Introduction & Theory

Images can be fuzzy from random noise and blurrings.

An image can be sharpened by:

1. detecting the edges

2. combining the edges with the original image

These steps are shown in the figure below.

Image sharpening steps.

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Edge detection

Edges can be detected using a Laplacian filter. The Laplacian \(L(x,y)\) is the second spatial derivative of the image intensity \(I(x,y)\). This means it highlights regions of rapid intensity change, i.e the edges.

\[ L(x,y) = \frac{\partial^2 I }{\partial x^2} + \frac{\partial^2 I }{\partial y^2} \]

In practice, the Laplacian also highlights the noise in the image and, therefore, it is sensible to apply smoothing to the image as a first step. Here we will apply a Guassian filter. The Guassian filter \(G(x,y)\) approximates each pixel as a weighted average of its neighbors.

\[ G(x,y) = \frac{1}{2 \pi \sigma^2} e^{- (x^2+y^2)/(2 \sigma^2)} \]

The two operations can be combined to give the Laplacian of Gaussian filter \(L \circ G(x,y)\).

\[ L \circ G(x,y) = -\frac{1}{\pi \sigma^4} \left( 1 - \frac{x^2+y^2}{2 \sigma^2}\right) e^{- (x^2+y^2)/(2 \sigma^2)} \]

These two functions \(G(x,y)\) and \(L \circ G(x,y)\) are graphed below.

Gaussian and Laplacian of Gaussian filters.

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Implementation

To apply the \(L \circ G\) filter to an image the \(L \circ G\) filter must be turned into a discrete mask, that is a matrix of size 2d+1 x 2d+1 where d is an integer. We use d=8, therefore the \(L \circ G\) filter is a 17x17 square, it looks like this:

\(L \circ G\) filter as a discrete mask.

To perform the convolution of this filter with the original image, the following operation is performed on each pixel,

\[ \text{edges}(i,j) = \sum_{k=-d}^d \sum_{l=-d}^d \text{image}(i + k, j + l) \times \text{filter}(k,l), \]

the sharpened image is then created by adding the edges to the original image, with a scaling factor (See the source code for the full details).