A Unified Theory of Image Scaling: Point Spread Functions, Discrete Kernels, and Scale-Spaces

Assaf Shocher

We derive the optically aligned system of image-scaling operators $\mathbf{K}_s$ that reproduces optical zoom (exact zoom-out, canonical zoom-in), by identifying a Huygens-like consistency principle in which each pixel acts as a secondary source. The system is determined in closed form by the camera's point spread function (PSF), itself the operator $\mathbf{K}_0$. The operators compose, $\mathbf{K}_{s'} \circ \mathbf{K}_s = \mathbf{K}_{ss'}$ (with band projection for down-up), and account for both upscaling and downscaling, which form a Moore-Penrose pseudoinverse pair, $\mathbf{K}_{1/s} = \mathbf{K}_s^\dagger$. We show that image scaling, as routinely practiced, is neither optically faithful nor internally consistent; for example, common anti-aliasing rules generally correspond to no camera, and two resizes by a factor of $1/2$ generally do not equal one resize by a factor of $1/4$. Gaussian scale-space kernels cannot be exact finite-aperture PSFs. Gaussian scale-space is an idealized special case of what emerges here: a generalized scale-space for sampled images, derived entirely from camera optics, bidirectional, and not limited to Gaussian kernels. The system goes beyond…