Gaussian Filter

The 1D Gaussian filter is:

$$g(x) = \frac{1}{\sqrt{2\pi}\sigma} e^{\frac{-x^2}{2\sigma^2}}$$ (1)

The impulse response of the 1D Gaussian Filter is given by:

$$g(x) = \frac{1}{\sqrt{2 \pi}}e^{\frac{-\sigma^2 u^2}{2}}$$ (2)

Properties of the Gaussian Filter

1. An important property of the Gaussian function is that the fourier of the Gaussian is itself a Gaussian
   

   $$G(u) = \frac{1}{\sqrt{2 \pi}}e^{\frac{-\sigma^2 u^2}{2}}$$ (3)

   
   

2. The width of the Gaussian increases as $\sigma$ increases

   Figure 3: Effect of parameter sigma on the Gaussian function

   

3. $\sigma_x$ and $\sigma_u$ are inversely related i.e, the bandwidth of the filter is inversely related to $\sigma_x$.

The impulse response of the 2D Gaussian filter is:

$$g(x,y) = \frac{1}{\sqrt{2\pi}\sigma} e^{-\frac{1}{2}{(\frac{x^2}{\sigma_x^2}+\frac{y^2}{\sigma_y^2})}}$$ (4)

The frequency response of the 2D Gaussian Filter is given by:

$$H(u,v) = 2 \pi \sigma_x \sigma_y [e^{-2 \pi^2[u^2 \sigma_x^2 + v^2 \sigma_y^2]}$$ (5)

= $\frac{1}{2 \pi \sigma_u \sigma_v} e^{- \frac{1}{2}[\frac{u^2}{\sigma_u^2} + \frac{v^2}{\sigma_v^2}]}$ where, $\sigma_u = \frac{1}{2 \pi \sigma_x}, \sigma_v = \frac{1}{2 \pi \sigma_y}$

Figure 4: Plot of frequency response of the 2D Gaussian

As can be seen from the figure, the frequency response of the 2D Gaussian, it is a low pass filter.