Sharpening

As the opposite of low-pass filtering for image smoothing and noise reduction, high-pass filtering can sharpen the image, thereby enhancing and emphasizing the detailed information (high spatial frequency components) in the image.

High-pass filtering can be carried out by subtracting the low-pass filtered image from its original version, which can be considered as all-pass filtered by a delta function kernel:

$$W_{ap}=\left[ \begin{array}{ccc} 0 & 0 & 0 0 & 1 & 0 \\ 0 & 0 & 0 \end{array} \right]$$

When an image is convolved with this delta function is not changed:

$$W_{ap} * I_o = I_o$$

As convolution is a linear operation, we have

$$I_{hp}&=& I_{ap}-I_{lp}=W_{ap}*I_o-W_{lp}*I_o =(W_{ap}-W_{lp})*I_o=W_{hp}*I_o$$

where $W_{hp}=W_{ap}-W_{lp}$ is the high-pass kernel corresponding to the low-pass kernel $W_{lp}$. Equivalently the frequency domain filtering is shown in the figure:

We can therefore obtain a high-pass filtering kernel corresponding to each of the low-pass filter kernels by subtracting the low-pass kernel from the all-pass kernel. The resulting kernels are various forms of high-pass filtering kernels, also called the Laplace operators.

• 
  
  $$W_{hp}=W_{ap}-W_{lp} =\left[ \begin{array}{ccc} 0 & 0 & 0 \\... ... & -1 & 0 -1 & 4 & -1 \\ 0 & -1 & 0 \end{array} \right]$$
  
  

• 
  
  $$W_{hp}=W_{ap}-W_{lp} =\left[ \begin{array}{ccc} 0 & 0 & 0 \\... ...-1 & -1 -1 & 8 & -1 \\ -1 & -1 & -1 \end{array} \right]$$
  
  

• 
  
  $$W_{hp}=W_{ap}-W_{lp} =\left[ \begin{array}{ccc} 0 & 0 & 0 \\... ...2 & -1 -2 & 12 & -2 \\ -1 & -2 & -1 \end{array} \right]$$
  
  

Note that the sum of all elements of the resulting high-pass filter is always zero. When such a high-pass kernel is convolved with a region of an image where all pixels have same gray level (constant or DC component), the result is zero, i.e., the zero spatial frequency component is totally suppressed by the high-pass filter.

Similar band-pass filters can be obtained by finding the difference between two low-pass filters of different cut-off frequencies.

The following figure shows the original image, a cat (left), and its low-pass (middle) and high-pass (right) filtered versions.