Computer Vision Foundations · free preview

Translation Invariance and Convolution

Why processing should not depend on where content sits, and the theorem that convolution is the only linear operator with that property.

1. Content can appear anywhere

Content can appear anywhere

2. The local average as the canonical example

The local average as the canonical example

3. Convolution, the only such operator

Convolution, the only such operator

4. Properties you get for free

Properties you get for free

5. What translation invariance buys

A filter is described as linear and translation-invariant. Which statement follows from the theory of this lesson?

6. Checking equivariance on an example

An image contains a bright dot, and a blur filter produces a small blob around it. The dot is moved 40 pixels to the right and the filter runs again. What does translation equivariance predict about the new output?

7. A tiny convolution

Convolve the kernel h=[1,1]h = [1,1] with the signal ℓin=[2,4]\ell_{\text{in}} = [2,4], computing h∘ℓin=[2,6,4]h \circ \ell_{\text{in}} = [2, 6, 4]. Write the middle value of the output.

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