平移不变性与卷积

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The local average as the canonical example

Convolution, the only such operator

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

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?

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