Converting Discrete Images to Partitioning Trees

The discrete space representation of most scientific datasets, 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 binary space partitioning tree 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 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. 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.

[1]  Arie E. Kaufman,et al.  Discrete ray tracing , 1992, IEEE Computer Graphics and Applications.

[2]  Kalpathi R. Subramanian,et al.  Interactive Segmentation and Analysis of Fetal Ultrasound Images , 1997, Visualization in Scientific Computing.

[3]  Marc Levoy,et al.  A hybrid ray tracer for rendering polygon and volume data , 1990, IEEE Computer Graphics and Applications.

[4]  John Amanatides,et al.  Merging BSP trees yields polyhedral set operations , 1990, SIGGRAPH.

[5]  William E. Lorensen,et al.  Decimation of triangle meshes , 1992, SIGGRAPH.

[6]  Richard O. Duda,et al.  Use of the Hough transformation to detect lines and curves in pictures , 1972, CACM.

[7]  Alan Norton,et al.  Generation and display of geometric fractals in 3-D , 1982, SIGGRAPH.

[8]  Georgios Sakas,et al.  Preprocessing and volume rendering of 3D ultrasonic data , 1995, IEEE Computer Graphics and Applications.

[9]  Jane Wilhelms,et al.  Multi-dimensional trees for controlled volume rendering and compression , 1994, VVS '94.

[10]  Tom Duff,et al.  Compositing digital images , 1984, SIGGRAPH.

[11]  Loren C. Carpenter,et al.  The A -buffer, an antialiased hidden surface method , 1984, SIGGRAPH.

[12]  John F. Canny,et al.  A Computational Approach to Edge Detection , 1986, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[13]  Bruce F. Naylor,et al.  Set operations on polyhedra using binary space partitioning trees , 1987, SIGGRAPH.

[14]  Bruce F. Naylor,et al.  Interactive solid geometry via partitioning trees , 1992 .

[15]  Bruce F. Naylor.  Binary space partitioning trees as an alternative representation of polytopes , 1990, Comput. Aided Des..

[16]  Hanan Samet,et al.  Applications of spatial data structures , 1989 .

[17]  Henry Fuchs,et al.  Image rendering by adaptive refinement , 1986, SIGGRAPH.

[18]  Kalpathi R. Subramanian,et al.  Representing medical images with partitioning trees , 1992, Proceedings Visualization '92.

[19]  Bruce F. Naylor,et al.  Constructing good partitioning trees , 1993 .

[20]  Josef Kittler,et al.  A survey of the hough transform , 1988, Comput. Vis. Graph. Image Process..

[21]  Abraham Mammen,et al.  Transparency and antialiasing algorithms implemented with the virtual pixel maps technique , 1989, IEEE Computer Graphics and Applications.

[22]  Marc Levoy,et al.  Display of surfaces from volume data , 1988, IEEE Computer Graphics and Applications.

[23]  Daniel Cohen-Or,et al.  Volume graphics , 1993, Computer.

[24]  Henry Fuchs,et al.  On visible surface generation by a priori tree structures , 1980, SIGGRAPH '80.

[25]  Pat Hanrahan,et al.  Hierarchical splatting: a progressive refinement algorithm for volume rendering , 1991, SIGGRAPH.

[26]  Ken Turkowski.  Properties of surface-normal transformations , 1990 .

[27]  Kalpathi R. Subramanian,et al.  Applying space subdivision techniques to volume rendering , 1990, Proceedings of the First IEEE Conference on Visualization: Visualization `90.

[28]  Hayder M. S. Radha,et al.  Efficient image representation using binary space partitioning trees , 1994, Signal Process..

[29]  Josef Kittler,et al.  A comparison of Hough transform methods , 1989 .

[30]  B. Naylor.  A priori based techniques for determining visibility priority for 3-d scenes , 1981 .

[31]  Pat Hanrahan,et al.  Volume Rendering , 2020, Definitions.

[32]  P. Hanrahan.  Three-pass affine transforms for volume rendering , 1990, VVS.

[33]  Marc Levoy,et al.  Gaze-directed volume rendering , 1990, I3D '90.

[34]  Pat Hanrahan,et al.  Fast algorithms for volume ray tracing , 1992, VVS.

[35]  J. Canny.  Finding Edges and Lines in Images , 1983 .

[36]  William E. Lorensen,et al.  Marching cubes: A high resolution 3D surface construction algorithm , 1987, SIGGRAPH.

[37]  Jan Cornelis,et al.  Segmentation of medical images , 1993, Image Vis. Comput..

[38]  L O Hall,et al.  Review of MR image segmentation techniques using pattern recognition. , 1993, Medical physics.

[39]  Donald S. Fussell,et al.  Automatic Termination Criteria for Ray Tracing Hierarchies , 1991 .

[40]  Tomaso A. Poggio,et al.  On Edge Detection , 1984, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[41]  Hanan Samet,et al.  The Design and Analysis of Spatial Data Structures , 1989 .