the mathematics of digital photography

Research Matters

February 25, 2009

Nick HighamDirector of Research

School of Mathematics

The Mathematics ofDigital Photography

Professor Nick HighamDirector of Research

School of Mathematics

nick.higham@manchester.ac.ukhttp://www.manchester.ac.uk/~higham/

What is Color?

Human perception; depends on light source.

Retina has 3 types of cones⇒ trichromatic theory.

Why does yellow appear so bright?

Nick Higham Digital Photography 2 / 46

What is Color?

Human perception; depends on light source.Retina has 3 types of cones⇒ trichromatic theory.

What is Color?

Human perception; depends on light source.Retina has 3 types of cones⇒ trichromatic theory.

Colour Blindness

John Dalton (1766–1844).Described his own c.b. inlecture to M/cr Lit & PhilSoc, 1794.He was a deuteranope.

Vector Space Model of ColourModel responses of the 3 cones as

∫ λmax

si(λ)f (λ)dλ, i = 1 : 3,

where f = spectral distrib. of light, si = sensitivity of i thcone, [λmin, λmax] = wavelengths of visible spectrum.

Discretizing gives

c = ST f , c ∈ R3, S ∈ Rn×3, f ∈ Rn.

For standardized S, c is the tristimulus vector.

Vector Space Model of ColourModel responses of the 3 cones as

∫ λmax

si(λ)f (λ)dλ, i = 1 : 3,

where f = spectral distrib. of light, si = sensitivity of i thcone, [λmin, λmax] = wavelengths of visible spectrum.Discretizing gives

c = ST f , c ∈ R3, S ∈ Rn×3, f ∈ Rn.

For standardized S, c is the tristimulus vector.

StandardizationCommission Internationale de l’Éclairage (CIE) definedstandard colour matching functions si(λ) (1931, 1964).CIE RGB space.CIE XYZ space: nonnegative si(λ), Y corresponds toperceived brightness.

Projective Transformation

CIE Chromacity Coordinates

X + Y + Z, y =

YX + Y + Z

(z = 1− x − y).

(x , y) chromacity diagram:

Perceptual Uniformity: LAB Space

XYZ and RGB far from perceptually uniform.Search for (non)linear transformations that give moreuniform colour spaces.CIE L*a*b* (LAB, 1976) is more uniform:L = lightness, A = green–magenta, B = blue–yellow.LAB supported by Adobe Photoshop, MATLAB ImageProcessing Toolbox.

Dan Margulis on LAB (2006)

Printers use subtractive colour model: dyes absorb powerfrom spectrum. To produce wide range of colours need cyan,yellow, magenta primaries.

But C + M + Y = K = black : why do we need K?

Printing 3 layers makes the paper very wet.Black as 3 layers requires accurate registration.C + M + Y will not give a true, deep black due toimperfections.Coloured ink is more expensive.

Printers use subtractive colour model: dyes absorb powerfrom spectrum. To produce wide range of colours need cyan,yellow, magenta primaries.

But C + M + Y = K = black : why do we need K?

Printing 3 layers makes the paper very wet.Black as 3 layers requires accurate registration.C + M + Y will not give a true, deep black due toimperfections.Coloured ink is more expensive.

Bayer Filter

Sensor has 2green filters foreach red andblue.

Raw files are the unprocessed data off the sensor.

Demosaicing

Converts toRGB colourimage byinterpolation.

Compression

Original number string1 2 0 0 0 0 8 7 7 7 7 7 6 13

Lossless compression1 2 0 *4 8 7 *5 6 8

Lossy compression0 *6 7 *7 4

Jpeg is a lossy compression scheme.

Compressed RGB file.Filesizes reduced by orders of magnitude.Used by all digital cameras and imaging software.

tif (LZW) 12111 kjpg 12 1892 kjpg 8 917 kjpg 0 221 k

Jpeg 200× 200 px

Quality 8 Quality 0

Colour Space

Jpeg compression first converts from RGB to YCbCr colourspace where Y = luminance, Cb,Cr = blue, redchrominances, by Y

0.299 0.587 0.114−0.1687 −0.3313 0.5

0.5 −0.4187 −0.0813

.Vision has poor response to spatial detail in colouredareas of same luminance⇒ Cb, Cr can take greatercompression.Note:

∑row1 = 1,

∑row2 =

∑row3 = 0.

Discrete Cosine Transform

Algorithm breaks image into 8× 8 blocks.For each block luminance values expressed as linearcombination of cosine functions of increasing frequency

`x ,y =7∑

7∑j=0

fij cos((2x + 1)iπ

((2y + 1)iπ

where fij computed by a discrete cosine transform:

fij =7∑

7∑y=0

`x ,y cos((2x + 1)iπ

((2y + 1)iπ

Coefficients rounded, higher freqs de-emphasized.Same for Cb,Cr but more aggressive compression done.

Fingerprints—FBIDigitized at 500dpi⇒ 10Mb. Compression >∼ 10:1 req’d.Standardized on wavelet compression (1993).Jpeg: resonance of 8-pixel tiling w/ 500dpi scans, manyedges.Wavelets: gradual blurring as compression increased.

Panoramic Stitching

Nonlinear least squares, Levenberg–Marquardt:

(JT J + λD)d = JT r , J ∈ R3200×32 for 8 images.

Panoramic Stitching

Cloning/Healing

Photoshop blends the source into the target by solving thebiharmonic equation

∂4f∂x4 + 2

∂4f∂x2∂y2 +

∂4f∂y4 = 0.

Originally used in mapping, contouring (1950s).

Cloning/Healing

Photoshop blends the source into the target by solving thebiharmonic equation

∂4f∂x4 + 2

∂4f∂x2∂y2 +

∂4f∂y4 = 0.

Originally used in mapping, contouring (1950s).

Transformations to improve images

UoM turquoise is (L,A,B) ≈ (85,−12,−3).Convert to LAB then A← −A.

Now have (L,A,B) ≈ (85,12,−3).

Median

Variance

Summary

Mathematics is intrinsic to digital imaging: modelling theeye’s response to colour, colour spaces, capturingimages, storing and processing them.Modern developments in Photoshop, Lightroom, etc., relyon clever mathematical algorithms as well as exploitingfaster processors—and, increasingly, GPUs.Most of the relevant mathematics is covered in ourhonours degree programme.

Talk, including references, available athttp://www.maths.manchester.ac.uk/~higham/talks/digphot.pdf

Acknowledgements for Graphics

Wikipedia:http://en.wikipedia.org/wiki/Image:CIE1931_XYZCMF.pnghttp://upload.wikimedia.org/wikipedia/commons/b/b0/CIExy1931.pnghttp://en.wikipedia.org/wiki/Bayer_filter

http://www2.cmp.uea.ac.uk/Research/compvis/ColourIntro/ColourIntro.htm

Fraser [4].

References I

D. Austin.What is . . . JPEG?Notices Amer. Math. Soc., 55(2):226–229, 2008.

C. M. Brislawn.Fingerprints go digital.Notices Amer. Math. Soc., 42(11):1278–1283, 1995.

J. B. Cohen.Visual Color and Color Mixture: The Fundamental ColorSpace.University of Illinois Press, Urbana and Chicago, USA,2001.

References II

B. Fraser.Raw capture, linear gamma, and exposure.www.adobe.com/products/photoshop/pdfs/linear_gamma.pdf.

B. Fraser.Understanding digital raw capture.www.adobe.com/products/photoshop/pdfs/understanding_digitalrawcapture.pdf.

References III

N. J. Higham.Color spaces and digital imaging.In N. J. Higham, M. R. Dennis, P. Glendinning, P. A.Martin, F. Santosa, and J. Tanner, editors, The PrincetonCompanion to Applied Mathematics, pages 808–813.Princeton University Press, Princeton, NJ, USA, 2015.

A. R. Hill.How we see colour.In R. McDonald, editor, Colour Physics for Industry, pages211–281. Society of Dyers and Colourists, Bradford,England, 1987.

References IV

D. M. Hunt, K. S. Dulai, J. K. Bowmaker, and J. D. Mollon.The chemistry of John Dalton’s color blindness.Science, 267:984–988, 1995.

JPEG file interchange format, version 1.02.http://www.w3.org/Graphics/JPEG/jfif3.pdf.

D. Margulis.Photoshop LAB Color: The Canyon Conundrum andOther Adventures in the Most Powerful Colorspace.Peachpit Press, Berkeley, CA, USA, 2006.

C. Poynton.A guided tour of color space, 1997.www.poynton.com/PDFs/Guided_tour.pdf.

References V

G. Sharma and H. J. Trussell.Digital color imaging.IEEE Trans. Image Processing, 6(7):901–932, 1997.

S. Westland and C. Ripamonti.Computational Colour Science Using MATLAB.Wiley, New York, 2004.

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