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<p><b>Abstract</b>—The discrete space representation of most scientific datasets (pixels, voxels, etc.), generated through instruments or by sampling continuously defined fields, while being simple, is also verbose and structureless. We propose the use of a particular spatial structure, the <it>binary space partitioning tree</it>, or, simply, <it>partitioning tree</it>, as a new representation to perform efficient geometric computation in discretely defined domains. The ease of performing affine transformations, set operations between objects, and correct implementation of transparency (exploiting the visibility ordering inherent to the representation) makes the partitioning tree a good candidate for probing and analyzing medical reconstructions, in such applications as surgery planning and prostheses design. The multiresolution characteristics of the representation can be exploited to perform such operations at interactive rates by smooth variation of the amount of geometry. Application to ultrasound data segmentation and visualization is proposed.</p><tf>$$\sum\limits^{d \log h}_{i=1} \tilde O\!\!\left({{n}\over{k^3_i}}\right) = \sum\limits^{d \log h}_{i=1}\tilde O\!\!\left({{n}\over{{\left(k/2^i\right)}^3\right)}}= \tilde\!\!{O} (n).$$</tf><p>The paper describes methods for constructing partitioning trees from a discrete image/volume data set. Discrete space operators developed for edge detection are used to locate discontinuities in the image from which lines/planes containing the discontinuities are fitted by using either the Hough transform or a hyperplane sort. A multiresolution representation can be generated by ordering the choice of hyperplanes by the magnitude of the discontinuities. Various approximations can be obtained by pruning the tree according to an error metric. The segmentation of the image into edgeless regions can yield significant data compression. A hierarchical encoding schema for both lossless and lossy encodings is described.</p>
Partitioning trees, BSP trees, space partitioning, miltiresolution representations, image reconstruction, image coding, scientific visualization, MRI visualization.

K. R. Subramanian and B. F. Naylor, "Converting Discrete Images to Partitioning Trees," in IEEE Transactions on Visualization & Computer Graphics, vol. 3, no. , pp. 273-288, 1997.
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