{"id":357,"date":"2010-12-30T15:34:54","date_gmt":"2010-12-30T20:34:54","guid":{"rendered":"https:\/\/blogs.mathworks.com\/steve\/2010\/12\/30\/a-and-b\/"},"modified":"2021-04-27T17:11:05","modified_gmt":"2021-04-27T21:11:05","slug":"a-and-b","status":"publish","type":"post","link":"https:\/\/blogs.mathworks.com\/steve\/2010\/12\/30\/a-and-b\/","title":{"rendered":"a* and b*"},"content":{"rendered":"
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Note added 27-Apr-2021:<\/strong> See the updated discussion of this topic in the 27-Apr-2021 post, \"Plotting a* and b* colors.\"<\/a><\/em><\/p>\r\n

Last week<\/a> I showed how to compute a two-dimensional histogram. Specifically, I computed the two-dimensional histogram of the a*-b*\r\n values of this image:\r\n <\/p>\r\n

<\/p>\r\n

This was the result:<\/p>\r\n

<\/p>\r\n

Blog reader Vanessa naturally wanted to know<\/a> what this all means. (And blog reader Cmac gave a pretty good answer<\/a>.)\r\n <\/p>\r\n

Let me elaborate a bit. By L*a*b* I refer to the CIE 1976 L*, a*, b* color space. Here's a description of the three dimensions\r\n (from the Wikipedia article<\/a>):\r\n <\/p>\r\n

\"The three coordinates of CIELAB represent the lightness of the color (L* = 0 yields black and L* = 100 indicates diffuse\r\n white; specular white may be higher), its position between red\/magenta and green (a*, negative values indicate green while\r\n positive values indicate magenta) and its position between yellow and blue (b*, negative values indicate blue and positive\r\n values indicate yellow). The asterisk after L, a and b are part of the full name, since they represent L*, a* and b*, to distinguish\r\n them from Hunter's L, a and b, [an earlier color space].\"\r\n <\/p>\r\n

Bright spots in the two-dimensional histogram indicate that a relatively large number of pixels in the original image have\r\n the corresponding a* and b* values.\r\n <\/p>\r\n

Consider, for example, the lowest bright spot on the two-dimensional histogram. By using impixelinfo<\/tt> and my mouse I was able to see that the a* and b* values for this spot are about -12 and 85. What color is that? To convert\r\n back to RGB values that we can display I have to pick some value for L*, whose natural range is 0 to 100. I'll pick 90, a\r\n fairly bright value. Then I use applycform<\/tt> to convert from L*a*b* to sRGB.\r\n <\/p>

lab = [90 -12 85];\r\nrgb = applycform(lab, makecform('lab2srgb'<\/span>))<\/pre>
\r\nrgb =\r\n\r\n    0.9297    0.9066    0.0971\r\n\r\n<\/pre>
How can we look at this color? There are a lot of different ways. One procedure is to display a 1-by-1 indexed image with\r\n      a one-color colormap.\r\n   <\/p>
image(1)\r\ncolormap(rgb)<\/pre> 
Looks like we found yellow! How about some of the other colors? The left-most bright spot is at about (-68,70). The right-most\r\n      spot is at about (60,70), and the top-most spot is at about (-4,-47).\r\n   <\/p>
lab = [90 -68 70; 90 60 70; 90 -4 -47];\r\nrgb = applycform(lab, makecform('lab2srgb'<\/span>))<\/pre>
\r\nrgb =\r\n\r\n    0.3737    1.0000    0.2844\r\n    1.0000    0.6832    0.3652\r\n    0.6637    0.9077    1.0000\r\n\r\n<\/pre>
image([1 2 3])\r\ncolormap(rgb)<\/pre> 
The two colors on the right don't look that much like any of our M&Ms. Maybe I guessed wrong about L*.<\/p>
lab(:,1) = 50<\/pre>
\r\nlab =\r\n\r\n    50   -68    70\r\n    50    60    70\r\n    50    -4   -47\r\n\r\n<\/pre>
rgb = applycform(lab, makecform('lab2srgb'<\/span>));\r\nimage([1 2 3])\r\ncolormap(rgb)<\/pre> 
That's better. We found our green, red, and blue M&M colors.<\/p>\r\n\r\n
Note added 27-Apr-2021:<\/strong> The following explanation fails to note that many of the colors in the a*-b* plane are actually out of gamut. See the updated discussion of this topic in the 27-Apr-2021 post, \"Plotting a* and b* colors.\"<\/a><\/em><\/p>\r\n\r\n   
Here's a crude way to visualize all the colors of the a*-b* plane. We make an \"image\" containing L*a*b* values and then convert\r\n      the whole thing to RGB.\r\n   <\/p>
[a,b] = meshgrid(-100:100);\r\nL = 90*ones(size(a));\r\nlab = cat(3, L, a, b);\r\nrgb = applycform(lab, makecform('lab2srgb'<\/span>));\r\nimshow(rgb, 'XData'<\/span>,[-100 100], 'YData'<\/span>, [-100 100])\r\nxlabel('a*'<\/span>)\r\nylabel('b*'<\/span>)\r\naxis on<\/span><\/pre> 
I will continue for a while longer to use this image as an excuse for a pseudorandom walk through some image processing computational\r\n      and visualization techniques.  We've looked at color-space conversion, two-dimensional histograms, and ways to show individual\r\n      color patches.\r\n   <\/p>\r\n   
I think that next time I'll start to look at segmentation based on color.<\/p>\r\n   
In the meantime, Happy New Year everyone!<\/p>