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Published in Rick S. Blum, Zheng Liu, Multi-Sensor Image Fusion and Its Applications, 2018
To describe the HMM for the wavelet coefficients, we need to use graphs and trees.23 An undirected graph consists of a set of nodes {v1, v2,…, vN} and a set of edges linking the nodes. A path is a set of edges connecting two nodes. A rooted tree is an undirected acyclic graph. All nodes that lie on a path from vi to the root are called ancestors of vi. All nodes that lie on paths from vi going away from the root node are called descendants of vi. A node is called the parent of vi if it is the immediate ancestor of vi. The parent of vi is denoted by vρ(i). A node is called the child of vi if vi is its parent. The children of vi are denoted by {vj}j∈c(j). Each node in the rooted tree has only one parent but may have several children. The root has no parent node. Nodes with no children are called leaves of the tree. In a rooted tree, if each node that is not a leaf has four children, this tree will be called a quadtree. A collection of quadtrees is called a forest of quadtrees.
Graph theory concepts and definitions used in image processing and analysis
Published in Olivier Lézoray, Leo Grady, Image Processing and Analysis with Graphs, 2012
One very well-known irregular image tessellation is the region quadtree tessellation [25]. A region quadtree is a hierarchical image representation that is derived from recursively subdividing the 2D space into four quadrants of equal size until every square contains one homogeneous region (based on appearance properties of every region of the underlying rectangular grid) or contains a maximum of one pixel. A region quadtree tesselation can be easily represented by a tree where each node of the tree corresponds to a square. The set of all nodes at a given depth provides a given level of decomposition of the 2D space. Figure 1.13 presents a simple example on an image (Figure 1.13(a)) with the associated tree (Figure 1.13(b)), as well as an example on a real image (Figure 1.13(c)). Quadtrees easily generalize to higher dimensions. For instance, in three dimensions, the 3D space is recursively subdivided into eight octants, and thus the tree is called an octree, where each node has eight children.
Pictorial Data Preprocessing and Shape Analysis
Published in Sing-Tze Bow, Pattern Recognition and Image Preprocessing, 2002
A quadtree is a popular data structure in both graphics and image processing. It is a hierarchical data structure that provides quick access to memory for data retrieval. A quadtree is based on the principle of recursive decomposition of pictures. This technique is best used when the picture is a square matrix A, with dimensions of a power of 2, say 2n. Then matrix A can be divided into four equal-sized quadrants, A0, Ax, A2, A3, whose dimensions are half that of A. This process is repeated until blocks (possibly single pixels) are reached that consist entirely of either l’s or O’s. In this process of successive matrix decomposition, quadrants consisting of all white or all black pixels remain untouched. Only quadrants consisting of both black and white pixels are to be decomposed further. In other words, terminal nodes correspond to those blocks of the array for which further subdivision is unnecessary. The levels can be numbered starting with zero for the entire picture down to n for single pixels, as shown in Figures 13.1 and 13.2. Figure 13.3 shows a simple object coded with this quadtree structure. Black and white square nodes represent blocks consisting entirely of l’s and O’s, respectively. Circular nodes, also termed gray nodes, denote nonterminal nodes.
Multi-Fidelity Machine Learning Applied to Steady Fluid Flows
Published in International Journal of Computational Fluid Dynamics, 2022
Kazuko W. Fuchi, Eric M. Wolf, David S. Makhija, Christopher R. Schrock, Philip S. Beran
We propose an adaptive quadtree sampling method to achieve effective resolution of flow features in our training data distribution. A quadtree is a tree data structure with four child nodes at each parent node. This structure is utilised here to subdivide the fluid domain recursively and selectively to generate sampling points distributed across the domain. Figure 4 illustrates a general quadtree refinement process, and the algorithm is provided in Algorithm 1c, Appendix 1. The problem domain is recursively divided into quadrants with new child nodes placed at the centre of each quadrant in each pass until a desired refinement level is reached. Selective refinement applies the partitioning process on selected nodes that satisfy specified criteria; derefinement may be applied to nodes that fall under certain conditions to remove subdivisions. In our work, the norm of difference in the high-fidelity field quantities between parent and child nodes is used as the refinement and derefinement metric, expressed as: for child node cn. Here and refer to the coordinates of parent and child nodes, respectively. The child nodes are ranked according to this metric, and a certain percentage of top ranking nodes are refined, while bottom ranking nodes are derefined.
Osteoarthritis detection by applying quadtree analysis to human joint knee X-ray imagery
Published in International Journal of Computers and Applications, 2022
Sophal Chan, Kwankamon Dittakan, Subhieh El Salhi
In the context of the knee X-ray image data considered in this work, the application of quadtree decomposition to ROI was applied. Quadtree decomposes an object into four equal quadrants. Figure 5 presents the example of quadtree decomposition process. Figure 5(a) illustrates the original image of the region of interest, Figure 5(b) presents a binary image array where are pixel insight the region, while are the pixels outside the region. Figure 5(c) illustrates the result of applying quadtree decomposition to the region. In the quadtree, the whole image is referred to the root node, while the immediate child nodes of the root nodes each represent as a region quadrant. Figure 5 illustrates the processes of quadtree decomposition, the process stops when the process arrives at the homogeneous region (region consist of all black pixels or all white pixels). Finally, from the Figure 5(c) the quadtree representation can be illustrated in Figure 5(d).
A hierarchical indexing strategy for optimizing Apache Spark with HDFS to efficiently query big geospatial raster data
Published in International Journal of Digital Earth, 2020
Fei Hu, Chaowei Yang, Yongyao Jiang, Yun Li, Weiwei Song, Daniel Q. Duffy, John L. Schnase, Tsengdar Lee
For tree-like approaches, there are four common structures: multiple-key index, k-d tree, quad-tree and r-tree. The multiple-key index is an index of indexes, where nodes at each level are indexes for an attribute. A k-d tree (k-dimensional tree) is a main-memory binary search tree in which interior nodes use their attribute values to split their child nodes into two parts, and nodes at different tree levels have different attributes (Ooi, McDonell, and Sacks-Davis 1987; Zhou et al. 2008). A k-d tree is best at partial-match, range and nearest-neighbour queries (Robinson 1981). A Quad tree divides a space into a square two-dimensional region, and each interior node splits its region into four quadrants (Finkel and Bentley 1974; Eppstein, Goodrich, and Sun 2005). The r-tree is similar to b-tree, but its internal nodes represent the region in any shape (Beckmann et al. 1990). It is efficient for point location and overlapped region query but is complicated to insert/delete data.