Exercise $7\mathrm{.}5$: Normalized Co$-$occurrence.

Compute the normalized co$-$occurrence matrix $N_{[1,1]}$ for the image of Figure $7\mathrm{.}7$ assuming that the black pixels have gray$-$tone $0,$ the gray pixels have gray$-$tone $1,$ and the white pixels have gray$-$tone $2.$ How does it represent the texture pattern of the image?

Figure $7\mathrm{.}7$ An image with a diagonal texture pattern.

Co$-$occurrence matrices capture properties of a texture, but they are not directly useful for further analysis, such as comparing two textures. Instead, numeric features are computed from the co$-$occurrence matrix that can be used to represent the texture more compactly. The following are standard features derivable from a normalized co$-$occurrence matrix.

Energy $={\displaystyle \underset{i}{\overset{?}{}}}{\displaystyle \underset{j}{\overset{?}{}}}{N}_d^2[i,j]$

Entropy $=-{\displaystyle \underset{i}{\overset{?}{}}}{\displaystyle \underset{j}{\overset{?}{}}}{N}_d[i,j]lo{g}_2{N}_d[i,j]$

Contrast $={\displaystyle \underset{i}{\overset{?}{}}}{\displaystyle \underset{j}{\overset{?}{}}}(i-j{)}^2{N}_d[i,j]$

Homogeneity $={\displaystyle \underset{i}{\overset{?}{}}}{\displaystyle \underset{j}{\overset{?}{}}}\frac{N_d[i,j]}{1+|i-j|}$

Correlation $=\frac{{\displaystyle \underset{i}{\overset{?}{}}}{\displaystyle \underset{j}{\overset{?}{}}}(i-{}_i)(j-{}_j)N_d[i,j]}{{}_i{}_j}$