A point response becomes a contrast transfer rule

For mutually incoherent object points, their time-averaged image intensities add. Let h be the intensity point-spread function after factoring out overall throughput, with its spatial integral equal to one. A displaced object point produces a displaced copy of h in appropriately scaled image coordinates. Summing these copies produces the convolution formula. Shift invariance is an approximation over a limited field: a strongly aberrated wide-field instrument may need a different response at different positions.

The normalized PSF is nonnegative because it represents intensity. Its width measures one aspect of blur, while sidelobes describe light redistributed away from a point. Two PSFs with equal widths can transmit different contrasts. The coherent amplitude impulse response is a different object: it can be complex and can change sign. In scalar quasi-monochromatic imaging the intensity PSF is proportional to its modulus squared, with normalization chosen separately. Squaring a PSF already defined as intensity would count the detection nonlinearity twice.

Define the normalized optical transfer function as the Fourier transform of the intensity PSF. The zero-frequency value is one, expressing conservation of mean intensity after the throughput factor is removed. Its modulus is the modulation transfer function, which multiplies the contrast of a sinusoidal intensity object. Its argument gives a phase shift of that sinusoid. An asymmetric shifted PSF may therefore have a complex OTF even though the PSF itself is entirely real and nonnegative.

For a coherent pupil function P written in spatial-frequency coordinates, the intensity PSF is the squared modulus of the inverse transform of P. The convolution theorem makes its transform a normalized autocorrelation of P. A clear one-dimensional rectangular pupil consequently gives a triangular incoherent OTF: shifted pupils overlap less as their separation grows. Aberrations enter through pupil phase, making the overlap a complex weighted integral rather than merely an overlap area. The coherent amplitude transfer function remains the pupil itself under the same scaling.

A sampled teaching model makes these ideas concrete. Give neighboring sample offsets minus one, zero, and plus one the PSF weights one quarter, one half, and one quarter. The weights sum to one, so a constant intensity remains unchanged. Adding the two conjugate frequency factors gives an OTF equal to one half plus one half cosine of twice pi times frequency in cycles per sample. This is a specified discrete convolution model, not a universal diffraction PSF or a claim about detector sampling without aliasing.

The spectral instrument helps distinguish coarse from fine spatial content. Reset and set both controls to mode count one and coefficient decay one; predict a single sine. Reset again and set mode count four and decay one; predict added shorter spatial periods with smaller coefficients. It computes a finite sum of sin(nx−t) divided by n to the selected decay power. It does not convolve an optical PSF, measure an OTF, or model the physical temporal frequencies of those spatial modes.