Step 1: Write the convolution integral. For two continuous images $f(x,y)$ and $h(x,y)$, spatial convolution is defined as \[ g(x,y) = f(x,y) * h(x,y) = \int\int f(\alpha,\beta)\, h(x-\alpha, y-\beta)\, d\alpha\, d\beta \]
Step 2: Take the Fourier transform of both sides. Taking the 2D Fourier transform of $g(x,y)$ and using the shift property of the transform, the double integral separates into a product of the individual transforms: \[ G(u,v) = \mathcal{F}\{f * h\} = F(u,v)\, H(u,v) \]
Step 3: Physical interpretation. Multiplying two Fourier spectra means that at every spatial frequency $(u,v)$, the amplitude and phase contributions of the image and the filter are multiplied together, which is exactly how ideal, Gaussian and Butterworth frequency-domain filters attenuate or pass specific frequency bands.
Step 4: Eliminate the distractors. "Convolution" (B) would apply only for combining operations within the same domain repeatedly. "Addition" (C) is what happens to the transforms only if the spatial operation itself was addition, by linearity. "Cross-correlation" (D) transforms to $F(u,v)H^*(u,v)$, involving a complex conjugate, a distinct operation used for template matching. \[ \boxed{\text{Spatial convolution} \; \Longleftrightarrow \; \text{Frequency-domain multiplication}} \]
