Developing Image Processing Meta-Algorithms with Data Mining of Multiple Metrics

Table 2

Some intensity-based and statistical image metrics. In the Difference metrics, the index 𝑥 ranges over voxel positions. In the Correlation and Woods metrics, the index 𝑖 ranges over intensity values, 𝑁(𝑖) is the number of voxels in ℛ having intensity 𝑖, and 𝜇(𝑖) and 𝜎²(𝑖) are the mean and variance of intensities of 𝒮 in the same voxel positions. Normalized Cross-Correlation is voxel-wise correlation, with means and standard deviations computed over the entire image.

(1) Mean Square Difference of Intensities
1msd(ℛ,𝒮)=𝑁∑𝑥(ℛ(𝑥)−𝒮(𝑥))^2
(2) Absolute Difference of Intensities
1adi(ℛ,𝒮)=𝑁∑𝑥|ℛ(𝑥)−𝒮(𝑥)|
(3) Shannon Entropy of Difference of Intensities
1edi(ℛ,𝒮)=𝑁∑𝑥𝑝(ℛ(𝑥)−𝒮(𝑥))log𝑝(ℛ(𝑥)−𝒮(𝑥))
(4) Mutual Information
mif(ℛ,𝒮)=ℐ(ℛ,𝒮)=ℋ(ℛ)+ℋ(𝒮)−ℋ(ℛ,𝒮)
(5) Normalized Mutual Information
nmi(ℛ,𝒮)=ℐ(ℛ,𝒮)ℋ(ℛ,𝒮)+1=ℋ(ℛ)+ℋ(𝒮)ℋ(ℛ,𝒮)
(6) Normalized Cross-Correlation
()ncc(ℛ,𝒮)=covℛ,𝒮𝜎ℛ𝜎𝒮
(7) Correlation
()1corℛ,𝒮=1−𝑁∑𝑖𝑁(𝑖)𝜎^2(𝑖)𝜎^2
(8) Woods
()1wooℛ,𝒮=1−𝑁∑𝑖𝑁(𝑖)𝜎(𝑖)𝜇(𝑖)
(9) Redundancy
red(ℛ,𝒮)=ℐ(ℛ,𝒮)(ℋ(ℛ)+ℋ(𝒮))=1−ℋ(ℛ,𝒮)(ℋ(ℛ)+ℋ(𝒮))
(10) A Universal Metric
ℐ()uni(ℛ,𝒮)=1−ℛ,𝒮ℋ()(ℋ(ℛ)(𝒮ℛ,𝒮=2−+ℋ))ℋ(ℛ,𝒮)
(11) Another Universal Metric
ℐ()aum(ℛ,𝒮)=1−ℛ,𝒮max(ℋ(ℛ),ℋ(𝒮))