Computer Vision Foundations · free preview

The Gaussian Filter

The Gaussian kernel as the smooth low-pass without the box's ripples, discretized and truncated, separable for speed, and closed under convolution in the continuous world.

1. A bell instead of a plateau

A bell instead of a plateau

2. Discretizing, normalizing, truncating

Discretizing, normalizing, truncating

3. A Gaussian transforms into a Gaussian

A Gaussian transforms into a Gaussian

4. Blurs that add, in the continuous world

Blurs that add, in the continuous world

5. Where to cut the kernel

A discrete Gaussian is built with $\sigma = 2$ using the unnormalized form $g[n,m] = e^{-(n^2+m^2)/2\sigma^2}$ and truncated at $\pm 3\sigma$ . Which half-width is used, and why that bound?

6. Widths that trade off

Two Gaussian filters have $\sigma_1 = 1$ and $\sigma_2 = 2$ . Comparing their frequency responses $G = e^{-w^2\sigma^2/2}$ , which statement is right?

7. Adding blur variances

An image is blurred with a Gaussian of variance $\sigma_1^2 = 4$ and then with an independent Gaussian of variance $\sigma_2^2 = 9$ . Using the closure law $\sigma_3^2 = \sigma_1^2 + \sigma_2^2$ , compute the combined variance $\sigma_3^2$ and write it as an integer.

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