Inverse Relation: Let R be a relation from a set A to another set B. Then, the i

Question

Inverse Relation:

Let R be a relation from a set A to another set B. Then, the inverse relation is a relation from B to A. It is denoted by R-1. Thus, if R = {(a, b) : a ? A, b ? B}, Then

Also,

Domain of R-1

= Range of R

Range of R-1

= Domain of R.

Given the inverse relation. Consider writing pseudocode to obtain a lookup table for the inverse relation. Will there be only one entry in the lookup table? Will each entry in the lookup table necessarily have a value? Write the pesudocode. For each value in the initial range how much space is it necessary to provide in the lookup table.

Domain of R-1

= Range of R

Range of R-1

= Domain of R.

Explanation / Answer

A method, apparatus, and computer program product implementing techniques to calculate the hedron that contains an input point. An input color value representing a color having n color components is received. A color lookup table organized as an n-dimensional lattice is received, where n is the number of input channels in the color lookup table, each entry in the color lookup table being a vertex in the lattice, the color lookup table having 2n adjacent lattice points that form an n-dimensional cube around the input value, and the color lookup table defining an output color value at each lattice point. n+1 vertices from the 2n vertices of the n-dimensional cube are calculated, the n+1 vertices defining an enclosing hedron that encloses the input value, where the instructions to calculate n+1 vertices perform no branching operations. he enclosing hedron is used to calculate an output value from the input value.

When working with digital images, it is frequently necessary to transform an image from one color space to a second color space; for example, from an RGB (Red Green Blue) color space to a CMYK (Cyan Magenta Yellow blacK) color space. Many methods used for color conversion use a lookup table that is organized as an n-dimensional lattice, with n representing the number of input color channels. In order to determine output values for input values that are between two entries in the look-up table, some sort of interpolation technique is desirable. In such a table, 2" adjacent lattice points form an n-dimensional cube around a provided input value, for which an output value is desired. To facilitate computation, most methods transform the input values so that 0 in all channels maps to the vertex of the cube with the lowest values in all input channels (which will be referred to as the starting point), and 1 in all channels maps to the vertex with the highest values in all input channels (which will be referred to as the end point). A method commonly known as linear interpolation uses all 2" adjacent lattice points to calculate an output value. However, this method is quite slow, as the number of terms that must be calculated is m x 2" , where m is the number of output channels. hi hedral methods, the cube is subdivided into n subvolumes, known as hedra, each with n+1 vertices. For example, six tetrahedra are used for three input channels, and 24 hedra are used for four input channels. The starting point, the end point, and the line between these two points (the diagonal of the cube) are shared by all hedra. The hedron that encloses the input point is used to calculate the output value. Because only n+ vertices are used, the output value can be obtained by only calculating

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