GAUSSIAN FILTER

Gaussian filter is an LTE operator with a 2D Gaussian function as a impulse response (with $\mu =0$ and a constant diagonal covariance matrix)

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

The effect of increasing $\sigma$ can be seen as reducing the importance of the closest points and increasing the importance of farther points smoothing the image

CHOOSING KERNEL SIZE

In a real implementation the Gaussian function need to be approximated by a discrete representation by sampling the function in a kernel matrix, so the matrix size need to be set

• with higher size matrix becomes more accurate
• computational cost grows with matrix size

so a rule of thumb is to select the kernel size based on the $\sigma$ value as

$$kernelsize=(2k+1)\times (2k+1)wherek=3\sigma$$

as $3\sigma$ covers the 99% of the area of the function (energy)

EXPLOITING SEPARABILITY TO IMPROVE PERFORMANCE

To improve the computational complexity, the separability of the 2D Gaussian function can be used as $G(x,y)=G(x)G(y)$

$$I(x,y)\ast G(x,y)=I(x,y)\ast G(x)\ast G(y)=I(x,y)\ast G(y)\ast G(x)$$

The speedup improvements of this process can be expressed as

$$S=\frac{2(2k+1{)}^2}{4(2k+1)}=k+\frac{1}{2}with$$ $$NOP{S}_{2Dfilter}=2(2k+1{)}^2$$ $$NOP{S}_{1Dfilter}=4(2k+1)$$

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