1. Denoising photographs using dark frames optimized by quadratic programming
Manuel Gomez-Rodriguez*
Jens Kober†
Bernhard Schölkopf†
†Max Planck Institute for Biological Cybernetics
Tübingen
*Electrical Engineering Department
Stanford University

2. Long exposure photographs
Long exposure photographs (e.g., astronomical photographs) contain substantial amounts of noise.
Dark current noise is the dominant source of noise in long exposure photographs.
We have access to samples of the joint distribution of the noise of our camera using bias frames and dark frames

3. Noise profile
A bias frame – raw image taken with closed shutter and exposure time ~ 0 seconds. The bias value is cased by the readout noise.
A dark frame – raw image taken with closed shutter and nonzero exposure time. It contains a bias frame plus a component that increases with exposure time, in a way that depends on several other factors (i.e. temperature, ISO setting, …)
A light frame – raw image to denoise.

4. The problem
Given the observed image I + D and a few points sampled from the multidimensional noise distribution, we want to estimate I. X1 X2 XN D I
Noise distribution

5. The problem
We want to include the joint statistics of the sensor noise in our denoising method
How should we combine the dark frames?
Does it generalize to different conditions?
Is the problem computationally tractable?
Given a noisy
long exposure setting
A library of dark
frames
Denoised image

6. Naïve approach
Single dark frame: record a dark frame of matching exposure time after each long exposure. This dark frame is subtracted from the light frame
It is implemented on commercial cameras
It doubles the amount of time
The temperature tends to change
One-point sample from the joint distribution of the noise

7. Average of dark frames approach
Average of dark frames: a set of dark frames under conditions matching the ones of the light frame. The mean of the set is substracted from the light frame.
Used, for example, in astrophotography
Works well for professional cooled CCDs with precise temperature control
Better estimate of the expected noise (multi-point sample)

8. Our approach
The distribution of noise for a given camera depends on various conditions, including temperature, ISO settings and exposure time.
If we knew the conditions for the image to be denoised, we should ideally use a library that matches the conditions of the image. But,
The exact temperature is usually unknown
We cannot store dark frames for every possible condition

9. Our approach
Our method generates a synthetic dark frame from the convex hull of the dark frames D(1)…D(N), taken under different conditions,
such that subtracting it from a noisy image optimizes a quality measure or prior for the class of images to denoise X1 X2 XN D I
Noise distribution
Image prior

10. Optimization problem
If the quality measure is the smoothness of the image (i.e. discrete derivative), the convex optimization problem can be formulated as,
where
is the variable, is a real convex cost function, is a set of evaluation points and is the 8-neighbor set of the location in the raw image

11. Quadratic programming problem
If a quadratic penalty function, , is chosen, the optimization problem is equivalent to the following quadratic program (QP)
where

12. Solution of the QP
A solution that generalizes well to the full image should be sparse because only the dark frames that were taking under similar conditions as the noisy image should be used for denoising; this is enforced by the constraints and , implying
= 1
Our method also allows to estimate in an indirect way the exposure time, temperature and ISO of a photograph

13. Evaluation points
As evaluation points, we use points that have high variance between dark frames and,
The selection of evaluation points is done only once for a specific camera and a relatively low number of evaluation points (~1000) is enough
The complexity does not depend on the size of the images but the # of dark frames
As the solution is usually sparse, we only need to load a few full dark frames to denoise

14. Evaluation
The same evaluation metric in the training set S and the test set T to numerically evaluate the performance; however S and T are disjoint → True generalization performance
Dark frames taken with a Canon EOS 1Ds with,
ISO of 800, 1000, 1250
Exposure times 1, 2, 4, 8,… 128 seconds, and 21 seconds
Various temperature conditions
have been used for the analysis

15. Evaluation
Three problem instances in increasing order of difficulty are proposed to validate our method,
Temperature
Exposure time
1st problem
Constant and matching the noisy image
Variable, including the same exposure time as the noisy image
2nd problem
Variable
3rd problem
Variable, not including the same exposure time as the noisy image

16. Evaluation: 1st case
Light frame with ISO 800, 16 seconds of exposure time
18 dark frames: constant temperature, variable exposure time
Correct exposure time!
Not used!

17. Evaluation: 2nd case
Light frame: ISO 1000, 16 seconds of exposure time
175 dark frames: variable temperature, variable exposure time
Not used!
Correct exposure time!

18. Evaluation: 3rd case
Light frame: ISO 1000, 21 seconds of exposure time
175 dark frames: variable temperature, variable exposure time (not inc. 21 sec)
200 evaluation points!

19. Noisy image
Our method
Horsehead nebula Barnard 33 in nebula IC 434, flame nebula NGC 2024, Canon EOS 5D with 300mm f/2.8 lens

20. Bilateral filter Our method
Horsehead nebula Barnard 33 in nebula IC 434, flame nebula NGC 2024, Canon EOS 5D with 300mm f/2.8 lens

21. Wavelet denoising Our method
Horsehead nebula Barnard 33 in nebula IC 434, flame nebula NGC 2024, Canon EOS 5D with 300mm f/2.8 lens

22. Our method + wavelet denoising
Horsehead nebula Barnard 33 in nebula IC 434, flame nebula NGC 2024, Canon EOS 5D with 300mm f/2.8 lens

23. Part of Orion constellation. Combination of ca
Part of Orion constellation. Combination of ca. 10 R, G, and   B images, denoised using the proposed method. Canon 200mm lens, SBIG CCD camera using Kodak KAF CCD chip

24. Magnified detailed (Running Man Nebula)

25. Conclusions
A relatively simple method with low complexity can help denoise long exposure images in raw format
Our method can beneficially be combined with image-based noise reduction methods
If available, our method could use evaluation points from the "optical black” (an area around the main image portion of the sensor which does not get light).
We believe that the proposed method can become a practical tool for digital photography

26. Thank You!