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In computer graphics, marching squares is an algorithm that generates contours for a two-dimensional scalar field (rectangular array of individual numerical values). A similar method can be used to contour 2D triangle meshes. The contours can be of two kinds: Isolines – lines following a single data level, or isovalue.
The Z-ordering can be used to efficiently build a quadtree (2D) or octree (3D) for a set of points. [4] [5] The basic idea is to sort the input set according to Z-order.Once sorted, the points can either be stored in a binary search tree and used directly, which is called a linear quadtree, [6] or they can be used to build a pointer based quadtree.
Then, this initial bounding box is partitioned into a grid of smaller cubes, and grid points near the boundary of the convex hull of the input are used as a coreset, a small set of points whose optimum bounding box approximates the optimum bounding box of the original input. Finally, O'Rourke's algorithm is applied to find the exact optimum ...
Trilinear interpolation as two bilinear interpolations followed by a linear interpolation. Trilinear interpolation is a method of multivariate interpolation on a 3-dimensional regular grid . It approximates the value of a function at an intermediate point ( x , y , z ) {\displaystyle (x,y,z)} within the local axial rectangular prism linearly ...
To find either of the single derivatives, or , using that method, find the slope between the two surrounding points in the appropriate axis. For example, to calculate f x {\displaystyle f_{x}} for one of the points, find f ( x , y ) {\displaystyle f(x,y)} for the points to the left and right of the target point and calculate their slope, and ...
The exchange of two rows multiplies the determinant by −1. Multiplying a row by a number multiplies the determinant by this number. Adding a multiple of one row to another row does not change the determinant. The above properties relating to rows (properties 2–4) may be replaced by the corresponding statements with respect to columns.
Head and cerebral structures (hidden) extracted from 150 MRI slices using marching cubes (about 150,000 triangles). Marching cubes is a computer graphics algorithm, published in the 1987 SIGGRAPH proceedings by Lorensen and Cline, [1] for extracting a polygonal mesh of an isosurface from a three-dimensional discrete scalar field (the elements of which are sometimes called voxels).
In the worst case, i = 1 or i = n − 2 at each recursive invocation yields a running time of O(n 2). In the best case, i = n / 2 or i = n ± 1 / 2 at each recursive invocation yields a running time of O(n log n). Using (fully or semi-) dynamic convex hull data structures, the simplification performed by the algorithm can be ...