I'm trying to figure out the math in a python program im creating that can simulate mixing paint in real life, but I cannot find any info about any CMYK equations on the internet.
Here's an example of what I need to do...
If the CMYK value of Paint Bucket A is (C=10, M=20, Y= 60, k=10)
and the CMYK value of Paint Bucket B is (C=30, m=60, y=100, k=15)
and these two paint buckets were mixed together, what would the CMYK value be of the resulting bucket, and how could I figure that out mathematically?
Also, say Bucket A has twice as much paint as Bucket B, how would that change the equation? And also, what if say three different buckets were mixed together?
First I thought averaging each value together would work. For example if I mixed
(0,100,0,0)
with
(100,0,0,0)
and that would give me
(50,50,0,0)
But then I thought "basically that's mixing pure cyan with pure magenta and it should create"
(100,100,0,0)
So I guess the reason it's complicated for me is because the percentages of each color don't have to add up to 100.
Any ideas or hints?
Related
I have a background on mathematics and Machine Learning, but I'm quite new on image compression. The other way I was thinking in the optimal way to compress an image just using a lookup table. This means, given an original image which has N unique values, change it to a new image with M unique values being M<N. Given a fixed value of M, my question was how to pick those values. I realized that if we take as figure of merit the total error (MSE) of all the pixels, all the information has to be in the histogram of the pixel intensities. Somehow, the most common values should be mapped to a closer value than the uncommon values, making the higher regions of the histogram more "dense" in the new values that the low regions.Hence I was wondering if it exists a mathematical formula that:
-Given the histogram h(x) of all the pixels intensities
-Given the number of uniques new values M
Defines the set of new M values {X_new} that minimizes the total error.
I tried to define the loss function and take the derivative, but it appeared some argmax operations that I don't know how to derivate them. However, my intution tells me that it should exist a closed formula.....
Example:
Say we have an image with just 10 pixels, with values {1,1,1,1,2,2,2,2,3,3}. We initially have N=3
and we are asked to select the M=2 unique values that minimizes the error. It is clear, that we have to pick the 2 most common ones, so {X_new}={1,2} and the new image will be "compressed" as {1,1,1,1,2,2,2,2,2,2}. If we are asked to pick M=1, we will pick {X_new}=2 to minimize the error.
Thanks!
This is called color quantization or palettization. It is essentially a clustering problem, usually in the 3D RGB space. Each cluster becomes a single color in the downsampled image. The GIF and PNG image formats both support palettes.
There are many clustering algorithms out there, with a lot of research behind them. For this, I would first try k-means and DBSCAN.
Note that palettization would only be one part of an effective image compression approach. You would also want to take advantage of both the spatial correlation of pixels (often done with a 2-D spatial frequency analysis such as a discrete cosine transform or wavelet transform), as well as taking advantage of the lower resolution of the human eye in color discrimination as opposed to grayscale acuity.
Unless you want to embark on a few years of research to improve the state of the art, I recommend that you use existing image compression algorithms and formats.
I am playing around with rgl and I have created a 3D rendering of the mouse brain, in which structures can be isolated and coloured separately.
The original data is a 3D array containing evenly spaced voxels.
Every voxel is coded with a structure ID.
Every structure is rendered separately as a mesh by marching cubes, and smoothed using Laplacian smoothing as implemented by Rvcg.
Some of these structures can be quite small, and it would make sense to look at them within the context of the whole brain structure.
One of the options is to create a low-threshold mesh of the whole set of voxels, so that only the outer surface of the brain is included in the mesh.
This surface can be smoothed and represented using a low alpha in rgl::shade3d colouring faces. This however seems to be quite taxing for the viewport as it slows down rotation etc especially when alpha levels are quite low.
I was wondering if there is any way to implement some sort of cel shading in rgl, e.g. outlining in solid colours the alpha hull of the 2D projection to the viewport in real time.
In case my description was not clear, here's a photoshopped example of what I'd need.
Ideally I would not render the gray transparent shell, only the outline.
Cel shading example
Does anybody know how to do that without getting deep into OpenGL?
Rendering transparent surfaces is slow because OpenGL requires the triangles making them up to be sorted from back to front. The sort order changes as you rotate, so you'll be doing a lot of sorting.
I can't think of any fast way to render the outline you want. One thing that might work given that you are starting from evenly spaced voxels is to render the outside surface using front="points", back="points", size = 1. Doing this with the ?surface3d example gives this fake transparency:
If that's not transparent enough, you might be able to improve it by getting rid of lighting (lit = FALSE), plotting in a colour close to the background (color = "gray90"), or some other thing like that. Doing both of those gives this:
You may also be able to cull your data so the surface has fewer vertices.
I understand that domain or color wheel plotting is typical for complex functions.
Incredibly, I can't find a million + returns on a web search to easily allow me to reproduce some piece of art as this one in Wikipedia:
There is this online resource that reproduces plots with zeros in black - not bad at all... However, I'd like to ask for some simple annotated code in Octave to produce color plots of functions of complex numbers.
Here is an example:
I see here code to plot a complex function. However, it uses a different technique with the height representing the Re part of the image of the function, and the color representing the imaginary part:
Peter Kovesi has some fantastic color maps. He provides a MATLAB function, called colorcet, that we can use here to get the cyclic color map we need to represent the phase. Download this function before running the code below.
Let's start with creating a complex-valued test function f, where the magnitude increases from the center, and the phase is equal to the angle around the center. Much like the example you show:
% A test function
[xx,yy] = meshgrid(-128:128,-128:128);
z = xx + yy*1i;
f = z;
Next, we'll get its phase, convert it into an index into the colorcet C2 color map (which is cyclic), and finally reshape that back into the original function's shape. out here has 3 dimensions, the first two are the original dimensions, and the last one is RGB. imshow shows such a 3D matrix as a color image.
% Create a color image according to phase
cm = colorcet('C2');
phase = floor((angle(f) + pi) * ((size(cm,1)-1e-6) / (2*pi))) + 1;
out = cm(phase,:);
out = reshape(out,[size(f),3]);
The last part is to modulate the intensity of these colors using the magnitude of f. To make the discontinuities at powers of two, we take the base 2 logarithm, apply the modulo operation, and compute the power of two again. A simple multiplication with out decreases the intensity of the color where necessary:
% Compute the intensity, with discontinuities for |f|=2^n
magnitude = 0.5 * 2.^mod(log2(abs(f)),1);
out = out .* magnitude;
That last multiplication works in Octave and in the later versions of MATLAB. For older versions of MATLAB you need to use bsxfun instead:
out = bsxfun(#times,out,magnitude);
Finally, display using imshow:
% Display
imshow(out)
Note that the colors here are more muted than in your example. The colorcet color maps are perceptually uniform. That means that the same change in angle leads to the same perceptual change in color. In the example you posted, for example yellow is a very narrow, bright band. Such a band leads to false highlighting of certain features in the function, which might not be relevant at all. Perceptually uniform color maps are very important for proper interpretation of the data. Note also that this particular color map has easily-named colors (purple, blue, green, yellow) in the four cardinal directions. A purely real value is green (positive) or purple (negative), and a purely imaginary value is blue (positive) or yellow (negative).
There is also a great online tool made by Juan Carlos Ponce Campuzano for color wheel plotting.
In my experience it is much easier to use than the Octave solution. The downside is that you cannot use perceptually uniform coloring.
I use the R colorspace package to convert a three-dimensional point into a LAB color. The LAB color is defined with three coordinates, the first one ranges from 0 to 100 and the two other ones range from -100 to 100.
But searching with Google I do not find a cuboidal representation of the LAB color space. Why ?
Short answer
The LAB color space, a.k.a. gamut, contain colors that are impossible to reproduce in nature or on a screen (according to this page).
Elaboration on converting RGB to LAB
I guess the reason you ask is that you want to make some kind of printed material and want to be sure the colors turn out right. I am merely an enthusiastic amateur in this field, but think this paragraph from the wikipedia article on lab color space explains some of the complications.
There are no simple formulas for conversion between RGB or CMYK values
and L*a*b*, because the RGB and CMYK color models are device
dependent. The RGB or CMYK values first need to be transformed to a
specific absolute color space, such as sRGB or Adobe RGB. This
adjustment will be device dependent, but the resulting data from the
transform will be device independent, allowing data to be transformed
to the CIE 1931 color space and then transformed into L*a*b*.
That is, in order to create a lab color cube, you must first find the transformation from your monitor specific color space into absolute color space. This is surprisingly difficult since the mapping is not linear or on any other simple form. The transformation is not likely to be perfect either since the RGB and LAB spaces do not span the same subspace (speculating here). I once talked to a printmaker about this and he said altough the human eye only has 4 types of color receptors (RGB + light intensity) you need about 17 color components on generate the full spectrum of visible colors on paper. Both RGB and LAB compromises on that, optimized for different purposes.
Bottom line
You can calibrate your screen to set up the transformation needed to convert the RGB of the screen to the LAB colors of human eyes, and then go on to make a color cube. However, it will only apply to your very monitor and not be perfect. You are best off test printing different color profiles and choose the one you like best.
Because there is no such thing. The CIELAB colour space has a Cartesian representation (of infinite size), but the (finite) gamut that we can perceive is not cubic, it has a complicated shape. Varying the two coordinates a* and b* independently in a pre-defined range may seem convenient, but this is fundamentally not the way human perception works.
I'm writing a program that works with images and at some point I need to posterize the image. This means I need to bin the colors, but I'm having trouble deciding how to tell how close one color is to another.
Given a color in RGB, I can think of at least 2 ways to see how different they are:
|r1 - r2| + |g1 - g2| + |b1 - b2|
sqrt((r1 - r2)^2 + (g1 - g2)^2 + (b1 - b2)^2)
And if I move into HSV, I can think of other ways of doing it.
So I ask, ignoring speed, what is the best way to tell how similar two colors are? Best meaning most accurate to the human eye.
Well, if speed is not an issue, the most accurate way would be to take some sample images and apply the filter to them using various cutoff values for the distance (distance being determined by one of the equations on the Color_difference page that astander linked to, meaning you'd have to use one of those color spaces listed there with the calculations, then convert to sRGB or something [which also means that you'd need to convert the image into the other color space first if it's not in it to begin with]), and then have a large number of people examine the images to see what looks best to them, then go with the cutoff value for the images that the majority agrees looks best.
Basically, it's largely a matter of subjectiveness; in fact, it also depends on how stylized you want the images, and you might even want to add in some sort of control so that you can alter the cutoff distance on the fly.
If speed does become a bit of an issue and/or you want more simplicity, then just use your second choice for distance calculation (which is simply the CIE76 equation; just make sure to use the Lab* color space) with the cutoff being around 2 or 2.3.
What do you mean by "posterize the image"?
If you're trying to cluster the colors into bins, you should look at
cluster analysis
Just a comment if you are going to move to HSV (or similar spaces):
Diffing on H: difference between 0° and 359° is numerically big but perceptually is negligible.
H difference if V or S are small - is small.
For computer vision apps, more important not perceptual difference (used mostly by paint manufacturers) but are these colors belong to the same object/segment or not. Which means that we might partially ignore V, which can change from lighting conditions.