Autofocus Algorithm

1. Derivative Based Algorithms:

• Thresholded absolute gradient: $$f=\sum _x\sum _y|I(x+1,y)-I(x,y)|$$  where   $|I(x+1,y)-I(x,y)|>\text{threshold};$
• Squared gradient: $$f=\sum _x\sum _y(I(x+1,y)-I(x,y){)}^2$$ where $(I(x+1,y)-I(x,y){)}^2>\text{threshold};$
• Brenner gradient: $$f=\sum _x\sum _y(I(x+2,y)-I(x,y){)}^2$$ where $(I(x+2,y)-I(x,y){)}^2>\text{threshold};$
• Tenenbaum gradient: $$f=\sum _x\sum _y({\text{SobelX}}^2(x,y)+{\text{SobelY}}^2(x,y));$$
• Energy Laplace: $$f=\sum _x\sum _y((L\ast I)(x,y){)}^2$$  where $$L=\left[\begin{array}{ccc}-1& -4& -1\\ -4& 20& -4\\ -1& -4& -1\end{array}\right]$$ 
• Sum of modified Laplace: $$f=\sum _x\sum _y(\text{LaplaceX}(x,y)|+|\text{LaplaceY}(x,y)|)$$
• Sum of squared Gaussian derivatives $$f=\frac{1}{\text{total pixels}}\sum _x\sum _y((\text{Gaussian derivative X}(x,y){)}^2+(\text{Gaussian derivative Y}(x,y){)}^2)$$ where $\sigma =d/(2\sqrt{3})$, $d$ = dimension of the smallest feature;

 

2. Statistical Algorithms:

•  Variance Measure ; $$f=\frac{1}{\text{total pixels}}\sum _x\sum _y(I(x,y)-\bar{I}{)}^2;$$ where $\bar{I}$ is the mean of image.
• Normalized Variance Measure ; $$f=\frac{1}{\text{total pixels}\times \bar{I}}\sum _x\sum _y(I(x,y)-\bar{I}{)}^2;$$
• Auto-correlation Measure: $$f=\sum _x\sum _{y}I(x,y)I(x+1,y)-\sum _x\sum _{y}I(x,y)I(x+2,y);$$
• Standard Deviation-based Correlation: $$f=\sum _x\sum _{y}I(x,y)I(x+1,y)-{\bar{I}}^2(x,y)\times \text{total pixels};$$

3. Histogram-based Algorithms:

• Range Algorithm: $$f=\text{max}\{i|\text{histogram}(i)>0\}-\text{min}\{i|\text{histogram}(i)>0\};$$
• Entropy Algorithm: $$f=-\sum _{i=0}^{255}p(i){\log }_{2}p(i)$$ where $p(i)=\text{histogram}(i)/\text{total pixels};$

4. Other Algorithms:

• Threshold Contents: $$f=\sum _x\sum _{y}I(x,y)$$ where $I(x,y)\ge \text{threshold};$
• Thresholded Pixel Count: $$f=\sum _x\sum _y\theta (\text{threshold}-I(x,y));$$ where $\theta (x)$ is the step-function.
• Image Power: $$f=\sum _x\sum _y{I}^2(x,y)$$ where $I(x,y)\ge \text{threshold};$

Ref: Dynamic evaluation of autofocusing for automated microscopic analysis of blood smear and pap smear, J. Microscopy,227, 15(2007).