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: $$f = \sum_x \sum_y ( (L * I) (x, y))^2$$  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 ; $$f = \frac{1}{\text{total pixels}}\sum _x \sum _y ( I  (x, y) - \overline{I})^2;$$ where $\overline{I}$ is the mean of image.
• Normalized Variance Measure ; $$f = \frac{1}{\text{total pixels}\times \overline{I}} \sum_x \sum_y ( I(x, y)- \overline{I})^2;$$
• 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: $$f = \sum_x \sum_y I(x, y) I(x+1, y) - \overline{I}^2(x, y)\times \text{total pixels};$$

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).