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98ˆσ 2 j =∑ Ni=1 ˆQ ij (Y i − ˆµ j ) 2∑i ˆQ ij(5.74)Significant computational savings can be achieved by calculating these estimatesin a slightly different way. With equations 5.72, 5.73, and 5.74, an iterationover all N pixels is needed. This will make the algorithm take time proportional tothe number of pixels. However, this time can be reduced to a time proportional tothe number of unique values in the data. This is done by creating a list of uniquedata values and counting the number of pixels with each value (like a histogram).The parameter estimates can be computed by iterating over the unique data valuesinstead of the pixels. This makes the algorithm operate in constant time withrespect to the number of pixels; the time is linear in the number of unique datavalues. The time saved by this change is enormous. For instance, with a 256-levelgreyscale image, we will have at most 256 unique values, regardless of whetherthere are thousands or millions of pixels. The equations for computing estimatesin this way are shown in equations 5.75, 5.76, and 5.77. My Splus implementationand my XV implementation both use this time-saving approach.Suppose there are C unique data values (e.g. C = 256 grey levels). Let Vdenote the unique data values, so V is a vector of length C. Let H i be the numberof pixels with the ith unique data value, so H is like a histogram of the data.Define R the same way as Q, except that each row is for a unique data valuerather than a particular pixel. In other words, R ij is the probability that the ithunique data value being generated by the jth component. As with Q, each row ofR must sum to 1. The sum of all elements of R will be equal to C.ˆP j = 1 NC∑H i ˆRij (5.75)i=1