Foundations of Colour Science by Alexander D. Logvinenko & Vladimir L. Levin

Author:Alexander D. Logvinenko & Vladimir L. Levin [Logvinenko, Alexander D. & Levin, Vladimir L.]
Language: eng
Format: epub
ISBN: 9781119885948
Published: 2022-11-14T00:00:00+00:00

Example 10.1

Let be a ‐dimensional continuous Grassmann structure, and its fundamental colour mechanisms. We will assume that a noise is added to the response of each mechanism. Specifically, the noisy response of the th mechanism to a light is assumed to be represented as a random variable , where is a random variable distributed as , i.e., a normally distributed random variable with zero mean and variance . We will also assume that the matching criterion region is defined as follows

(10.2)

where is a fixed positive real number, , and is the Euclidian norm.

When and are the noisy colour signals in response to not metameric lights and respectively, that is, is a normal random vector with mean and the diagonal covariance matrix with on the diagonal, has the noncentral chi‐square distribution with degrees of freedom, and noncentrality parameter

(10.3)

Thus, if denotes the cumulative distribution function of the noncentral chi‐square distribution with degrees of freedom and noncentrality parameter , the probability of the response ‘same’, on condition that the not metameric lights and are presented, is expressed as

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