Of all the ways to decompose a numerical (floating point) matrix, my favorite is the singular value decomposition (SVD).  There are a lot of applications of the SVD (see my dissertation for one related to polynomial algebra) but my favorite ones are probably two applications related to image processing.

The first one I want to talk about comes from the cover of James Demmel's book "Applied Numerical Linear Algebra": image compression.  This example gives a really cool intuitive understanding of the Rank of Matrix and is also nice excuse to play with Maple's ImageTools package.

So, the first thing you need a test image. I used the classic image compression benchmark of a Mandrill.

Read this in with:

mandrill:=ImageTools:-Read("4.2.03.tiff");

The result is a 512x512x3 array.  In order to do something with this, we need to make it into a matrix so, call

manmat:=convert(ArrayTools:-Reshape(mandrill, 512*3, 512), Matrix);

Now we can compute a singular value decomposition of the image:

(U, S, V) := LinearAlgebra:-SingularValues(manmat, output = ['U', 'S', 'Vt']);

Now we can zero-out small singular values and multiply things back together to create low-rank approximations of the matrix that are also compressed versions of the image.
Rank 32 will give us 1/8 of the data (64 dimension 512 vectors: 32 rows of U, 32 columns of V, and the 32 corresponding singular values) but still a pretty good image:

rank32approx:=MatrixMatrixMultiply(`.`(U, DiagonalMatrix(S[1..32], 3*512, 512)), V, outputoptions = [order = C_order]);

This reshape it back to an image and display:

Preview((Reshape(rank32approx, 512, 512, 3)));

Taking things down to rank 8, is leaving only 1/32 of the data, but it is amazing how what is left resembles the original image:

rank8approx:=MatrixMatrixMultiply(`.`(U, DiagonalMatrix(S[1..8], 3*512, 512)), V, outputoptions = [order = C_order]);
Preview((Reshape(rank8approx, 512, 512, 3)));

To look at more images in order of descending rank, take a look at my worksheet:
Download 5480_SVD-face-colour-improved.mw
View file details

Next time: eigenfaces

A colleague showed this to me earlier this afternoon. I can explain, and accept, most of Maple's responses. I do have one case where I believe Maple could do better.

This arose during the creation of some Maple materials to support the derivation of the Integral Test for series convergence. Consider:

restart;
I1 := Int( 1/x^p, x=1..infinity );
                                /infinity      
                               |          1    
                               |          -- dx
                               |           p   
                              /1          x    

The time for Easter eggs will soon be here.

And using LibraryTools to browse and poke about in Maple's .mla files can show a few undocumented items.

Here's one below, that's an interesting part of a package.

I would like Maple to assist me with the following definite double integral:

int(int(x/(x^2+y^2+z^2)^(3/2), y = -b .. b), z = c .. a+c)

so far, I have failed.  Can anybody help?

Here is my worksheet:

View 4937_Page92.mw on MapleNet or Download 4937_Page92.mw
View file details

I'm asking Maple for simple things only, yet still (5 years later) I get nothing.

I'm creating 3 contour plots for potential functions in 3 bordering regions of 3-space,

because a continuous potential function isn't possible to define. 

After an entire worksheet worth of variable definitions and series expansions, i say simply:

CPA:=contourplot3d(VI(a,b,c,d,e)......)

CPB:=contourplot3d(VII(a,b,c,d,e)......)

CPC:=contourplot3d(VIII(a,b,c,d,e).......)

Does anyone here use this?

Would it be useful for someone new to topology/homological algebra learning wise ?

comments/opinions very welcomed.

thanks,

dc

http://wwwb.math.rwth-aachen.de/homalg/

Why is the following function such a problem to differentiate?

A := 1/(8*cos(`θ`)*(sin(`θ`))(sin(`θ`)+cos(`θ`))^2)

Here is a worksheet:

View 4937_page84.mw on MapleNet or Download 4937_page84.mw
View file details

I posed an integration problem and Maple solved it so easily I decided I would like to get Maple.  I'm waiting to hear whether Maplesoft will sell me the student version.  While I'm waiting, I would like to ask this group a question I've always had about systems like Maple.  I will start by posing a problem:

Let C be a unit circle and let P be a point on C.  Draw a bigger circle B with P as center which crosses C at two diametrically opposed points of C.  What is the area of the crescent-shaped region inside C and outside B?

Hello Everyone:

The e-mail notification feature that was previously available before the last update to the site has been re-enabled.

Please post any issues with this feature as comments on this blog entry. If you have comments about any other features, please add them to the MaplePrimes Suggestions forum

I used to receive emails each time when a comment/reply was made to my questions on mapleprimes. Very convenient. Now this service either stopped to exist or something else, as I receive nothing. Can it be resumed somehow? Thanks.

I am thinking about buying Maple, but I would like to know how it does with the following problem:

f(x):=log(2-cos(x)+sqrt(3+cos(x)*(cos(x)-4))) ;
r(n,p):=(1/%pi)*integrate((1-exp(-abs(n)*f(x))*cos(p*x))/sinh(f(x)),x,0,%pi) ;
r(1,3) ;

These are Maxima commands--let me know if it is not clear what they are doing.  I am wondering if someone with Maple could translate these commands into Maple-ese and try them. (Maxima just gives up on this integral.)

The following give very different answers.

> evalf( int( ln(2+2*cos(t)^2+cos(t)^4), t = 0 .. Pi));

   -8.710344354+19.73920881*I;

> evalf( Int( ln(2+2*cos(t)^2+cos(t)^4), t = 0 .. Pi));

Hello to All,

I am just in testing process of new version Mechanics of Materials 2.0 ToolBox.

Some of Maple pictures are so good that I decided to put here one from them.

What a great Maple...

Sorry for my some immodesty...

Hello,

I have a question about e-mail notification from Mapleprimes. Some month ago when somone answered my post it was generated an e-mail notificaion, but now it is not (I think).

I looked for notification e-mails in Spam bin as well, but it was empty.

Does Mapleprimes still create notifications?

Thank you very much.

Vladimir