Latest Trends on Systems - Volume I Pyramid Method for Reversible Discrete Wavelet Transformation of 3D Image Eustache Muteba Ayumba Abstract— In this research, the pyramidal structure and algorithm are proposed for computing the reversible discrete wavelet transformation of 3D Image. The concept of formal language is used to emphasis the modeling of a pyramidal structure of forward and reverse transform. With the assumption, the pyramid representation can help the 3D Image representation and can facilitate the processing. Keywords— Discret Wavelet Transform, Pyramid Method, Pyramidal Data Structure and Algorithm, Tree-Dimension. I I. INTRODUCTION MAGE processing based on Discrete Wavelet Transformation (DWT) and using pyramidal techniques, as reveal Adelson et al [1], is nowadays essential. Discrete Wavelet Transformation (DWT), as stated in [2], proceed a signal has a cut-of-frequency and it is computed by successive Lowpass and Highpass filtering of the discrete time-domain signal. Filters are signal processing functions. Furthermore, a measure of the amount of detail information signal is determined by the filtering operation (Lowpass and Highpass) and the scale is detrmined by Upsampling and Downsampling. The pyramidal technique is a type of multi-scale signal processing in which a signal or an image is subject to repeated smoothing and subsampling [3]. Practically, for representing a three dimensional image (3D Image) , we can refer to a three dimensional Euclidian space as stated in [4]. In this research, the pyramidal structure and algorithm are proposed for computing the reversible discrete wavelet transformation of 3D Image. The concept of formal language is used to emphasis the modeling of a pyramidal structure of forward and reverse transform. With the assumption, the pyramid representation can help the 3D Image representation and can facilitate the processing. II. PYRAMID METHOD OF 3D DWT A. Overview For our purpose, the 3D DWT is based on the method of inferring 3D image information into a smaller non-overlapping tiles on which 2D DWT can be applied. The reversible DWT as show in figure 1, has two main phases: Decomposition and reconstruction phases. ISBN: 978-1-61804-243-9 In decomposition phase, the DWT can be done by iteration of filtering and Downsampling operation. In reconstruction phase, the DWT is the reverse process of decomposition in that the original signal is then obtained by iteration of Upsampling and filtering operation A 3D image is a set of sequence of sample values I=[ xi, yj, zk], where 0 ≤ xi ≤ N1, 0 ≤ y j ≤ N2, 0 ≤ z k ≤ N3; xi, yj and zk are the integers that having finite extents, N1, N2 and N3, in the horizontal, vertical and depth directions, respectively. A 2D image is a set of sequence of sample values I’=[ xi, yj], where 0 ≤ x i ≤ N1, 0 ≤ y j≤ N2. A digitized 3D image size is in N1 X N2 X N3 and a sub block is in size 1/2n X 1/2n X 1/2n. With respect to a fixed global reference frame, it is possible to make correspondence between points in 3D space and their projected images in a 2D image plane, with respect to a local coordinate frame. 3D Original Image Forward DWT 2D DWT Reverse DWT 3D Restore Image Fig. 1 Reversible DWT of 3D B. Formalization For representing the DWT of 3D Image, we refer to formal language and regular images [5]. We assume that the computer has finite memory, so there are only finitely many states the computer can be at, and the previous state of the computer determines the next state, so the machine has deterministic state transitions. The concept of language theory serves as basis to emphasis the problem related to reversible 3D DWT. We consider the following definitions: Definition 1: Language The formal language L is composed by an alphabet ∑ and a specific grammar G. Definition 2: Alphabet Let an alphabet ∑ be a set of element, ∑ ={H, L}, that represents the high and low frequencies. Definition 3: Grammar The grammar is given by a 5-tuple: G={VN, VT, I, F,C} with VN={I, W}, VT={Hx, Hy, Hz, Lx, Ly, Lz}, where G denotes a grammar; VN denotes a Non Terminal Verb; VT denotes a 382 Latest Trends on Systems - Volume I Terminal nodes expressing low and high frequencies in all direction, X, Y and Z; I denotes an input image; F denotes the formula with a set of constraints C attached to each terminal node to code the word. In order to achieve perfect reconstruction the analysis and synthesis filters have to satisfy certain conditions. Definition 4: Formula The formula generates word from a terminal verb: F={I → W}; W=w Wm , m>=1 Thus, there is a specification language L for representing 3D DWT as a text that is a set of sentences S. The respective text should be described as follows: Text := S{S}* S := W{W}* W := w{w}* w := ‘Hx’ |‘Hy’| ‘Hz’ | ‘Lx’ | ‘Ly’ | ‘Lz’ It is possible by means of sentences to keep track of image information. Downsample or Upsample along Direction X Downsample or Upsample along Direction Y Downsample or Upsample along Direction Z LxLyLz Lowpass Filtering / Scaling LxLy LxLyHz Lx LxHyLz LxHy Original Image LxHyHz HxLyLz HxLy III. PYRAMIDAL DATA STRUCTURES AND ALGORITHMS HxLyHz Hx Highpass Filtering / Scaling A. Data Structures The data structure used to represent image information can be critical to the successful completion of an image processing task. The structure of reversible 3D DWT as showed in figure 2 express all operations of filtering and scaling along the three direction saying width direction, height direction and depth direction at decomposition and reconstruction phases. This pyramidal structure representation is suitable to a tree structure. Therefore, it is, also, possible to use pointers, as data type, at each node in order to improve performance in repetitive operations. Given an input 3D image, with respect to our language L, we can organize it in hierarchical tree structure where each terminal node corresponds to a sub block as shown bellow. Thus, any word in the language can be parsed by the grammar that has three levels of related image information to the start node, the original image. HxHyLz HxHy HxHyHz Fig. 2 Structure of the reversible 3D DWT B. Algorithms Our algorithm is based on the purpose of overview in section 2 and is a consequence of the pyramidal data structure presented in section 3. For a given image I=[ xi, yj, zk], where 0 ≤ xi ≤ N1, 0 ≤ y j ≤ N2, 0≤ z k ≤ N3; Let dwt (discrete wavelet transform) be a recursive function that computes I on l level, l>=1: m (1) dwt = ∑ dwt ( I ) l =1 Firstly, we use the function dwt and process a low-pass decomposition and high-pass decomposition in the width direction (horizontal details), after in the height direction (vertical details) and at last in depth direction. Filtering and scaling along Direction X: Lx, Hx N N N N (2) N1 N1 dwt1 = 3 2 1 ∑ (∑ ((∑ L z =1 y =1 x =1 x ( 1 2 , N 2 , N 3 ))(∑ H x ( 2 , N 2 , N 3 )))) Filtering and scaling along Direction Y: LxLy, LxHy, HxLy, HxHy N1 N2 dwt 2 = (∑ ( ∑ L x L y ( x =1 y=2 N1 2 , N2 2 N2 , N 3 )))( ∑ L x H y ( y=2 N1 2 , N2 2 N1 N2 , N 3 ))))(∑ (∑ (H x H y ( x =1 y =1 N1 2 , N2 2 N2 , N 3 )))(∑ H x L y ( y =1 N1 2 , N2 2 (3) , N 3 )))) Filtering and scaling along Direction Z: LxLyLz, LxLyHz, LxHyLz, LxHyHz, HxLyLz, HxLyHz, HxHyLz, HxHyHz (4) N3 N1 N2 N N N N N N N N N N N N dwt 3 = (∑ (∑ (∑ ( L x L y L z ( 1 , 2 , 3 ))( L x L y H z ( 1 , 2 , 3 )))(∑ (∑ (∑ ( L x H y L z ( 1 , 2 , 3 ))( L x H y H z ( 1 , 2 , 3 )))) 2 2 2 2 2 2 2 2 2 2 2 2 x =1 y =1 z =1 x =1 y =1 z =1 N1 N2 N1 N2 N3 x =1 y =1 z =1 N3 (∑ ( ∑ (∑ ( H x L y L z ( N3 N1 N2 N1 N 2 N 3 N N N N N N N N N , , ))( H x L y H z ( 1 , 2 , 3 )))(∑ (∑ (∑ ( H x H y L z ( 1 , 2 , 3 ))( H x H y H z ( 1 , 2 , 3 )))) 2 2 2 2 2 2 2 2 2 2 2 2 x =1 y =1 z =1 Secondly, we use the reverse function dwt' and process a lowpass reconstruction and high-pass reconstruction in the width ISBN: 978-1-61804-243-9 383 Latest Trends on Systems - Volume I direction (horizontal details), after in the height direction (vertical details), and at last in depth direction. The algorithm is highly memory efficient due to fact that the 3D size is limited. Also, the computation complexity has acceptable processing time. IV. CONCLUSION The image coding methods using wavelet transform and pyramidal technique is essential and really efficient in image processing [6], [7]. One of the important contributions of our paper is the pyramidal data structure and algorithm for 3D (DWT) image processing. This is method can be used efficient in the filed of medical science and Biotechnology. REFERENCES [1] E.H. Adelson, C.H. Anderson, J.R. Bergen, P.J. Burt & J.M. Ogden, “Pyramid methods in image processing”, 1984. Available: http://persci.mit.edu/pub_pdfs/RCA84.pdf [2] T. Edwards, “Discrete Wavelets Transforms: Theory and Implementation”, 1992. Available: http://qss.stanford.edu/~godfrey/wavelets/wave_paper.pdf [3] Wikipedia, ”Pyramid (image processing)”. Available: http://en.wikipedia.org/wiki/Pyramid_(image_processing) [4] Y.H. Lee, S.J. Horng, J. Seitzer, “Parallel Computation of the Euclidean Distance Transform on Tree-Dimensional Image array” in IEEE Transactions on Parallel and Distributed Systems, vol. 14, NO. 3, March 2003, pp 203-212. [5] S. Wintner, “Formal Language Theory for Natural Language Processing”, 2001. Available: http://www.helsinki.fi/esslli/courses/readers/K10.pdf [6] E. H. Adelson, C. H. Anderson, J. R. Bergen, P. J. Burt, J. M. Ogden, “Pyramid methods in image processing”. Available: https://alliance.seas.upenn.edu/~cis581/wiki/Lectures/Pyramid.pdf [7] B. K. Choudhary, N. K. Sinha, P. Shankera, “Pyramid Method in Image Processing”, in Journal of Information Systems and Communication, Volume 3, Issue 1, 2012, pp.- 269-273. Available: http://www.bioinfo.in/contents.php?id=45 ISBN: 978-1-61804-243-9 384