Tag Archives: linear

Wavefront to PSF to MTF: Physical Units

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

Obtaining the 2D DFTs is easy: simply feed MxN numbers representing the two dimensional complex image of the pupil function in its uv space to a fast fourier transform routine and, presto, it produces MxN numbers that represent the amplitude of the PSF on the xy sensing plane, as shown below for the pupil function of a perfect lens with a circular aperture and MxN = 1024×1024.

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?

Figure 2. A slice through the center of the PSFshown in figure 1b.

Less easy – and the subject of this article as seen from a photographic perspective.

Continue reading Wavefront to PSF to MTF: Physical Units

Aberrated Wave to Image Intensity to MTF

Goodman, in his excellent Introduction to Fourier Optics[1], describes how an image is formed on a camera sensing plane starting from first principles, that is electromagnetic propagation according to Maxwell’s wave equation.  If you want the play by play account I highly recommend his math intensive book.  But for the budding photographer it is sufficient to know what happens at the exit pupil of the lens because after that the transformations to Point Spread and Modulation Transfer Functions are straightforward, as we will show in this article.

The following diagram exemplifies the last few millimeters of the journey that light from the scene has to travel in order to smash itself against our camera’s sensing medium.  Light from the scene in the form of  field  U arrives at the front of the lens.  It goes through the lens being partly blocked and distorted by it (we’ll call this blocking/distorting function P) and finally arrives at its back end, the exit pupil.   The complex light field at the exit pupil’s two dimensional uv plane is now  U\cdot P as shown below:

Figure 1. Simplified schematic diagram of the space between the exit of a camera lens and its sensing plane. The space is filled with air.

Continue reading Aberrated Wave to Image Intensity to MTF

A Simple Model for Sharpness in Digital Cameras – I

The next few posts will describe a linear spatial resolution model that can help a photographer better understand the main variables involved in evaluating the ‘sharpness’ of photographic equipment and related captures. I will show numerically that the combined spectral frequency response (MTF) of a perfect AAless monochrome digital camera and lens in two dimensions can be described as the normalized multiplication of the Fourier Transform (FT) of the lens Point Spread Function by the FT of the (square) pixel footprint, convolved with the FT of a rectangular grid of Dirac delta functions centered at each  pixel, as better described in the article

    \[ MTF_{2D} = \left|(\widehat{ PSF_{lens} }\cdot \widehat{PIX_{ap} })\ast\ast\: \delta\widehat{\delta_{pitch}}\right|_{pu} \]

With a few simplifying assumptions we will see that the effect of the lens and sensor on the spatial resolution of the continuous image on the sensing plane can be broken down into these simple components.  The overall ‘sharpness’ of the captured digital image can then be estimated by combining the ‘sharpness’ of each of them. Continue reading A Simple Model for Sharpness in Digital Cameras – I

Linearity in the Frequency Domain

For the purposes of ‘sharpness’ spatial resolution measurement in photography  cameras can be considered shift-invariant, linear systems.

Shift invariant means that the imaging system should respond exactly the same way no matter where light from the scene falls on the sensing medium .  We know that in a strict sense this is not true because for instance a pixel has a square area so it cannot have an isotropic response by definition.  However when using the slanted edge method of linear spatial resolution measurement  we can effectively make it shift invariant by careful preparation of the testing setup.  For example the edges should be slanted no more than this and no less than that. Continue reading Linearity in the Frequency Domain