point_spread_function

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Definition

The point spread function (PSF) is the response of an imaging system to a point source. It is the optical equivalent of the impulse_response in systems theory.

For an imaging operator $\mathcal{L}$, the point spread function $h(x; x_0)$ is:

$$ \mathcal{L}\{\delta(x - x_0)\} = h(x; x_0) $$

where $\delta(x - x_0)$ represents a point source at position $x_0$.

Physical Interpretation

  • The PSF describes how a single point of light is “spread out” by the imaging system.
  • A perfect imaging system would have $h(x; x_0) = \delta(x - x_0)$ (no spreading).
  • Real systems have finite-width PSFs due to diffraction, aberrations, or defocus.

Shift-Invariant Systems

For shift-invariant (isoplanatic) systems, the PSF depends only on relative position:

$$ h(x; x_0) = h(x - x_0) $$

The image $g(x)$ of an object $f(x)$ is then given by convolution:

$$ g(x) = f(x) * h(x) = \int f(x_0) \, h(x - x_0) \, dx_0 $$

Examples of Shift-Invariant Effects

Examples of Non-Shift-Invariant Effects

  • Coma
  • Astigmatism
  • Distortion