Fast Colour2GreyFast Colour2Grey

Ali Alsam and Mark S. Drew The Scientific Department School of Computing Science

The National Gallery London Simon Fraser Universityali.alsam@gmail.com mark@cs.sfu.ca

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What’s the colour2grey problem?• A black and white image has one channel.

• Ok, so make from RGB: use some form of Luminance – e.g., L*, Y(IQ), Y(UV), Y(CbCr),… or use PCA to find main direction

• What about a multi-spectral satellite image: 200 channels?

• Well, start with RGB colour images – how about averaging, say:

3.0

yx,B+yx,G+yx,R=yx,g

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Oops – equi-luminous images:

Very common situation, from graphics, so need more sophistication.

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The problem geometrically: say in L*a*b*:

Obviously, very different colours project to identical grey values.

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Many approaches:

• [Bala & Eschbach, CIC2004] -- a component related to edges of the chromatic 2-vector is added to the brightness: spatially adaptive.

• [Gooch et al., SIGGRAPH05] – Color2Gray: iterative optimization (with multiple local minima!) to map colours to grey maintaining chromatic plus luminance difference as well as possible.

• [Grundland & Dodgson, Patt.Rec. 2007] – a single achromatic channel to optimize chromatic contrast: a combination of luminance and a global “predominant” chromatic change

coordinate must set several parameters, need user input.

E.g.,

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Color2Gray is likely the most well-known. Again, needs several parameters to be set by the user:

-- so-called “aesthetic decisions”:• choose a colour direction for “cooler” to “hotter” colours• choose a window size for the optimization• choose a clamping value to clip the effect of a*,b* on L*

Algorithm is very slow (unconstrained optimization over thousands of variables)… Also,

Parameters make trouble:

This paper

Input

Gooch

Lum.

Uh-oh! For this image (© Jay Neitz) only colorblind people should see the “45”!

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Another example:Lum.

This paper

Gooch

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We could of course improve the Gooch et al. output by fixing the parameters manually…[and, some aspects of these algorithms are indeed compelling and could be added to the method presented here!]

But, can we devise a parameter-free method? (and also, let’s not forget higher-D colour than RGB)

Basic tenet of such algorithms is that we wish to map colour contrast into greyscale contrast.

But what is contrast ? --

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[Socolinsky & Wolff, IEEE Trans. Image Proc., 2002] – use math definition of what is gradient of an n-dimensional image:

the colour-channel gradient 2-vectors ▽R, ▽G, ▽B, combine to induce a metric for the underlying geometry. The resulting greyscale gradient is in the direction of maximal change.

Contrast:

Details:

[Di Zenzo, 1986]: Define a 2x2 structure tensor (first fundamental form, in math) --- main eigenvector points in the direction of maximal rate of change.

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What does this actually look like:

* ▽g based on strong mathematical guarantee on results *

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Details:

In the case of a 3-band color image = {R, G, B}, we consider the 2x3 array of color-channel gradient 2-vectors

▽ = {▽R, ▽G, ▽B} .

Now form the 2x2 array Structure Tensor Z:

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200

Let’s not forget satellite images…

200

200

Back to RGB:Z is the matrix of outer products of the 2D color-channel gradient components:

(2x3) * (3x2)

Z is real symmetric, so its eigenvectors form an orthogonal matrix V,

The (normalized) eigenvectors of Z point in the direction of minimum- and maximum-contrast, for an underlying grayscale image with metric induced by the structure tensor.

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Colour to Grey-Gradient:

Let V={u, v} with 2-vectors u v. Then the maximum-eigenvalue direction v is associated with maximum contrast in the grey image, with norm

Grey-Gradient to Grey:

Re-integrating gray-gradient -- take another derivative and solve Poisson’s equation:

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Done ?

Problem: eigenvectors are defined only up to sign:

±

If we get sign wrong, we have integrability error:

If we make a ± mistake,leads to bends, folds,halos!

Wolff solution: use sign of gradient of Luminance.

Problem: what if image is iso-luminant (common issue).

If we make a ± mistake,leads to bends, folds,halos!

Problem: Iso-luminant:

WolffRGB This paper

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Our solution: use max’s of ▽R, ▽G, ▽B, in each compass direction:

No

need

to c

onsi

der

± si

gns!

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Learn the weights ={,,,} by optimizing closeness of colourZ to resulting grey Z:

…

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Median weights ={0.530, 0.430, 0.537, 0.443};

Can we make faster?

Use Frankot-Chellappa algorithm in Fourier space to solve for grey g from gradient ▽g:

Takes a 2nd derivative, by multiplying by the transform of thederivative operator in the frequency domain, and solvesthe resulting Poisson equation by going back to the spatial domain. Uses ={1,0,1,0}.

Can use FFTW – fast!

and then solving for g:

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Results:Input Lum. Fast Colour2Grey

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…Results

Input Lum. This paper

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…Results

Input

Lum.

This paper

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…Results

Input

Lum.

This paper

Lum.

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…Results

Input Lum. Wolff This paper

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Thanks!To Natural Sciences and

Engineering Research Council of Canada