# Bayer CFA Effect on Sharpness

In this article we shall find that the effect of a Bayer CFA on the spatial frequencies and hence the ‘sharpness’ captured by a sensor compared to those from a corresponding monochrome imager can go from nothing to halving the potentially unaliased range based on the chrominance content of the image projected on the sensing plane and the direction in which the spatial frequencies are being stressed.

#### A Little Sampling Theory

We know from Goodman[1] and previous articles that the sampled image ( ) captured in the raw data by a typical current digital camera can be represented mathematically as  the continuous image on the sensing plane ( ) multiplied by a rectangular lattice of Dirac delta functions positioned at the center of each pixel:

(1)

with the functions representing the two dimensional grid of delta functions, sampling pitch apart horizontally and vertically.  To keep things simple the sensing plane is considered here to be the imager’s silicon itself, which sits below microlenses and other filters so the continuous image is assumed to incorporate their as well as pixel aperture’s effects. Continue reading Bayer CFA Effect on Sharpness

# 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 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 space to a fast fourier transform routine and, presto, it produces MxN numbers that represent the amplitude of the PSF on the sensing plane, as shown below for the pupil function of a perfect lens with a circular aperture and MxN = 1024×1024.

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?

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

# 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   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 ) and finally arrives at its back end, the exit pupil.   The complex light field at the exit pupil’s two dimensional plane is now   as shown below:

# A Simple Model for Sharpness in Digital Cameras – Aliasing

Having shown that our simple two dimensional MTF model is able to predict the performance of the combination of a perfect lens and square monochrome pixel we now turn to the effect of the sampling interval on spatial resolution according to the guiding formula:

(1)

The hats in this case mean the Fourier Transform of the relative component normalized to 1 at the origin (), that is the individual MTFs of the perfect lens PSF, the perfect square pixel and the delta grid.

#### Sampling in the Spatial and Frequency Domains

Sampling is expressed mathematically as a Dirac delta function at the center of each pixel (the red dots below).

# 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

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

# Sub Bit Signal

My camera has a 14-bit ADC.  Can it accurately record information lower than 14 stops below full scale? Can it store sub-LSB signals in the raw data?

With a well designed sensor the answer, unsurprisingly if you’ve followed the last few posts, is yes it can.  The key to being able to capture such tiny visual information in the raw data is a well behaved imaging system with a properly dithered ADCContinue reading Sub Bit Signal