Category Archives: Quantization

illuminant, Information Science, Quantization, Sensors, SPD, variation

Photons, Shot Noise and Poisson Processes

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Every digital photographer soon discovers that there are three main sources of visible random noise that affect pictures taken in normal conditions: Shot, pixel response non-uniformities (PRNU) and Read noise.[1]

Shot noise (sometimes referred to as Photon Shot Noise or Photon Noise) we learn is ‘inherent in light’; PRNU is per pixel gain variation proportional to light, mainly affecting the brighter portions of our pictures; Read Noise is instead independent of light, introduced by the electronics and visible in the darker shadows.  You can read in this earlier post a little more detail on how they interact.

Read Noise Shot Photon PRNU Photo Resonse Non Uniformity

However, shot noise is omnipresent and arguably the dominant source of visible noise in typical captures.  This article’s objective is to  dig deeper into the sources of Shot Noise that we see in our photographs: is it really ‘inherent in the incoming light’?  What about if the incoming light went through clouds or was reflected by some object at the scene?  And what happens to the character of the noise as light goes through the lens and is turned into photoelectrons by a pixel’s photodiode?

Fish, dear reader, fish and more fish.

Continue reading Photons, Shot Noise and Poisson Processes →

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 →

Optics, Photometry, Quantization, Radiometry, Sensors, Spatial Resolution

Aberrated Wave to Image Intensity to MTF

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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 be absorbed by a 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 as it arrives at its virtual back end, the Exit Pupil, we’ll call this blocking/distorting function P.   Other than in very simple cases, the Exit Pupil does not necessarily coincide with a specific physical element or Principal surface.[iv]  It is a convenient mathematical construct which condenses all of the light transforming properties of a lens into a single plane.

The complex light field at the Exit Pupil’s two dimensional uv plane is then  U\cdot P as shown below (not to scale, the product of the two arrays is element-by-element):

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

Continue reading Aberrated Wave to Image Intensity to MTF →

Quantization, Sensors, Spatial Resolution

The Units of Discrete Fourier Transforms

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This article is about specifying the units of the Discrete Fourier Transform of an image and the various ways that they can be expressed.  This apparently simple task can be fiendishly unintuitive.

The image we will use as an example is the familiar Airy Disk from the last few posts, at f/16 with light of mean 530nm wavelength. Zoomed in to the left in Figure 1; and as it looks in its 1024×1024 sample image to the right:

Airy Mesh and Intensity
Figure 1. Airy disc image I(x,y). Left, 1a, 3D representation, zoomed in. Right, 1b, as it would appear on the sensing plane (yes, the rings are there but you need to squint to see them).

Continue reading The Units of Discrete Fourier Transforms →

Information Science, IQ, Quantization, Sensors

Sub Bit Signal

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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 ADC.  Continue reading Sub Bit Signal →

Posterization, Quantization

Sub LSB Quantization

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This article is a little esoteric so one may want to skip it unless one is interested in the underlying mechanisms that cause quantization error as photographic signal and noise approach the darkest levels of acceptable dynamic range in our digital cameras: one least significant bit in the raw data.  We will use our simplified camera model and deal with Poissonian Signal and Gaussian Read Noise separately – then attempt to bring them together.

Continue reading Sub LSB Quantization →

Quantization, Sensors

Photographic Sensor Simulation

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Physicists and mathematicians over the last few centuries have spent a lot of their time studying light and electrons, the key ingredients of digital photography.  In so doing they have left us with a wealth of theories to explain their behavior in nature and in our equipment.  In this article I will describe how to simulate the information generated by a uniformly illuminated imaging system using open source Octave (or equivalently Matlab) utilizing some of these theories.

Since as you will see the simulations are incredibly (to me) accurate, understanding how the simulator works goes a long way in explaining the inner workings of a digital sensor at its lowest levels; and simulated data can be used to further our understanding of photographic science without having to run down the shutter count of our favorite SLRs.  This approach is usually referred to as Monte Carlo simulation.

Continue reading Photographic Sensor Simulation →

Banding, Information Science, IQ, Perceptual, Posterization, Quantization

Smooth Gradients and the Weber-Fechner Fraction

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Whether the human visual system perceives a displayed slow changing gradient of tones, such as a vast expanse of sky, as smooth or posterized depends mainly on two well known variables: the Weber-Fechner Fraction of the ‘steps’ in the reflected/produced light intensity (the subject of this article); and spatial dithering of the light intensity as a result of noise (the subject of a future one).

Continue reading Smooth Gradients and the Weber-Fechner Fraction →

Banding, Information Science, IQ, Perceptual, Processing, Quantization, Sensors, Sharpening, USM

Image Quality: Raising ISO vs Pushing in Conversion

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In the last few posts I have made the case that Image Quality in a digital camera is entirely dependent on the light Information collected at a sensor’s photosites during Exposure.  Any subsequent processing – whether analog amplification and conversion to digital in-camera and/or further processing in-computer – effectively applies a set of Information Transfer Functions to the signal  that when multiplied together result in the data from which the final photograph is produced.  Each step of the way can at best maintain the original Information Quality (IQ) but in most cases it will degrade it somewhat.

IQ: Only as Good as at Photosites’ Output

This point is key: in a well designed imaging system** the final image IQ is only as good as the scene information collected at the sensor’s photosites, independently of how this information is stored in the working data along the processing chain, on its way to being transformed into a pleasing photograph.  As long as scene information is properly encoded by the system early on, before being written to the raw file – and information transfer is maintained in the data throughout the imaging and processing chain – final photograph IQ will be virtually the same independently of how its data’s histogram looks along the way.

Continue reading Image Quality: Raising ISO vs Pushing in Conversion →

Information Science, IQ, Quantization, Sensors

How Many Bits to Fully Encode My Image

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My camera sports a 14 stop Engineering Dynamic Range.  What bit depth do I need to safely fully encode all of the captured tones from the scene with a linear sensor?  As we will see the answer is not 14 bits because that’s the eDR, but it’s not too far from that either – for other reasons, as information science will show us in this article.

When photographers talk about grayscale ‘tones’ they typically refer to the number of distinct gray levels present in a displayed image.  They don’t want to see distinct levels in a natural slow changing gradient like a dark sky: if it’s smooth they want to perceive it as smooth when looking at their photograph.  So they want to make sure that all possible tonal  information from the scene has been captured and stored in the raw data by their imaging system.

Continue reading How Many Bits to Fully Encode My Image →

Banding, Perceptual, Quantization

I See Banding in the Sky. Is my Camera Faulty?

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This is a recurring nightmare for a new photographer: they head out with their brand new state-of-the art digital camera, capture a set of images with a vast expanse of sky or smoothly changing background, come home, fire them up on their computer, play with a few sliders and … gasp! … there are visible bands (posterization, stairstepping, quantization) all over the smoothly changing gradient.  ‘Is my new camera broken?!’, they wonder in horror.

Relax, chances are very (very) good that the camera is fine.  I am going to show you in this post how to make sure that that is indeed the case and hone in on the real culprit(s). Continue reading I See Banding in the Sky. Is my Camera Faulty? →