Last week I posted an introduction to logical indexing. This week I want to continue with a brief discussion of linear indexing and its connection to image processing in MATLAB.
Lets start with a small matrix. Breaking with tradition, I'll use a Hilbert matrix instead of a magic square:
A = hilb(5)
A =
1.0000 0.5000 0.3333 0.2500 0.2000
0.5000 0.3333 0.2500 0.2000 0.1667
0.3333 0.2500 0.2000 0.1667 0.1429
0.2500 0.2000 0.1667 0.1429 0.1250
0.2000 0.1667 0.1429 0.1250 0.1111
POP QUIZ: What does the MATLAB expression A(17) mean?
A(17)
ans =
0.2000
MATLAB interprets a single subscript as an index into a vector containing all the values of A in column order. So A(17) is the same as A(2, 4).
A(2, 4)
ans =
0.2000
This is called linear indexing.
So what's the connection to image processing? Suppose A is our image, and some previous computation has told us that we are interested in the pixel values at these row and column coordinates:
row = [2 1 5]; col = [1 3 4];
Can we extract the desired pixel values in a single expression? Let's try:
A(row, col)
ans =
0.5000 0.2500 0.2000
1.0000 0.3333 0.2500
0.2000 0.1429 0.1250
Well, we got nine values out, not three, so that certainly isn't what we are looking for. MATLAB computes A(row, col) as the submatrix of A formed by the intersections of the specified rows and columns.
Instead of indexing into A using row and column subscripts, we need to index using a single subscript.
Here's how to convert from row-column subscripts into linear indices:
M = size(A, 1); idx = M * (col - 1) + row
idx =
2 11 20
The expression A(idx) pulls out just the three values we want:
A(idx)
ans =
0.5000 0.3333 0.1250
Or we can assign to just those elements:
A(idx) = Inf
A =
1.0000 0.5000 Inf 0.2500 0.2000
Inf 0.3333 0.2500 0.2000 0.1667
0.3333 0.2500 0.2000 0.1667 0.1429
0.2500 0.2000 0.1667 0.1429 0.1250
0.2000 0.1667 0.1429 Inf 0.1111
The functions sub2ind ("subscript" to "index") and ind2sub convert back and forth between row-column subscripts and linear indices.
idx = sub2ind(size(A), row, col)
idx =
2 11 20
[r, c] = ind2sub(size(A), idx)
r =
2 1 5
c =
1 3 4
For a couple of detailed image processing examples illustrating linear indexing, see these previous posts:
Get
the MATLAB code
Published with MATLAB® 7.5
The “subplot” command seems to use a sort of linear indexing, but in Row order. Why?
Daisuke—I have no idea. That behavior was established probably 20 years ago. I’ve been at MathWorks at long time but not that long. Why does it matter to you?
I get confused between col-order (matrix) and row-order (subplot) sometimes. I thought it’d be cleaner (less confusing) if they are consistent, plus it’d be nice to be able to use something built-in like sub2ind to specify col & row of subplots, instead of N*(row-1)+col (and mess up). Not a big deal, though.
Linear indexing is great for dense matrices, but use it with caution for sparse matrices. It’s very slow (the problem is intrinsically difficult), and it breaks down if prod(size(A)) is greater than 2^31. It’s quite easy to work with sparse matrices that large, but not with linear indexing.
It would be really easy to implement a kind of indexing where you could refer to the kth nonzero element in the matrix (in column order). That is, the kth nonzero would be the kth entry in i, j, and x where [i j x] = find (A), not the A(row,col) index where k=M*(col-1)+row. That would be extremely fast, as compared with linear indexing based on row/column indices, but M doesn’t support that kind of indexing.
Tim—Thanks for the clarification. Almost all of my image processing work is with dense matrices.
Hello Steve,
I am trying to analyze a scanning electron microscope image which has many layers bounded very closely which makes the edges/boundaries of the layers very unclear (but can still tell by naked eyes). It also has some obvious cracks as well. I tried a couple of edge functions and they didn’t give me satisfactory results, either too much distaction on surrounding or no edges except for cracks. If you have hard time imagining the image, it just looks a page of lined notepad with the lines very much blended to the background color and with a few obvious crack spreaded on it. Is there a way that I can detect the edges accurately and clearly?
Thank you!
From your description, especially the “lines very much blended to the background color” part, it sounds challenging. If the lines are parallel, however, you might try computing a projection along the direction parallel to the lines. The line locations would show up as peaks in the projection.
They are not quite parallel but I will try that, thank you. And I just bought your book “Digital Image Processing using Matlab” to try to solve one of my recent projects. Which part of the book should I read to solve pattern tracking/detection/recognition problems? I basically have a light microscope image that has some fingerprint-like patterns that I would like to detect and mark. Or what methods would you recommend? Thank you!
Kevin—The later chapters are about image analysis and object recognition. There’s nothing in the book about tracking.
Hi, Steve:
As you are dealing with indexing, maybe my question fits in this topic: is there a fast way to do template matching in Matlab? I am thinking in something related to xcorr/xcorr2 or something related to it (as in the case of the summation of the absolute/squared difference).
Best regards from,
JanKees
JanKees—See normxcorr2.