Tag Archives: image

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 aberration function \Delta W(u,v) as earlier described.

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.

Continue reading Fourier Optics and the Complex Pupil Function →

Deconvolution, Processing, Sharpening

Elements of Richardson-Lucy Deconvolution

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We have seen that deconvolution by naive division in the frequency domain only works in ideal conditions not typically found in normal photographic settings, in part because of shot noise inherent in light from the scene. Half a century ago William Richardson (1972)[1] and Leon Lucy (1974)[2] independently came up with a better way to deconvolve blurring introduced by an imaging system in the presence of shot noise. Continue reading Elements of Richardson-Lucy Deconvolution →

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 →

adobe camera raw, Color, LR, Processing, Sharpening

How Is a Raw Image Rendered?

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What are the basic low level steps involved in raw file conversion?  In this article I will discuss what happens under the hood of digital camera raw converters in order to turn raw file data into a viewable image, a process sometimes referred to as ‘rendering’.  We will use the following raw capture by a Nikon D610 to show how image information is transformed at every step along the way:

Nikon D610 with AF-S 24-120mm f/4 lens at 24mm f/8 ISO100, minimally rendered from raw as outlined in the article.
Figure 1. Nikon D610 with AF-S 24-120mm f/4 lens at 24mm f/8 ISO100, minimally rendered from raw by Octave/Matlab following the steps outlined in the article.

Rendering = Raw Conversion + Editing

Continue reading How Is a Raw Image Rendered? →

Equivalence, Exposure, Formats, IQ, Sensors

Comparing Sensor SNR

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We’ve seen how SNR curves can help us analyze digital camera IQ:

SNR-Photon-Transfer-Model-D610-4

In this post we will use them to help us compare digital cameras, independently of format size. Continue reading Comparing Sensor SNR →

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, assuming a well set up imaging system in good working conditions. Continue reading MTF50 and Perceived Sharpness →

Exposure, Information Science, IQ, Sensors

Exposure and ISO

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The in-camera ISO dial is a ballpark milkshake of an indicator to help choose parameters that will result in a ‘good’ perceived picture. Key ingredients to obtain a ‘good’ perceived picture are 1) ‘good’ Exposure and 2) ‘good’ in-camera or in-computer processing. It’s easier to think about them as independent processes and that comes naturally to you because you shoot raw in manual mode and you like to PP, right? Continue reading Exposure and ISO →