Gaussian Blur

As we saw in Example 6.8, an obvious use for masking is to implement a smoothing or blurring. Such an effect is easy to see on an image. We now approximate a normal, or Gaussian blur, obtained perhaps using a ``soft'' lens so the light spreads out on the focal plane, rather than all going to the ``correct'' spot. Such a Gaussian, of variance $\sigma^{2}_{}$ is of the form

f (j, k) = $${\frac{1}{\sqrt{2\pi\sigma}}}$$exp$$\left(\vphantom{\frac{-(j^2 + k^2)}{2\sigma^2}}\right.$$$${\frac{-(j^2 + k^2)}{2\sigma^2}}$$$$\left.\vphantom{\frac{-(j^2 + k^2)}{2\sigma^2}}\right)$$,

where conventionally we assume the function is zero outside a neighbourhood of radius 3$\sigma$. If we take $\sigma$ = 1 we get the mask

$${\textstyle\frac{1}{1003}}$$ $$\begin{pmatrix} 0& 0& 1& 2& 1& 0& 0\\ 0& 3&13& 22&13& 3& 0 ... ... 97&59&13& 1\\ 0& 3&13& 22&13& 3& 0\\ 0& 0& 1& 2& 1& 0& 0 \end{pmatrix}$$.

The result of applying this mask, considered as centered on the largest value, to the image of Fig 6.1 is shown in Fig. 6.2. It is clear that a noticeable blur has been obtained as was expected.

Figure 6.1: Input image.

Figure 6.2: Output after Gaussian filtering.

Figure 6.3: The zero-crossings of the image filtered by a difference of Gaussians.

It may be less clear why there is any interest in blurring an image. However it can form a useful intermediate step. The image shown in Fig 6.1 was filtered using a difference of Gaussians (DoG filter) at two different scales. The resulting filter is supposed to respond in a way very simlarly to the receptive fields in our eyes. The output is no longer always positive, so does not represent an image, but the pixels where the intensity changes sign are marked in Fig 6.3. The overall effect is that of a neurophysiologically plausible edge detector.