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in a scale with fewer quantization levels, possibly producing a new image with K = 64 or lower. Although some of the image information is eliminated, texture information can be retained and texture metrics derived from the co-occurrence matrices remain valuable, if the mapping is appropriate.

Linear mapping, however, may not be always the best choice. For example, if the structure of interest has pixels distributed approximately evenly between 128 and 138 in an 8-bit image, a linear mapping to a 5-bit image will map about 80% of the pixels in this structure to the same new gray level. This will obliterate the statistical information within that structure by severely reducing the pixel variations. However, a nonlinear mapping that uses larger steps at gray levels with insignificant information and small or unchanged steps at critical gray levels imparts the desired reduction in K without adversely affecting the texture metrics.

Many texture metrics can be derived from the co-occurrence matrices [44], some of which are described below. The angular second moment, also known as energy,

The number of pixel pairs P' used to build H(d, 6) is always smaller than P because, for each choice of d and 6, pixels on some edges do not form pairs. In the above example, the 11 pixels in the top and right edge do not contribute pairs and P' is 25. The value of P' gets smaller as the selected distance increases because a larger number of rows and columns along the edges are excluded from the counts.

In the most general case, assuming an image with M rows and N columns, when 6 is 45°, there will be M — d/V^ pairs along each row and only N — d/ V2 rows will contribute, resulting in a total of p' = (m — d /v2)(n — d /V2) pairs to be used in forming the co-occurrence matrix. Expressions of P' for the four main orientations are:

quantifies homogeneity. In a homogeneous region, there are few gray level transitions and most of the pixel pairs have the same or close values. This concentrates most of the probability on and around the diagonal of C(d, 6), leaving most of the cij elements to be zero. On the other hand, an inhomogeneous image has many gray level transitions, and the probability is more evenly distributed across the entire co-occurrence matrix, resulting in very small values for each cij. Consequently, an inhomogeneous region has an angular second moment that is lower than that ofa homogeneous region. This metric is sensitive to intensity transitions but insensitive to the magnitude of the transitions. That is, higher local intensity contrast within the inhomoge-neous region does not increase this metric.

The inertia measure quantifies the texture contrast and is given by

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