Tag Archives: Gaussian

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

Photons, Shot Noise and Poisson Processes

May 17, 2023 Jack Leave a comment

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.

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 →

Fourier Optics, Geometrical Optics, MTF, PSF, wavefront

Fourier Optics and the Complex Pupil Function

April 13, 2023 Jack Leave a comment

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.

Continue reading Fourier Optics and the Complex Pupil Function →

IQ, Raspberry Pi, Sensors

Pi HQ Cam Sensor Performance

May 17, 2020 Jack 13 Comments

Now that we know how to open 12-bit raw files captured with the new Raspberry Pi High Quality Camera, we can learn a bit more about the capabilities of its 1/2.3″ Sony IMX477 sensor from a keen photographer’s perspective. The subject is a bit dry, so I will give you the summary upfront. These figures were obtained with my HQ module at room temperature and the raspistill – -raw (-r) command:

Raspberry Pi HQ Camera	raspistill --raw -ag 1	Comments
Black Level	256.3 DN	256.0 - 257.3 based on gain
White Level	4095	Constant throughout
Analog Gain	1	Gain Range 1 - 16
Read Noise	3 e-, gain 1 1.5 e-, gain 16	1.53 DN from black frame 11.50 DN
Clipping (FWC)	8180 e-	at base gain, 3400e-/um^2
Dynamic Range	11.15 stops 11.3 stops	SNR = 1 to Clipping Read Noise to Clipping
System Gain	0.47 DN/e-	at base analog gain
Star Eater Algorithm	Partly Defeatable	All channels - from base gain and from min shutter speed
Low Pass Filter	Yes	All channels - from base gain and from min shutter speed

Continue reading Pi HQ Cam Sensor Performance →

IQ, Sensors, Sharpness, Spatial Resolution

Canon’s High-Res Optical Low Pass Filter

January 15, 2020 Jack 7 Comments

Canon recently introduced its EOS-1D X Mark III Digital Single-Lens Reflex [Edit: and now also possibly the R5 Mirrorless ILC] touting a new and improved Anti-Aliasing filter, which they call a High-Res Gaussian Distribution LPF, claiming that

“This not only helps to suppress moiré and color distortion,
but also improves resolution.”

Figure 1. Artist’s rendition of new High-res Low Pass Filter, courtesy of Canon USA

In this article we will try to dissect the marketing speak and understand a bit better the theoretical implications of the new AA. For the abridged version, jump to the Conclusions at the bottom. In a picture:

Canon High-Res Anti-Aliasing filter — Figure 16: The less psychedelic, the better.

Continue reading Canon’s High-Res Optical Low Pass Filter →

Banding, IQ, Sensors

The HV Spectrogram

November 16, 2019 Jack 3 Comments

A spectrogram, also sometimes referred to as a periodogram, is a visual representation of the Power Spectrum of a signal. Power Spectrum answers the question “How much power is contained in the frequency components of the signal”. In digital photography a Power Spectrum can show the relative strength of repeating patterns in captures and whether processing has been applied.

In this article I will describe how you can construct a spectrogram and how to interpret it, using dark field raw images taken with the lens cap on as an example. This can tell us much about the performance of our imaging devices in the darkest shadows and how well tuned their sensors are there.

Pixel level noise spectrum — Figure 1. Horizontal and Vertical Spectrogram of noise captured in the raw data by a Nikon Z7 at base ISO with the lens cap on. The plot shows clear evidence of low-pass filtering in the blue CFA color plane and pattern noise repeating every 6 rows there and in one of the green ones.

Continue reading The HV Spectrogram →

Deconvolution, Sharpening, unsharp mask, USM

The Richardson-Lucy Algorithm

May 23, 2019 Jack 10 Comments

Deconvolution by the Richardson-Lucy algorithm is achieved by minimizing the convex loss function derived in the last article

(1) $\begin{equation*} J(O) = \sum \bigg (O**PSF - I\cdot ln(O**PSF) \bigg) \end{equation*}$

with

$J$ , the scalar quantity to minimize, function of ideal image $O(x,y)$
$I(x,y)$ , linear captured image intensity laid out in $M$ rows and $N$ columns, corrupted by Poisson noise and blurring by the $PSF$
$PSF(x,y)$ , the known two-dimensional Point Spread Function that should be deconvolved out of $I$
$O(x,y)$ , the output image resulting from deconvolution, ideally without shot noise and blurring introduced by the $PSF$
** two-dimensional convolution
$\cdot$ element-wise product
$ln$ , element-wise natural logarithm

In what follows indices $x$ and $y$ , from zero to $M$ -1 and $N$ -1 respectively, are dropped for readability. Articles about algorithms are by definition dry so continue at your own peril.

So, given captured raw image $I$ blurred by known function $PSF$ , how do we find the minimum value of $J$ yielding the deconvolved image $O$ that we are after?

Continue reading The Richardson-Lucy Algorithm →

Optics, Quantization, Sensors, Spatial Resolution

Wavefront to PSF to MTF: Physical Units

February 19, 2017 Jack 2 Comments

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]

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 →

Information Science, IQ, Quantization, Sensors

Sub Bit Signal

January 19, 2016 Jack Leave a comment

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 →

Deconvolution, IQ, Sharpening, Spatial Resolution

What Radius to Use for Deconvolution Capture Sharpening

August 29, 2014 Jack Leave a comment

The following approach will work if you know the spatial frequency at which a certain MTF relative energy level (e.g. MTF50) is achieved by your camera/lens combination as set up at the time that the capture was taken.

The process by which our hardware captures images and stores them in the raw data inevitably blurs detail information from the scene. Continue reading What Radius to Use for Deconvolution Capture Sharpening →

Strolls with my Dog