19.2 Exhaustive enumeration 19.2.1 Definition and construction methods Exhaustive enumeration is a representation by means of nonoverlapping cubes of uniform size and orientation, see Figure 19.1. An object is represented by a three dimensional boolean array Each cell represents a cubic volume of space. If a cell intersects with the region of interest it has a true value. Otherwise, the value is false. This can be pictured as a box divided into 3D cubical pixels, with 0 assigned if empty and 1 assigned if full. This representation involves lar subdivision of 3D space within a cube of given size which is partitioned and oriented in a certain way within a global coordinate system. The subdivision is made up of sub-cubes (3D pixels)of given size. Reference and access to each sub-cube is made by three integer indices i, 3, h For fixed space of interest we need a 3-D array, Cik of binary data if the sub-cube i, j, k intersects the solid if the sub-cube i,j, k is empty 19.1) Construction of exhaustive enumeration models requires an alternate representation or measurements(eg. digital tomograghy, medical scanning, sonar data, acoustic tomography data,etc). Usually the primary data type for such construction is a B-Rep or a CSg model or another exhaustive enumeration model at different resolution and cube location and orien- tation Operations on exhaustive enumeration models are easy. Boolean operations for example (especially for models within the same cube at the same resolution) are direct. Similarly visualization and integral computations are very easy. However, for higher quality rendering filtering methods to estimate accurate surface normals may be involved 16 The binary matrix(19. 1)typically represents a valid solid. However disconnected cells or cells with low degree of connectivity as in Figure 19.2 are undesirable. For the results of Boolean operations, filtering may be needed to maintain connectivity of cells. Strict connectivity occurs when each full cell has at least one full neighbor across a face 19.2.2 Applications Applications of exhaustive enumeration methods include Underwater environment representation Finite element meshing(first step in an algorithm to build such a mesh) Medical 3D data representation Preprocessing representation for speeding up operations on other representations(eg approximating integral properties such as volume, center of gravity, moments of inertia19.2 Exhaustive enumeration 19.2.1 Definition and construction methods Exhaustive enumeration is a representation by means of nonoverlapping cubes of uniform size and orientation, see Figure 19.1. An object is represented by a three dimensional Boolean array. Each cell represents a cubic volume of space. If a cell intersects with the region of interest it has a true value. Otherwise, the value is false. This can be pictured as a box divided into 3D cubical pixels, with 0 assigned if empty and 1 assigned if full. This representation involves: • A regular subdivision of 3D space within a cube of given size which is partitioned and oriented in a certain way within a global coordinate system. The subdivision is made up of sub-cubes (3D pixels) of given size. Reference and access to each sub-cube is made by three integer indices i, j, k. • For fixed space of interest we need a 3-D array, Cijk of binary data: Cijk = ( 1 if the sub-cube i, j, k intersects the solid 0 if the sub-cube i, j, k is empty (19.1) Construction of exhaustive enumeration models requires an alternate representation or measurements (eg. digital tomograghy, medical scanning, sonar data, acoustic tomography data, etc). Usually the primary data type for such construction is a B-Rep or a CSG model or another exhaustive enumeration model at different resolution, and cube location and orien￾tation. Operations on exhaustive enumeration models are easy. Boolean operations for example (especially for models within the same cube at the same resolution) are direct. Similarly visualization and integral computations are very easy. However, for higher quality rendering, filtering methods to estimate accurate surface normals may be involved [16]. The binary matrix (19.1) typically represents a valid solid. However disconnected cells or cells with low degree of connectivity as in Figure 19.2 are undesirable. For the results of Boolean operations, filtering may be needed to maintain connectivity of cells. Strict connectivity occurs when each full cell has at least one full neighbor across a face. 19.2.2 Applications Applications of exhaustive enumeration methods include: • Underwater environment representation. • Finite element meshing (first step in an algorithm to build such a mesh). • Medical 3D data representation. • Preprocessing representation for speeding up operations on other representations (eg. approximating integral properties such as volume, center of gravity, moments of inertia). 3