Computer Vision Foundations · free preview

Binomial Filters

Rows of Pascal's triangle form a discrete blur family with exact closure and an exact zero on the checkerboard wave, making the small binomial the default kernel.

1. Pascal's triangle is a convolution engine

Pascal's triangle is a convolution engine

2. Closure, this time exact

Closure, this time exact

3. The checkerboard wave dies exactly

The checkerboard wave dies exactly

4. The two-dimensional default

The two-dimensional default

5. Reading closure off the triangle

The binomial kernels come from repeated convolution of one-one. Why does $b_2 \circ b_3 = b_5$ hold exactly while the same statement fails for sampled Gaussians?

6. The zero at Nyquist

Why is the exact zero of $[1,2,1]$ at the alternating wave more valuable than the sampled Gaussian's strong but incomplete attenuation there?

7. Combining binomial rows

The binomial kernel $b_2$ is convolved with the binomial kernel $b_3$ . Write the index of the resulting binomial row $b_n$ as a number.

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