Czekanowski distance

Androutsos et al. give the Czekanowski coefficient as follows:^[1]

${\displaystyle d_{z}(i,j)=1-{\frac {2\sum _{k=1}^{p}{\text{min}}(x_{ik},\ x_{jk})}{\sum _{k=1}^{p}(x_{ik}+x_{jk})}}}$

Where a pixel ${\displaystyle x_{i}}$ is being compared to a pixel ${\displaystyle x_{j}}$ on the k-th band of color – usually one for each of red, green and blue.

For a pixel matrix of size ${\displaystyle M\times N}$, the Czekanowski coefficient can be used in an arithmetic mean spanning all pixels to calculate the Czekanowski distance as follows:^[2][3]

${\displaystyle {\frac {1}{MN}}\sum _{i=0}^{M-1}\sum _{j=0}^{N-1}{\begin{pmatrix}1-{\frac {2\sum _{k=1}^{3}{\text{min}}(A_{k}(i,j),\ B_{k}(i,j))}{\sum _{k=1}^{3}(A_{k}(i,j)+B_{k}(i,j))}}\end{pmatrix}}}$

Where ${\displaystyle A_{k}(i,j)}$ is the (i, j)-th pixel of the k-th band of a color image and, similarly, ${\displaystyle B_{k}(i,j)}$ is the pixel that it is being compared to.

In the context of image forensics – for example, detecting if an image has been manipulated –, Rocha et al. report the Czekanowski distance is a popular choice for Color Filter Array (CFA) identification.^[2]