Code Transformation of DF-Expression between Bintree and Quadtree

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1 Code Transformation of DF-Expression between Bintree and Quadtree Chin-Chen Chang*, Chien-Fa Li*, and Yu-Chen Hu** *Department of Computer Science and Information Engineering, National Chung Cheng University ** Department of Computer Science and Information Management Providence University Abstract In this paper, algorithms for transforming code between bintree and quadtree are proposed. The quadtree and bintree are two commonly used representations for image manipulations. However, there exists no algorithm for transforming code between bintree and quadtree. When one receiver a DF-expression of a bintree or quadtree code and does not have the corresponding decoding program, she/he, with our new code transformation algorithms at hand, can transform the received code into a quadtree or bintree code that she/he can handle. After code transformation, the receiver can see whether the output code is smaller than the input code. From it, she/he can further decide which code is more economic in its memory space needed; therefore, she/he can store the code with smaller size for saving more memory space. Keywords: bintree, quadtree, code transformation 1. Introduction In the fields of computer graphics, geographic information systems, and image processing, the representation of spatial data, such as arbitrary regions, point data, collections of lines, curves, surfaces, volumes, and so on, is an important issue. Because of the huge storage requirements and the high computational complexity of image manipulations, the extensive study of efficient image representations has become an important research issue. Up to the present, several spatial data structures for representing compressed binary images and the related image manipulations [7,8,9,11,12] have been presented. Basically, these data structures can be classified into six categories, namely, chain-codes, run-length codes, trees, linear-trees, depth-first representation, and breadth-first representation. The first category, chain-codes [15], can be used to represent a series of numbers that record the boundary of each object with a given image. The second category, run-length codes [18], can be used to represent, for an image, a list of pixels that are ordered by rows. Trees [7] can be used to represent a tree structure and store it by using the pointer-type data structure, such as the octree, the quadtree, the bintree, and so on. Linear trees can be used to represent a set of codes which are formed by the black leaf nodes of the corresponding quadtree or bintree of the image, such as FD-locational codes, FL-locational codes, VL-locational codes [7], bincode [17], and so on. Depth-first representations can be used to represent an image by traversing the nodes of quadtree or bintree in depth-first order. Finally, breadth-first representations can be used to represent the nodes of a quadtree or bintree in breadth-first order [1,3,4,5,6,10]. Many important and efficient algorithms for these different spatial representations, for example, rotation, mirroring, hidden-surface elimination, ray tracing, bean tracing, medial axis transforms, set operations, neighbors finding, moments, connected component labeling, Euler number, area, perimeter, windowing, expansion, and dilation, have been presented in recent decades [7,8]. Thus, it is very important and interesting to design an efficient transformation among these representations. That is, if any two representations can be transformed by each other, the algorithms that were developed for transforming the two representations can be shared by the two representations. In addition, the storage space required in each representation is quite different. We can choose the one with less storage cost to save storage costs. The algorithm for converting a quadtree from chain codes [19] has been presented. In addition, the algorithm for converting a quadtree representation to its corresponding chain codes representation has been presented [14]. The algorithm for converting a quadtree representation to its corresponding run-length codes [20] has also been presented. Furthermore, the algorithm for converting a quadtree to run-length codes was presented in [21]. The transformation between chain codes and run-length codes are straightforward [7]. The code transformation between the DF linear quadtree and BF linear quadtree has already been proposed [2]. When a DF-expression code of a quadtree/bintree is received without a corresponding decoding program, this problem can be solved if the DF-expression code of a quadtree/bintree can be transformed into its corresponding DF-expression code of the bintree/quadtree. Taking advantage of this flexibility, the storage space of one image can also be reduced by changing it from one code to another. In this paper, we intend to provide the code transformation algorithm for two commonly used DF-expressions, which are the quadtree and bintree scheme. The remainder of this paper is organized as follows. In Section 2, we shall

2 review the DF-expression of a bintree and quadtree. In Section 3, our new code transformation algorithms for using the DF-expression between a bintree and quadtree will be presented. Finally, the conclusion will be given in the last section. 2. Previous Works In this section, we will review some related schemes of the DF-expression and the quadtree and the bintree algorithms. The DF-expression was introduced by Kawaguchi and Endo [16]. In Section 2.1, we will briefly describe one DF-expression quadtree representation. In Section 2.2, the proposed DF-expression bintree code will be briefly introduced. 2.1 The DF-expression Representations of a Quadtree A quadtree is a hierarchical data structure for representing a binary image. It is constructed by successively sub-dividing an image into four equal-sized sub-images in four quadrants: NW (northwest) quadrant, NE (northeast) quadrant, SW (southwest) quadrant, and SE (southeast) quadrant. A leaf node in the tree alone can represent a homogeneous quadrant of the image. On the other hand, an internal node represents a heterogeneous quadrant of the image. Each internal node is further divided into four sub-quadrants, and each sub-quadrant can be even further divided into four smaller sub-quadrants until each sub-quadrant has only one color. An external node whose color is black (white) is called a black (white) node, and an internal node is called a gray node. Fig. 1 shows an example of the quadtree segmentation process. The original binary image of pixels is shown in Fig. 1(a). The corresponding quadtree is depicted in Fig. 1(b). In 1980, Kawaguchi and Endo proposed an effective way for the DF-expression of a quadtree representation [16]. When an internal node is visited, the symbol ( is used to represent it. When a black leaf node and a white leaf node are encountered, the symbols B and W are used to represent them, respectively. The DF-expression quadtree representation of the image shown in Fig. 2 is ((BBWW((WBWWWW(BWWWB(W(WWBBBW. DF-expression codes:((bbww((wbwwww(bwwwb(w(wwbbbw Fig. 2 The DF-expression of quadtree codes for the image shown in Fig. 1(a) 2.2 The DF-expression Representations of a Bintree In this subsection, we will introduce the DF-expression of a bintree, which was proposed in [13]. A bintree is a hierarchical data structure used to represent a binary image. The root of the bintree represents the image, if the entire image is completely black or white. Otherwise, the root is grey. The image is splitted into two equal-sized sub-images, and its children corresponding to the two sub-images are added to the root. This subdivision process is then repeated recursively for each of the two sub-images until the sub-image is completely black or white. At each step, the partition alternates between the x- and y-axes. If a subdivision is either black or white, then its corresponding node is an external node; otherwise, it is an 3 3 internal node. Given a binary image of size 2 2 and shown in Fig. 1(a). Fig. 3 depicts an example of the corresponding bintree segmentation process. Some as the way for the DF-expression of a quadtree representation in[16], when an internal node is visited, the symbol ( is used to represent it. When a black leaf node and a white leaf node are encountered, the symbols B and W are used to represent them, respectively. The DF-expression for a bintree of the image shown in Fig. 1(a) is shown in Fig. 3. The result is ((((BW(BWB((((W(BWW(W((BWW((WB(((WB (WBW. (a)the binary image (b) The corresponding quadtree Fig. 1 Example of the quadtree segmentation process DF-expression:((((BW(BWB((((W(BWW(W((BWW((WB(((WB(WBW Fig. 3 The corresponding DF-expression codes of a bintree for the image in Fig. 1(a)

3 3. The Proposed Algorithms In this section, the algorithm that transforms a DF-expression of a quadtree into its corresponding DF-expression of a bintree shall be introduced. Next, the algorithms for the transformation of a DF-expression of a bintree into its corresponding DF-expression of a quadtree shall also be described. To transform the DF-expression of a quadtree into the corresponding DF-expression of a bintree, the following steps are executed. If an internal node, that is, the symbol (, is found, it will be transformed into the symbols (( S1S3(S2S4. The symbols S1, S2, S3, and S4 denote these four sub-tree expressions of a quadtree. When we scan the next quadtree symbol, we record the DF-expression expression of a bintree in reverse order, from the S2 and S3 subtree node of a quadtree. If the scanned symbol indicates that a leaf node, that is, the symbols B or W, is found, then the corresponding bintree codes, the symbols B or W, are produced. The above steps are recursively executed until all the symbols in the DF-expression of a quadtree are processed. After all the symbols of the DF quadtree expression are processed, the corresponding DF-expression of bintree is generated. (a) The 2 2 binary image (b) DF-expression of quadtree and bintree (c) DF-expression transformation from quadtree to bintree Fig. 4 A quadtree to a bintree transformation for a DF-expression A simple example of a transformation from the DF-expression of a quadtree to the DF-expression of a bintree is demonstrated in Fig. 4. The DF-expression of a bintree can be obtained by scanning the symbols in the DF quadtree expression. In Fig. 4(b), an internal node of a quadtree produces three internal nodes of a bintree. The second and third leaf nodes of a quadtree become the third and second leaf nodes of a bintree. Therefore, we must change the positions of these two nodes. In Fig. 4(c), the original DF-expression of a quadtree and its resultant DF bintree expression are given. When a DF-expression of a quadtree wants to transform into its DF-expression of bintree, the DF-expression of a quadtree is inputted and Procedure QTB is executed. Procedure QTB: /* Quadtree code to Bintree code transformation */ Step 1: Get the current DF-expression code of quadtree qc. If qc is NULL, then go to Step 6. Step 2: If qc= (, four recursive procedures of QTB are sequentially performed to process the next four expression codes, S 1, S 2, S 3, and S 4, respectively. Step 3: If qc= B or W, return qc to the caller. Step 4: If all the recursive calls generated by the current procedure are terminated, generate the DF-expression of a bintree for qc according to the following rules: (( S1S 3 ( S 2 S 4 if qc = '(', here S1, S 2, S 3, and S 4 denote the retruned results of these bintree code = four recursive calls, ' B' if qc = ' B', ' W' if qc = ' W'. Step 5: When an internal node records the symbol ( and the leaf nodes of its two children have the same color B or W, the symbol B or W is used to represent these three symbols. Step 6: Output the bintree code. A complete transformation example of the DF-expression of a quadtree to the DF-expression of a bintree code is given in Fig. 5. The DF-expression of a bintree representation is obtained by scanning the DF-expression code of a quadtree representation. In Fig. 5, the quadtree code without the underline denotes a sub-quadtree of the root node. A code with the underline is another sub-tree of the quadtree root node. When a symbol with the value ( is found in the DF-expression of a quadtree, the symbols (( ( are used to replace the previous one. At the same time, the program will start to generate four recursive procedure calls sequentially. Finally, it reads each code and appends the corresponding code to the previously output code list which gives the corresponding DF-expression of a bintree. Let us now look at the guidelines for transforming a DF-expression of a bintree into its corresponding DF-expression of a quadtree. First, if the scanned DF-expression symbols are two successive internal nodes

4 with the value ((, then the second symbol is removed to generate (. If the scanned code is a leaf node with the value B or W, then the corresponding quadtree codes are produced. The above steps are recursively processed until all the symbols in the DF-expression of the quadtree are scanned. Fig. 5 Example of a DF-expression from a quadtree to a bintree transformation (a) The 2 2 binary image (b) DF-expression of bintree and quadtree (c) DF-expression transformation from bintree to quadtree Fig. 6 A bintree to quadtree transformation for a DF-expression An example illustrating the transformation of a DF-expression of a bintree to a DF-expression of a quadtree is given in Fig. 6. In Fig. 6(b), the three internal nodes of the two-level bintree are transformed into a single level quadtree with one internal node. The second and third leaf nodes of a bintree become the third and second leaf nodes of the quadtree. For this reason, when the codes are transformed, we must change the positions of these two sub-trees. In Fig. 6(c), a DF-expression of a bintree and the resultant DF-expression of quadtree is given. When a DF-expression of a bintree wants to transform into a DF-expression of quadtree, Procedure BTQ described as follows is executed. Procedure BTQ: /* Bintree code to Quadtree code transformation */ Step 1: Get the current DF-expression code of bindtree bc. If bc is NULL, then go to Step 4. Step 2: Suppose bc= ( and the next bc= (. Four recursive procedures of BTQ are called sequentially to process the next four bintree code S 1, S 3, S 2, and S 4, respectively. Step 3: If bc= B or W, return bc to the caller. Step 4: If all the recursive calls generated by the current procedure are terminated, generate the DF-expression codes of the quadtree for the bc according to the following rules: ( S1 S 2 S 3 S 4 if bc = '(' and next bc = '(', here S1 S 2 S 3 S 4 denote the results of the quadtree code = four recursive procedures 'B' if bc = 'B', ' W' if bc = ' W'. Step 5: Output the quadtree code. A complete example for transforming the DF-expression of a bintree to the DF-expression of a quadtree is given in Fig. 7. In Fig. 7, the bintree code with a single underline is a grandchild sub-tree of the bintree root node; a code with a double underline is a grandchild sub-tree of the bintree root node. As it visits the continue symbol (( of code, the symbol ( is used to replace the previous one. At the same time, four recursive calls will be executed sequentially. 4. Conclusions Code transformation algorithms between different image representations are very important. The demand for code transformation algorithms between two different image representations is quite important because they provide flexibility for the usage of different image representations. Generally, we can choose the one requiring less storage cost to save space while transforming it to another image representation for effective image manipulation.

5 Quadtrees and bintrees are commonly used techniques to represent binary images because they can effectively reduce the required storage space. This paper presented algorithms for transforming codes between quadtree and bintree. If we receive a DF-expression code of a quadtree/bintree that we cannot handle, we can transform it into another DF-expression code of a bintree/quadtree with our algorithms. Combined with other algorithms for transforming code between a bincode [17] and a DF-expression of a bintree, the use of a quadtree and bintree representation would be very flexible. In addition, the new algorithms can also be used to reduce the storage space that an image requires, if necessary. Fig. 7 Example of a DF-expression of a bintree to quadtree transformation References [1] Chang, H. K. and Chang, J. W., The fixed binary linear quadtree coding scheme for spatial data, SPIE Proceedings -- Visual Communication and Image Processing, Vol. 2308, 1994, pp [2] Chen, P. M., Variant code transformations for linear quadtrees, Pattern Recognition Letters, Vol. 23, 2002, pp [3] Dyer, C., The space efficiency of quadtrees, Computing Graphics Image Process, Vol. 19, No. 4, 1982, pp [4] Gargantini, I., An effective way to represent quadtrees, Communications of the ACM, Vol. 25, No. 12, 1982, pp [5] Hunter, G. M. and Steiglitz, K., Operations on images using quadtrees, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 1, No. 2, 1979, pp [6] Lin, T. W., Set operations on constant bit-length linear quadtrees, Pattern Recognition, Vol. 30, No. 7, 1997, pp [7] Samet, H., Application of spatial data structures, Addison-Wesley, New York [8] Samet, H., Design and analysis of spatial data structure, Addison-Wesley, New York [9] Tsai, Y. H. and Chung K. L., Some image operations on S-tree-related spatial data structure, Image and Vision Computing, Vol. 17, 1999, pp [10] Wang, F., Relational-linear quadtree approach for two dimensional spatial representation and manipulation, IEEE Transactions on Knowledge and Data Engineering, Vol. 3, No. 1, 1991, pp [11] Wu, J. G. and Chung, K. L., The logical representation of bincode and its applications in manipulating binary images, Real-Time Imaging, Vol. 5, 1999, pp [12] Yang, Y. H., Chung K. L. and Tsai, Y. H., A compact improved quadtree representation with image manipulations, Image and Vision Computing, Vol. 18, 2000, pp [13] Jonge, W. De., Scheuermann, P. and Schijf, A., S+-trees: an efficient structure for the representation of large pictures, Computer Vision, Graphics & Image Processing: Image Understanding, Vol. 59,No. 3, 1994, pp [14] Dyer, C. R., Rosenfeld, A. and Samet, H., Region representation: boundary codes from quadtree, Communications of the ACM, Vol. 23, No. 3, 1980, pp [15] Freeman, H., Computer processing of line-drawing image, ACM Computing Surveys, Vol. 6, No. 1, 1974, pp [16] Kawaguchi, E. and Endo, T., On a method of binary picture representation and its application to data compression, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 2, No. 1, 1980, pp [17] Ouksel, M. A. and Yaagoub, A., The interpolation-based bintree and encoding of binary images, Computer Vision, Graphics & Image Processing: Graphical Models and Image Processing, Vol. 54, No. 1, 1992, pp [18] Rutovitz, D., Data structures for operations on digital images, Pattern Recognition, 1968, pp [19] Samet, H., Region representation: quadtrees from boundary codes, Communications of the ACM, Vol. 23, No. 3, 1980, pp [20] Samet, H., An algorithm for converting rasters to quadtrees, IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 3, No. 1, 1981, pp [21] Samet, H., Algorithm for conversion of quadtrees to rasters, Computer Vision, Graphics, and Image Processing, Vol. 26, No. 1, 1984, pp

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