Can I use PCA effectively on a greyscale image?

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Neo 2015-12-18

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回答： Aya Ahmed 2020-4-5

采纳的回答： Walter Roberson

Hello!

I found this code online:

I = double(imread('peppers.png'));
X = reshape(I,size(I,1)*size(I,2),3);
coeff = pca(X);
Itransformed = X*coeff;
Ipc1 = reshape(Itransformed(:,1),size(I,1),size(I,2));
Ipc2 = reshape(Itransformed(:,2),size(I,1),size(I,2));
Ipc3 = reshape(Itransformed(:,3),size(I,1),size(I,2));
figure, imshow(Ipc1,[]);
figure, imshow(Ipc2,[]);
figure, imshow(Ipc3,[]);

provided from another commentary form I was reading and I was wondering if there was any code that performed PCA that did not give the pc as a color channel. I don't want to do PCA on the colors of the image composite I want something else (not sure what else is but something not color), so applying PCA to a gray scale image.

Thanks for you any and all suggestions.

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Image Analyst 2015-12-21

If you think of a PCA coordinate system as a 3-D rotation of an original system, then I think that might help you. Imagine you have a cigar shape gamut (scatterplot of points in RGB color space). If the cigar didn't align with any of the R, G, or B axes but was at some crazy angle, then you could do PCA on it to get a new rotated system where one PC axis does through the main axis of the cigar, and the two others are perpendicular to that and lie in the small cross-sectional plane of the cigar. But with only 1 axis, how can you think that you can rotate it?

What about my last question where I said "better yet, just show your real image and tell us what you want to measure in it and we'll tell you if PCA is the best approach or if there is a better approach." Can you do that, or are you just trying to get a theoretical intuitive understanding of what PCA is and can do?

Neo 2015-12-21

编辑：Neo 2015-12-21

Oh I see, but just because an image is not RGB, it doesn't necessarily mean that it has only one axis, it might mean that the image has axes that are not related like in a RGB image. And let me get a picture for you to give an idea of what I am talking about actually, great suggestion. I read your response to quickly. Here ya go:

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Walter Roberson 2015-12-18

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Sure. For example,

I = double(imread('cameraman.tif'));
X = reshape(I,[],4);
coeff = pca(X);

This would correlate vertical quarters of the image.

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Image Analyst 2015-12-22

What the reshape does is to rearrange your image into a 4-column array. The leftmost column, column 1, is the left-most quarter of the image. For example, if you're using the cameraman image with 256 columns, then column 1 of X is columns 1-64 of the cameraman image, I, all strung together in one big long column vector. Then column 2 of X is columns 65-128 of I. And column 3 of X is columns 129 - 192 of I. And column 4 of X is columns 193-256 of I. So basically each column of X is a vertical quarter panel of the original image.

So, now that you understand what X in this example is, is there any reason to believe that there is any substantive, deterministic difference between any of the columns for your pattern image, or the cameraman image? Not really. Nothing that I would expect. So if you get 4 PCs from that, then the axes will probably be pretty much random. The PC1 may be somewhat correlated with how the mean brightness changes as you go from one vertical quarter-panel to the next, but who cares? What good is that? None really.

So you really need to figure out what you might expect to get out of doing PCA, rather than just do it and hope, without any basis, that what you get will actually mean something useful.

Image Analyst 2015-12-28

"PCA is only meaningful when the image being PCA'd is a RGB image" <-- I did not say that! I would not say that because it's not true. PCA can be used on features. The features can be individual R, G, or B color channels, OR they can be individual gray scale images (like a set of faces images), or they can be a feature vector of a bunch of measurements (like mean, circularity, solidity, and mean intensity, or whatever you want).

IF the features are color channels, then the main PC will lie along the major axis of the color gamut, and the two other PCs will lie perpendicular to them. It's like a rotation of coordinate system. See this visualization from the side:

and this visualization of the gamut from the top down:

The first PC would be along the V (vertical axis) and the other two would be radially extending out from there, one going out at 15 degrees and one going out at 65 degrees. So you have a rotated axis system where you could represent any RGB or HSV value in coordinates of that new rotated coordinate system instead of the original RGB coordinate system.

Doing PCA will give you as many PCs as features. If you're doing PCA on a set of 1000 face images, you have 1000 features because each grayscale face image is a feature. So you'd have 1000 PCs. Each PC image is called an eigenface in face recognition parlance. You could then reconstruct virtually any face in the database by taking 10 or 20 eigenface images. The more PCs you include in your reconstruction, the more accurate you're building the face. Of course using all 1000 would rebuild the face exactly to the exact gray level for every pixel.

Image Analyst 2015-12-29

By the way, here is one of the "average faces" of females constructed from eigenfaces. It's often asserted that the average face is the most attractive. However high fashion models that walk the catwalks seldom have faces that look like "average"

Neo 2015-12-29

Haha, thanks Analyst. But I am more concerned with how I can feed multiple images into the PCA code so that I can get more than one PC from the image. Do you have an average face answer to that? Cheers, Neo Cornel

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mugahid albadawy 2017-2-9

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i ve already used the same function for dicom image but it's not working properly

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Stefan Karlsson 2015-12-27

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I think the answer by Walter, while technically correct, confounds a VERY simple topic. Being technically correct here amounts to what exactly?

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Stefan Karlsson 2015-12-28

1. A single RGB image, of W x H size. Is there a way to perform PCA in a meaningful way?

answer: yes. the first PC-basis is usually the Brightness component in natural images. Of course, you can create images where this is not true, by forcing all pixel values to have the same brightness you will make the brightness component be the last PC instead of the first.

2. A single gray scale image, of W x H size. Is there any way to do PCA in a meaningful way?

Answer: no.

3. Performing multiple feature extractions, over local regions of a gray-scale image I can gather a "feature image". Can i do PCA on this image in a meaningful wau?

answer: yes. The easist example is if you gather 3 features over each local region. The resulting feature image can then even be visualized as an RGB image. On this image you can do PCA. As example, you can collect the TRIPLET: (mean, median, variance). While these feature are independent, you would expect a high correlation between mean and median.

Do not confuse this with the kind of PCA that is performed on databases of images. In that case, each image is an observable (what Image Analyst called a "feature" above). If you have 100 gray scale images, you can do PCA on the collection of those images as a whole. In this case, the dimensionality is equal to the number of pixels in the images, and the images must be normalized to the same number of pixels(by resizing usually).

Walter Roberson 2015-12-28

I have to disagree on the "Answer: no" of #2. As you explored in #3, feature extraction can be done over the single image and the resulting features can be PCA'd: that is a "meaningful" way of doing PCA on a single grayscale image. The features can be extracted in multiple ways, including some ways that come down to geometric subsets. It would be uncommon that vertical quarters would have useful information but finer grained localities certainly might.

For example, if an image happens to be a multiple of 8 pixels horizontally and vertically, then break down the image into 8 x 8 sub-blocks and reshape() that into N x 64 and pca() that. The information so obtained about how the pixels in the block tend to relate to each other can be used for image compression.

More generally, breaking into P x Q blocks, permute and reshape to N x (P*Q), PCA, and the information you get out can be interpreted as texture information; you can probably vary P and Q over a range to get an idea of what the texture size is. (Though perhaps an FFT approach might be easier for that, I am not sure.)

pca() of an entire grayscale image without reshaping or extraction could potentially be meaningful for detecting texture.

Image Analyst 2015-12-28

But Walter, that would be creating multiple features from an image, like measurements, or chunks of the image. And we agree with that - if you derive the right information from a single image, it can mean something. But if you don't, it's meaningless.

But I think Stefan was referring to doing PCA on just a single gray level image where the only feature is the gray levels - in that case you'd have a single PC and that is the gray level itself. If you divide the image up into meaningful regions like strips of 8 pixels wide, and that corresponds to some pattern in the image, then it might mean something. But if you're not smart about it and just arbitrarily divide, say, the cameraman demo image up into vertical strips 16 pixels wide, or the moon demo image up into quadrants, or the cell microscope image up into a checkerboard, then the PC of those things would probably not be meaningful -- you'd get something but it might not be interesting or useful to you. To be meaningful you have to create meaningful features from your input image in the first place. I know you'd agree with that.

Stefan Karlsson 2015-12-28

well.... I guess I will have to apologize for my inacuracies. You will get something useful from PCA over a single gray scale image:

you will get the global variance of the image, but that is the only thing. I do not disagree with anything that Walter says, it is technically correct.

Question: What is a Jumbo Jet made up of? I answer: mechanical parts, engines, fuel. While Walter answers by reference to the elements of the periodic table. What is the "right" answer depends only on the context the original question was posed...

(scale selection!) With that I leave this short but fun exchange. Maybe we should quickly note that PCA is actually not the best tool for what the question was about. I would guess Neo should use non-negative matrix factorization, or some other similar flavor. For photographic images, negative values are not allowed in neither basis nor coefficients....

sorry if i was harsh in my original reply.

Image Analyst 2015-12-28

Nah, you were fine. But is a little frustrating waiting for Neo to verbalize what features he wants to characterize in his image. It's almost like he heard about PCA and thought that it sounded cool and wanted to apply it to his image without considering if it was appropriate or not, or whether there might be better methods. I still don't know what kind of result he would want.

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Stefan Karlsson 2015-12-28

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... I guess one can also give another piece of advice to anyone who stumbles onto this looking for info on PCA. Read the posts by Image Analyst. They are as high in quality as they usually are.