Czenakowski distance

The Czenakowski distance (sometimes shortened as CZD) is a per-pixel quality metric that estimates quality or similarity by measuring differences between pixels. Because it compares vectors with strictly non-negative elements, it is often used to compare colored images, as color values cannot be negative. This different approach has a better correlation with subjective quality assessment than PSNR.

Definition

Androutsos et al. give the Czenakowski coefficient as follows:

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

Where a pixel $x_i$ is being compared to a pixel $x_j$ on the k-th band of color – usually one for each of red, green and blue.

For a pixel matrix of size $M\times N$, the Czenakowski coefficient can be used in an arithmetic mean spanning all pixels to calculate the Czenakowski distance as follows:

$\frac{1}{MN}\sum _{i=0}^{M-1}\sum _{j=0}^{N-1}\left(\begin{array}{c}1-\frac{2\sum _{k=1}^3\text{min}(A_k(i,j),B_k(i,j))}{\sum _{k=1}^3(A_k(i,j)+B_k(i,j))}\end{array}\right)$

Where $A_k(i,j)$ is the (i, j)-th pixel of the k-th band of a color image and, similarly, $B_k(i,j)$ is the pixel that it is being compared to.

Uses

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