1. Image enhancement algorithms: noise reduction
Anton Lytvynenko, Kyiv, Ukraine

2. Motivation
The best way to obtain good pictures is to shoot them with good quality. But:
Conditions of image shooting (or other ways to obtain the image) may be too bad to yield good picture at all;
Conditions of image shooting may require too profound equipment in order to yield good picture, which may be inappropriate due to:
High cost of the equipment;
Inability to deliver it, high risks of loss (e.g. heavy fragile lenses in mountains) or just absence when required;
Errare humanum est;
The picture of bad quality given to you may be unique and you may have no choice with somebody's restoration request or your own will.

3. General notes
The goal of image enhancement is not to create less noisy image, less blurry image etc. The goal is to create image perceived as more perfect by a human. E.g. Too severely denoised face looks „plastic“.
Image defects usually have some pattern. This makes the differentiation between defects and genuine image details easier. E.g. so called „hot pixels“ (broken matrix elements) have maximum channel value (#FF in 8-bit) in one or a few channels.
The information for image restoration is taken only from the image itself (so, no A0 posters from 100x100 picture!).
Anyway, some amount of genuine image details will always be lost after a restoration.

4. Notes about thermal noise
The main object of image denoising nowadays is thermal noise of the sensor;
May be considered as statistically distributed deviations from „genuine“ values of pixels.
It is coloured and ugly (monochrome noise looks better => sometimes only colour may be denoised);
When one uses cropped frame cameras, sensor noise may be noticeable from ISO 200 (on some cameras even with ISO 100).
Film grain is virtually no problem nowadays;
Any information that restricts noise pattern may be and should be used (e.g. for C-41 negative scans a blue channel is the most noisy).

5. The following FLOSS programs will be considered
GIMP (GPL3+);
Image Magick (Apache 2.0);
G'MIC (and it's predecessor GREYCStoration, CeCILL);
Krita (GPL2);
ufraw (GPL)

6. Blur + rescale Primitive and non-selective
Easy to implement in any image editor
Too many information is lost
Still may be useful if noise pattern is difficult to be understood (so more profound ways are not available)
Image Magick example:
newsize=`convert src.tif -format "%[fx:w/4]x%[fx:h/4]" info:`
convert src.tif -blur 4x4 -resize $newsize dst.tif

7. Dechroma
Severe ( px) blur of A and B (position in colour space) channel in LAB model.
The noise became monochrome.
White balance is shifted after the procedure (additional red shift is required);
May be useful if luminosity channel is preferred to leave untouched, e.g. if small noise amount is to be „hidden“ with no sharpness loss; or, on the contrary, on severe noise (if other methods are even worse).
Implemented in dechroma plugin for GIMP or manually in any editor that supports LAB colourspace.

8. GREYCStoration (anisotropic smoothing)
Based on partial differential equations;
Aimed to preserve image morphology;
May be used for image scaling (enlargement preserving morphology) or interpolation;
Leaves curled coloured spots or „plastic“ on severe noise;
Implemented in G'MIC (formerly GREYCStoration) as both separate application and GIMP plugin.
D. Tschumperlé. “Fast Anisotropic Smoothing of Multi-Valued Images using Curvature-Preserving PDE’s”, International Journal of Computer Vision, May 2006;
D. Tschumperlé, R. Deriche. “Vector-Valued Image Regularization with PDE’s : A Common Framework for Different Applications”. IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol 27, No 4, pp , April 2005.

9. Wavelet denoise Based on wavelet data processing;
Selects image elements with different characteristic sizes, suppressing some of them (noise reduction) and enhancing others (sharpening);
Deals finely with both slightly and severely noisy images; but on the latter it leaves spots (looking like face spots after usage of cosmetic powder);
Should be used on the earliest stages of processing if possible (e.g. better withing RAW conversion than after it);
Implemented in: GIMP, Krita, ufraw.

10. Example http://lvee.lan/dl/example.tif
Original image (1/1.6'' matrix, ISO 400);
Dechroma (Blur: A:50, B:50, red shift: 10);
GREYCStoration (Amplitude: 80, Method: Runge-Kutta, other: default);
Wavelet (Colourspace: LAB, threshold: (L, A, B): 0.75, 1.0, 1.0, softness: 0.0).

11. Notes about spots
Values of affected pixels don't depend on their „genuine“ values;
Origins:
dust and dirt (on film, sensors or lenses), scratches on film;
„hot“ and „broken“ pixels on sensors,
text or watermarks on the image etc.
General way to fix: find affected pixels and replace them by resynthesized ones (blurring the rest of the image or analyzing its morphology). The main task is to find affected pixels.

12. Critical values search
#FFFFFF, #FF0000 – is theoretically valid for image, but in fact shouldn't occur on real continuous tone image;
User sets white limit and black limit (e.g. #05-#FA), the pixel outside the boundary considered noisy.
Implemented: GIMP (despeckle plugin), ufraw and G'MIC („remove hot pixels“);
Krita and Image Magick – no ready implementation (but may be implemented manually).

13. Median filters
Based on the fact that noisy pixels significantly differ from their neighbors.
Each pixel is replaced by median value of some area around it;
The result is mixed with the original image;
Implemented: G'MIC, Fred's isonoise script over Image Magick.
Fred’s ImageMagick Scripts: Isonoise.

14. Specific methods
Dark frame in RAW processing (ufraw) – „darkness“ is shot and fed to RAW converter in order to exclude broken pixels;
Infrared scan in colour film scanning (various scanning software, mostly proprietary) – dust and scratches on C- 41 negatives and E-6 reversal films are recognized by additional scan with infrared irradiation;
Stamp and healing brush.

15. Conclusions
Free software delivers wide range of powerful theoretically grounded algorithms of image denoising, providing a big choice of tools for enhancement of images with insufficient quality.