i. High-boost filter is a sharpening second order derivative filter.

ii. High-boost filter image is obtained by subtracting LPF image from the scaled input image.

HBF image = k(original image) – LPF image

= (k-1)original image + original image – LPF

= (k-1)original image + HPF image

where k is any positive scaling factor.

For k-1, HBF image = HPF image, therefore for HBF image k > 1 let us derive HBF mask by considering a digital image F

$ F = \begin{bmatrix} z_1 & z_2 & z_3 \ z_4 & z_5 & z_6 \ z_7 & z_8 & z_9 \ \end{bmatrix} $

Step 1: Find LPF image using Low Pass Averaging filter mask

To find A(x=1, y=1)

A(1,1)=$\frac{1}{9}[z_1+z_2+z_3+z_4+z_5+z_6+z_7+z_8+z_9]$

Step 2: Find HBF image

HBF image = k(original image) – LPF image

To find B(x=1, y=1)

B(1,1)=kz_5-A(1,1)

B(1,1)=$kz_5-\frac{1}{9}[z_1+z_2+z_3+z_4+z_5+z_6+z_7+z_8+z_9]$

B(1,1)=$(k-\frac{1}{9}) z_5-\frac{z_1}{9}-\frac{z_2}{9}-\frac{z_3}{9}-\frac{z_4}{9}-\frac{z_5}{9}-\frac{z_6}{9}-\frac{z_7}{9}-\frac{z_8}{9}-\frac{z_9}{9}$

Step 3: Find HBF mask

$w_1=-\frac{1}{9} w_2=-\frac{1}{9} w_3=-\frac{1}{9}w_4=-\frac{1}{9} w_5=\frac{9k-1}{9}$

$w_6=-\frac{1}{9} w_7=-\frac{1}{9} w_8=-\frac{1}{9} w_9=-\frac{1}{9}$

The HBF mask is given by,