Tag Archives: magnification

Fourier Optics, Geometrical Optics, MTF, PSF, wavefront

Fourier Optics and the Complex Pupil Function

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In the last article we learned that a complex lens can be modeled as just an entrance pupil, an exit pupil and a geometrical optics black-box in between.  Goodman[1] suggests that all optical path errors for a given Gaussian point on the image plane can be thought of as being introduced by a custom phase plate at the pupil plane, delaying or advancing the light wavefront locally according to wavefront aberration function \Delta W(u,v) as described there.

The phase plate distorts the forming wavefront, introducing diffraction and aberrations, while otherwise allowing us to treat the rest of the lens as if it followed geometrical optics rules.  It can be associated with either the entrance or the exit pupil.  Photographers are usually concerned with the effects of the lens on the image plane so we will associate it with the adjacent Exit Pupil.

aberrations coded as phase plate in exit pupil generalized complex pupil function
Figure 1.  Aberrations can be fully described by distortions introduced by a fictitious phase plate inserted at the uv exit pupil plane.  The phase error distribution is the same as the path length error described by wavefront aberration function ΔW(u,v), introduced in the previous article.

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Optics, Quantization, Sensors, Spatial Resolution

Wavefront to PSF to MTF: Physical Units

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In the last article we saw that the intensity Point Spread Function and the Modulation Transfer Function of a lens could be easily approximated numerically by applying Discrete Fourier Transforms to its generalized exit pupil function \mathcal{P} twice in sequence.[1]

Numerical Fourier Optics: amplitude Point Spread Function, intensity PSF and MTF

Obtaining the 2D DFTs is easy: simply feed MxN numbers representing the two dimensional complex image of the Exit Pupil function in its uv space to a Fast Fourier Transform routine and, presto, it produces MxN numbers representing the amplitude of the PSF on the xy sensing plane.  Figure 1a shows a simple case where pupil function \mathcal{P} is a uniform disk representing the circular aperture of a perfect lens with MxN = 1024×1024.  Figure 1b is the resulting intensity PSF.

Figure 1a, left: A circular array of ones appearing as a white disk on a black background, representing a circular aperture. Figure 1b, right: Array of numbers representing the PSF of image 1a in the classic shape of an Airy Pattern.
Figure 1. 1a Left: Array of numbers representing a circular aperture (zeros for black and ones for white).  1b Right: Array of numbers representing the PSF of image 1a (contrast slightly boosted).

Simple and fast.  Wonderful.  Below is a slice through the center, the 513th row, zoomed in.  Hmm….  What are the physical units on the axes of displayed data produced by the DFT? Continue reading Wavefront to PSF to MTF: Physical Units →

Perceptual, Sharpness, Spatial Resolution

MTF50 and Perceived Sharpness

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Is MTF50 a good proxy for perceived sharpness?   In this article and those that follow MTF50 indicates the spatial frequency at which the Modulation Transfer Function of an imaging system is half (50%) of what it would be if the system did not degrade detail in the image painted by incoming light.

It makes intuitive sense that the spatial frequencies that are most closely related to our perception of sharpness vary with the size and viewing distance of the displayed image.

For instance if an image captured by a Full Frame camera is viewed at ‘standard’ distance (that is a distance equal to its diagonal), it turns out that the portion of the MTF curve most representative of perceived sharpness appears to be around MTF90.  On the other hand, when pixel peeping, the spatial frequencies around MTF50 look to be a decent, simple to calculate indicator of it with a current imaging system in good working conditions. Continue reading MTF50 and Perceived Sharpness →