] ,0,0 ≈γ∆≠β∆≠α∆ 0 1t3t2t1 ttt 2q (a 00 pc3/)ba()3/ -, (a 00000 +=+−= 0 ,b)3/ ([ )][ ( )] ([ γΦβΦαΦ )] − − t31t21t1 ttt ≠γ∆≈β∆≈α∆ 0,0,0 3 0 arccos[−=ϕ 0 )2/q( (| 0 ,])3/|p (39) ([ − )][ ( )] ([ γΦβΦαΦ −1t3t21t1 )] ttt ≈γ∆≠β∆≈α∆ 0,0,0 =λ 01 3/p2 cos()0 −ϕ 0 )3/a(3/ , ([ )][ ( − )] ([ γΦβΦαΦ −1t31t2t1 )] ,0 ,0 ttt ≈γ∆≈β∆≠α∆ 0 (40) 3,2 −=λ 0 3/p2 cos[( 0  −πϕ 0 )3/a(]3/) , ([ − )][ ( − )] ([ γΦβΦαΦ −1t31t21t1 )] ttt ≈γ∆≈β∆≈α∆ 0,0,0 k(A λ−= )[ −λ− kk)k(k ], • Calculation of the inverse RAC-KLT (IRAC- 0,60,40,30,20,50,30,30,3 KLT) by analogy with Eq. (15): k(A λ−= )[ −λ− kk)k(k 0,60,40,30,20,50,30,30,3 ], (41) -1 T   T   2 tt,s Φ=Φ= tt,s L][L][X t,s or tt,s +Φ= tt,s .mL][C (33) = k(kkk2[kD )] 0,50,20,10,60,50,40,62,0 k(k λ−−λ−− 0,20,2 ), Here the inverse transform matrix Φ ][ T and the 2 2 2 2 2 2 t 0,20,2 0,2 ++= 0,2 ,DBAP 0,30,3 0,3 ++= 0,3 .DBAP (42)    vectors m,C,L tt,st,s are calculated recursively: Step 3. Calculation of the trigonometric functions  T T T sin(.) и cos(.) for the angles α0, β0, γ0: t −1t +Φ=Φ ∆Φ t ,][][][ − ∆+= t,s1t,st,s ,LLL (34)   2 2 2 =α 0,30,20,30,20 + 0,3 ,BAPPDsin cos 0 α−=α 0 ,)(sin1 (43) − ∆+= t,s1t,st,s ,CCC ∆+= t1-tt .mmm (35) ∆Φ T 2 The difference matrix t ][ for the inverse cos =β 0,30,30 ,P/D 0 1sin (cosβ−=β 0 ,) (44) transform is calculated in correspondence with Eq. 2 2 2 (22), but using the inverse difference angles, cos =γ 0,30,30 + 0,3 ,BAB 0 1sin (cosγ−=γ 0 ,) (45)

( t),( t),( α∆−β∆−γ∆− t). Step 4. Calculation of the initial AC-KLT matrix

Φ0 ][ as a product of the corresponding rotation matrices: 4. Algorithm RAC-KLT ([][ )] ([ )][ ( 0101010 )] ,,([ γβαΦ=γΦβΦαΦ=Φ 000 )]. (46) On the basis of the above presented method was developed the corresponding algorithm (RAC-KLT) Step 5. Calculation of the initial mean color vector S for the processing of a group of N correlated   T 0= ∑ 0,s = R(E[C)S/1(m ), G(E ), B(E 0,s0,s0,s )] . (47) images. The consecutive number of the image is =1s given as t = 0,1,..,N-1. Step 6. Calculation of the transformed color vectors for the image with number t=0:    −Φ=Φ= 4.1. Algorithm for direct RAC-KLT [X][L ]( 00,s00,s00,s )mC for s=1,2,.,S. (48) For the first image in the group (with number t=0) is For each of the remaining images in the group executed the algorithm initialization, which (t=t+1 for t≤N-1) are executed steps, similar to these comprises the following steps: for the image with number t=0:

Step 1. Calculation of the elements of the initial Step 7. Calculation of the rotation angles ,, γβα ttt color covariance matrix [Kc,0]: for the image t=t+1 using relations, similar to Eqs. 2 2 2 2 2 2 (43)-(45), where the index 0 is replaced by t. 0,s0,1 −= 0 ,)R()R(Ek 0,s0,2 )G(Ek −= 0 ,)G( = 0,s0,3 -)B(Ek 0 ,)B( Step 8. Calculation of the difference rotation angles for the image t:

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Step 13. Calculation of the modified angles α α−α=∆ −1ttt , β−β=β∆ −1ttt , γ−γ=γ∆ −1ttt . (49) ,, γ∆β∆α∆ ∗∗∗ satisfying the rules: Step 9. Calculation of the approximated angles ttt  ,0 if δ<α∆ ;|| ˆ , ˆ ,γβα ˆ for the image t: ∗ t ttt t =α∆   α∆ t - in cases,other  αˆ −1t , if t δ<α∆ ;|| ˆ =α  t ∗  ,0 if t δ<β∆ ;|| ˆ − α∆+α - in cases,other  t1t t =β∆  (51)  β∆ t - in cases,other βˆ δ<β∆ ˆ  −1t , if t ;|| t =β  (50)  ,0 if δ<γ∆ ;|| ˆ −β∆+β ∗ t  − t1t in cases,other t =γ∆   γ∆ t - in cases.other γˆ −1t , if t δ<γ∆ ;|| ˆ t =γ  Step 14. Calculation of the difference vectors: ˆ − t1t −γ∆+γ in cases.other  ˆ ˆ ˆ  In the relations above, xˆ means the approximation L =∆ -L −1t,st,st,s ,L −=∆ 1-ttt .mmm (52) of x, and δ is a preset threshold (a small number, for When t reaches the value t=N, the execution of example, in the range 0,01-0,05), which defines the the direct RAC-KLT for the group of images is accuracy of the restored color vectors after the finished. As a result, the values of the vectors' inverse RAC-KLT, and the data which represents the     components of ,L ,m ∆ ˆ ,L ∆m and of the difference angles. 0,s 0 t,s t ∗∗∗ angles γβα 000 ,,, ,, γ∆β∆α∆ ttt for t=0,1,2,..,N-1 Step 10. The values of functions sin αˆ t , cosαˆ t, and s=1,2,..,S, are got. After that they could be βˆ βˆ γˆ γˆ sin t , cos t , sin t , cos t are set from the pre- efficiently coded using various methods, selected in calculated tables. After that, the matrix accordance with the application of the algorithm ˆ RAC-KLT. ([ ˆ , , γβαΦ ˆ tttt )] is calculated as the product of the

ˆ γΦ three matrices ([ αΦ ˆ t1 )], ([ βΦ t2 )], ([ ˆ t3 )] in correspondence with Eqs. (43-45), where the index 0 4.2. Algorithm of the inverse RAC-KLT is replaced by t.  The input parameters, needed for the inverse Step 11. Calculation of the mean color vector m t RAC-KLT are: for the image t, in correspondence with Eq. (47) - the initial angles ,, γβα 000 and the where the index 0 is replaced by t, and the vectors      components of the vectors L 0,s and m0 for the −= mCX ; tt,st,s image with group number t=0, when s=1,2,..,S; Step 12. Calculation of the restored vectors ∗∗∗   - the difference angles ,, γ∆β∆α∆ ttt and the ˆ ˆ  ([L ˆ , , γβαΦ= ˆ )]X t,stttt,s on the basis of Table 1 and  components of the vectors ∆L t,s and ∆m for each Eq. (50), for s=1,2,..,S, as given in Table 2. t image with number t=1,2,..,N-1, for s=1,2,..,S. Table 2 For the first image (with number t=0) the inverse ˆ Condition ([ ˆ , ,γβαΦ ˆ ttt )] RAC-KLT comprises the following steps: ˆ ˆ ([ ˆ )][ ( )] ([ γΦβΦαΦ ˆ t3t2t1 )] | ˆ t |,| |,| ˆ tt | δ≥γ∆δ≥β∆δ≥α∆ Step 1. Calculation of the matrix Φ0][ , using the ˆ ˆ ([ ˆ − )][ ( )] ([ γΦβΦαΦ ˆ t3t21t1 )] | ˆ t |,| |,| ˆ tt | δ≥γ∆δ≥β∆δ<α∆ values of ,, γβα 000 in correspondence with (46). ˆ ˆ ([ ˆ )][ ( − )] ([ γΦβΦαΦ ˆ t31t2t1 )] | ˆ t |,| |,| ˆ tt | δ≥γ∆δ<β∆δ≥α∆ Step 2. Calculation of the initial restored colors, got after the inverse RAC-KLT: ˆ ˆ γΦβΦαΦ ˆ ˆ ˆ ˆ δ<γ∆δ≥β∆δ≥α∆ ([ )][ ( )] ([ −1t3t2t1 )] | t |,| |,| tt |    ˆ T ˆ ˆ ˆ 00,s ][C L +Φ= mˆ 00,s for s=1,2,..,S. (53) ([ ˆ − )][ ( − )] ([ γΦβΦαΦ ˆ t31t21t1 )] | ˆ t |,| |,| ˆ tt | δ≥γ∆δ<β∆δ<α∆ ˆ ˆ For each of the next images with number t=t+1 ([ ˆ − )][ ( )] ([ γΦβΦαΦ ˆ −1t3t21t1 )] | ˆ t |,| |,| ˆ tt | δ<γ∆δ≥β∆δ<α∆ when t ≤ N-1, the inverse RAC-KLT comprises the ˆ ˆ γΦβΦαΦ ˆ ˆ ˆ ˆ δ<γ∆δ<β∆δ≥α∆ ([ )][ ( − )] ([ −1t31t2t1 )] | t |,| |,| tt | steps: ˆ ˆ ([ ˆ − )][ ( − )] ([ γΦβΦαΦ ˆ −1t31t21t1 )] | ˆ t |,| |,| ˆ tt | δ<γ∆δ<β∆δ<α∆

ISSN: 2367-8984 77 Volume 2, 2017 International Journal of Signal Processing Roumen Kountchev, Roumiana Kountcheva http://iaras.org/iaras/journals/ijsp

Step 3. Calculation of the difference matrix ∆Φ t ][ To execute the inverse color transform for each pixel s in the processed image should be calculated γ∆β∆α∆ ∗∗∗ based on the values of ,, ttt in the following parameters: correspondence with relations from Table 3. - for AC-KLT (for one image) - the parameters Table 3   s γβα ,,,,m,L which are represented by (4S+1)N ∆Φ Condition t ][ numbers in total (S is the number of pixels in the ([ )][ ( )] ([ γ∆Φβ∆Φα∆Φ )] ∗∗∗ − t3t21t1 ttt ≠γ∆≠β∆=α∆ 0,0,0 image, N - the number of images in the group);

∗∗∗ ([ )][ ( − )] ([ γ∆Φβ∆Φα∆Φ t31t2t1 )] ≠γ∆=β∆≠α∆ 0,0,0 - for RAC-KLT of the initial image t=0 - the ttt   ([ )][ ( )] ([ γ∆Φβ∆Φα∆Φ )] ∗∗∗ parameters ,,,m,L γβα . For each of the next −1t3t2t1 ttt =γ∆≠β∆≠α∆ 0,0,0 00000,s ([ )][ ( )] ([ γ∆Φβ∆Φα∆Φ )] ∗∗∗ images with number t=1,2,..,N-1 - the parameters − − t31t21t1 ttt ≠γ∆=β∆=α∆ 0,0,0   ([ )][ ( )] ([ γ∆Φβ∆Φα∆Φ )] ∗∗∗ ˆ ,m,L ˆ , ˆ , γ∆β∆α∆∆∆ ˆ . These parameters are − −1t3t21t1 ttt =γ∆≠β∆=α∆ 0,0,0 ttttt,s ∗∗∗ represented by (S+4)+(S+4)(N-1)=(S +4)N numbers ([ )][ ( − )] ([ γ∆Φβ∆Φα∆Φ −1t31t2t1 )] =γ∆=β∆≠α∆ 0,0,0 ttt for the whole group. ([ )][ ( )] ([ γ∆Φβ∆Φα∆Φ )] ∗∗∗ ≠γ∆≠β∆≠α∆ 0,0,0 t3t2t1 ttt Hence, there is no difference between both ([ )][ ( )] ([ γ∆Φβ∆Φα∆Φ )] ∗∗∗ − − −1t31t21t1 ttt =γ∆=β∆=α∆ 0,0,0 algorithms in respect of the needed parameters number. However, due to high mutual correlation T Step 4. Calculation of the matrix Φ t ][ for the between neighbor images, significant part of the inverse RAC-KLT of the image t: RAC-KLT parameters for the processed group are T T T close or equal to zero. For AC-KLT such − +Φ=Φ ∆Φ .][][][ (54) t 1t t  dependence does not exist and this is the main ˆ  difference between both algorithms. Step 5. Calculation of vectors L t,s and m t for the image t:   5.2. Computational complexity of algorithms AC- ˆ ˆ ∆+= ˆ ∆+= L L − t,s1t,st,s ,L t1-tt .mmm (55) KLT and RAC - KLT   Step 6. Restoration of color vectors C t,s and mt : The computational complexity is evaluated on the  T   basis of the mathematical operations (additions and +Φ= mL][C for s=1,2,..,S. (56) tt,s tt,s multiplications), needed for their execution. For this The execution of RAC-KLT stops when t=N is are used the generalized data, obtained after analysis satisfied. of the operations needed for the direct and inverse The presented algorithm RAC-KLT is asymmetric, color transforms for the compared algorithms. The because the number of calculations needed for the comparison is done for a group of N images, of S inverse transform is much lower than that for the pixels each: direct. - direct AC-KLT: (37S+124)N≈37SN; (57)

- direct RAC-KLT: 37S+124+[37S+129](N-1)≈ 5 Evaluation of RAC-KLT efficiency ≈37SN; (58) The efficiency of the algorithm RAC-KLT for a - inverse AC-KLT: (15S+23)N≈15SN; (59) group of correlated color images is evaluated here by comparison with that of the AC-KLT. For the - inverse RAC-KLT: (15S+47)N-24≈15SN. (60) evaluation are used the number of the parameters Hence, the computational complexity of the needed for the inverse color transform (which are direct and inverse RAC-KLT is approximately same added to the coded image information), and the as that of AC-KLT. computational complexity of the algorithm. The main advantage of RAC-KLT towards AC- KLT is the ability for higher reduction of the data needed for the representation of the color 5.1. Parameters of the inverse color transform parameters of the processed image group. Taking These parameters are got after the execution of the into account the high correlation between couples of direct transforms AC-KLT and RAC-KLT. sequential images, the values of the difference angles ,, γ∆β∆α∆ ttt and of the components of the

ISSN: 2367-8984 78 Volume 2, 2017 International Journal of Signal Processing Roumen Kountchev, Roumiana Kountcheva http://iaras.org/iaras/journals/ijsp

 applications of RAC-KLT should be developed its difference vectors ∆Lˆ for significant part of the t,s modification, based on the algorithm reversible pixels in each image are close, or equal to zero. As a integer KLT [7]. consequence, the volume of data representing the color components for each image in the processed RAC-KLT group could be reduced, by using various 7. Conclusions coding methods. However, the algorithm RAC-KLT needs larger memory to keep the color information A basic method for recursive color space transform for the preceding image in the group. The problems, of a group of correlated images is developed, based related to the larger memory volume are easily on the AC-KLT. It could be used as a basis for the surmountable by new contemporary computer development of: technologies. - New algorithms for parallel processing of the The selected value of δ influences the accuracy images in the group, based on the RAC-KLT, which of the restored color vectors of each image in the will enhance the calculation speed of the direct and processed group after the first one, and also - the inverse transforms; volume of the data which represents the image color - RAC-KLT algorithm with higher efficiency, parameters. For smaller values of δ the accuracy of based on the adaptive division of the processed the restored color vectors is higher, but the color group of images into groups of various length, data is increased. Therefore, the value of δ should be where the correlation between the neighbor image selected by compromise, depending on the RAC- couples is higher that a preset threshold. This KLT application. approach will ensure higher reduction of the color Experimental results for the comparison of AC- components number for each group; KLT with other algorithms for deterministic color - New algorithms, aimed at various application space transform are given in [15]. These results areas. confirm the higher efficiency of AC-KLT compared Maximum efficiency for each algorithm could be achieved when the basic parameters of RAC- for example, with the well-known RGB - YC C . BR KLT are adapted depending on the application Similar conclusions could be made for the algorithm requirements in respect of accuracy, reversibility, RAC-KLT. Its experimental investigation will be execution time, implementation, methods for color executed for various application areas, so that to fix parameters coding, etc. best values of the used parameters.

References 6. Application areas for RAC-KLT 1. T. Reed (Ed.), Digital Image Sequence On the basis of the RAC-KLT efficiency evaluation Processing, Compression, and Analysis, CRC Press given above, the following application areas are LLC, 2005. defined: 2. R. Lucas and K. Plataniotis, Color Image Processing: Methods and Applications, Taylor & - coding of video sequences obtained from TV cameras with fixed spatial position (for example - Francis Group, LLC, 2007. surveillance systems); 3. M. Celebi, M. Lecca, B. Smolka (Eds.), Color Image and Video Enhancement, Springer, 2015. - coding of groups of images got at the same moment from a group of TV cameras, aimed at 4. M. Drew and S. Bergner, Spatio-Chromatic same object but from difference spatial angle (multi- Decorrelation for Color Image Compression, Signal view); Processing: Image Communication, Vol. 23, No. 8, 2008, pp. 599-609. - search of closest contents in images from large databases, got from TV cameras with fixed 5. H. Abdi and L. Williams, Principal Component position; Analysis, Wiley Interdisciplinary Reviews: Computational Statistics, Vol. 2, No. 4, 2010, pp. - objects segmentation on the basis of their color in correlated group of images, etc. 433-459. RAC-KLT could be also used in digital TV 6. C. Maccone, A simple Introduction to the KLT systems, but in this case the difference color vectors (Karhunen-Loève Transform), In: Deep Space for each pixel and the difference rotation angles for Flight and Communications, C. Maccone (Ed.), the consecutive frames from the processed group Springer, 2009, pp. 151-179. should be calculated using corresponding methods 7. P. Hao and Q. Shi, Reversible Integer KLT for for movement compensation. For real-time Progressive-to-Lossless Compression of Multiple

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Component Images. Proc. of Intern. Conf. on Image Processing (ICIP), Vol. 1, 2003, pp. 633–636. 8. A. Abadpour and S. Kasaei, Color PCA Eigenimages and their Application to Compression and Watermarking, Image & Vision Computing, Vol. 26, No. 7, 2008, рр. 878-890. 9. M. Minervini, C. Rusu and S. Tsaftaris, Unsupervised and Supervised Approaches to Color Space Transformation for Image Coding, Proc. of 21st IEEE Intern. Conference on Image Processing (ICIP’14), France, Oct. 2014, pp. 5576–5580. 10. M. Fairchild, Color and Image Appearance Models. John Wiley & Sons, 2005. 11. S. Singh and S. Kumar, Novel Adaptive Color Space Transform and Application to Image Compression, Signal Processing: Image Communication, Vol. 26, No. 10, 2011, pp. 662- 672. 12. P. Farrelle, Recursive Block Coding for Image Data Compression, Springer-Verlag, 1990. 13. R. Kountchev, R. Kountcheva, Image Color Space Transform with Enhanced KLT, In: New Advances in Intelligent Decision Technologies, K. Nakamatsu, G. Philips-Wren, L. Jain, R. Howlett (Eds.), Springer, 2009, pp. 171-182. 14. Р. Ivanov and R. Kountchev, Comparative Analysis of Adaptive Color KLT and YCrCb for Representation of Color Images, Proc. of the Intern. Scientific Conf. on Information, Communication and Energy Systems and Technologies (ICEST’10), Macedonia, 2010, SP I, pp. 29-32.

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