Modulation Transfer Function (MTF) & Spatial Frequency: Contrast Transfer from 0 to 60 Cycles/Degree

The Optical Transfer Function (OTF) is the complex Fourier transform of the Point Spread Function ($PSF$). The MTF is defined as the absolute magnitude of the OTF:

$$OTF(f_x, f_y) = \mathcal{F}\{PSF(x, y)\} = MTF(f_x, f_y) \cdot e^{i \cdot PTF(f_x, f_y)}$$

For a sinusoidal luminance grating, contrast modulation $M$ is defined via Michelson contrast:

$$M = \frac{L_{\text{max}} - L_{\text{min}}}{L_{\text{max}} + L_{\text{min}}} \implies MTF(f) = \frac{M_{\text{image}}(f)}{M_{\text{object}}(f)}$$

In human visual optics, spatial frequency corresponds directly to Snellen optotype letter sizes:

• $3\text{ to } 6\ \text{cycles/degree}$ (cpd): Large coarse objects (faces, doorways, road contours). Peaks in the neural Contrast Sensitivity Function (CSF).
• $15\ \text{cpd}$: Equivalent to $20/40$ Snellen acuity (minimum legal standard for unrestricted driver's licensing).
• $30\ \text{cpd}$: Standard $20/20$ Snellen Acuity ($1\text{ minute of arc}$ stroke width).
• $60\ \text{cpd}$: $20/10$ 'Eagle Vision', the absolute anatomical Nyquist sampling limit of human foveal cone photoreceptors ($2.5\ \mu\text{m}$ cone spacing).

When spherical aberration or defocus is present, the MTF curve plunges rapidly to zero at medium frequencies. In severe defocus, the Phase Transfer Function (PTF) jumps by $\pi$ radians, causing spurious resolution where black and white bars invert—a phenomenon that causes double borders and visual migraine aura.