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Optics shift invariant
Many ISETCam calculations use a shift-invariant optical calculation. Shift invariance means that the image of a point in the scene spreads by the same amount in the image, no matter the position of the point in the scene. The spread in the image, called the point spread function (PSF), varies depending on the wavelength of the point light. The image region over which shift-invariance is a good approximation for a given lens is called its isoplanatic region.
In general, the PSF depends on the location of the point in the scene, both its location from the center of the image (field height) and most importantly the distance from the lens to the point. But the shift-invariant approximation is very useful for many scenes, for example when scenes with a modest (20 deg) field of view and with all the objects at approximately the same distance (in a plane). It is also useful when all the objects are far away, because for many lenses the PSF changes little for points beyond a certain distance. For the human eye or for the classical double Gauss lens, the distance is 10-20 focal lengths.
The PSF is a property of the optics, and thus we describe it with parameters in the optics structure. Over the years, we stored information about the PSF in various representations. Starting in 2023, we stored the information using wavefront aberrations. (To learn why, see the page on PSF representations).
The PSF is not stored explicitly in the optics struct; it is computed from the parameters stored there. We store the optics wavefront aberration as a set of polynomial coefficients, using the international standard Zernike representations. The polynomial coefficients correspond to aberrations with specific labels such as defocus, coma, and different types of astigmatism. We can compute the PSF from these polynomial coefficients, and this is done on-the-fly in oiCompute().
The value of using the polynomial representations is that they are continuous. Thus, we can start with them and create a PSF that is properly sampled in space for the resolution of the optical image (see opticsPSF). The spatial sampling resolution of the oi is determined by If we store the PSF, we need to pick a spatial sampling resolution, and that may not be appropriate for a specific scene's spatial resolution.
In the spectral irradiance calculations within oiCompute, the wavefront aberrations are converted into a point spread function that is sampled at the resolution of the scene. The PSF is convolved with the scene, wavelength-by-wavelength. The implementation requires paying attention to the details of the sampling rate for the scene spectral radiance. There are many tutorials and examples of the oiCompute calculations in the ISETCam toolbox.
There is an important special case of the perfect, shift-invariant lens with a circular aperture. The diffraction-limited calculation for a circular aperture has a formula for the point spread function. The calculation is shift-invariant, applying the classic point spread uniformly across the scene to create the optical image. The point spread is wavelength-dependent, but shift-invariant.
The diffraction-limited PSF depends only on the f-number (ratio of the focal length to the aperture) of the lens. We will explain the formula and the issues with spatial sampling here.
The wavefront calculation can also be diffraction limited, with a circular (or non-circular) aperture. The calulcations differ in how they handle the spatial sampling. The wavefront calculation,which is a bit slower, and it is more accurate for very high dynamic range scenes.
Many evaluations use a simple approximation of the scene and optics. The scene is modeled as a two-dimensional plane, and for each wavelength the optics is modeled as a shift-invariant linear system. In this way the calculation from the scene radiance to the optical image is no more than a set of convolutions, one for each wavelength. The convolution kernel, called the point spread function (PSF), varies with wavelength but (in this case) not with position in the visual field. Even systems that are not completely shift-invariant, are approximately shift-invariant over some region (isoplanatic).
Perfect optics is called diffraction-limited optics. The wavelength-dependent point spread function (PSF) of a diffraction-limited lens with circular optics can be calculated from a formula that depends only on the f-number (ratio of focal length to aperture diameter). This PSF is called the Airy Disk. This lens model is often used for fast, approximate, calculations.
ISETcam development is led by Brian Wandell's Vistalab group at Stanford University and supported by contributors from other research institutions and industry.