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So today we will start talking about D squared, and single value decomposition. D squared is a very important concept
used many times in our engineering world. It was used for example by Gauss. In his calculation of the orbits,
of earth around the sun. In computer vision D square is used in many, many different cases.
What we'll study through a sequence of examples?First, we're going to take a detour. So, a computation known as singular value decomposition or SVD. It is one of the most useful tools you will encounter in our computation of vision problems in the next few lectures. Singular value decomposition, is a way of taking a matrix a, and decomposing to three separate components. So matrix A, could be of many forms, it comes from many different reasons, and physical meanings. The end results is we have a matrix a of size m, by n. Given any matrix of A, which you can think of as a transformation themselves. We can decompose this matrix, into three components U, D, and V transpose, meaning that U, multiplied by D, multiplied by V transpose will reconstruct the matrix exactly. And this is a very interesting fact. Like any matrix A can be decomposed in this following fashion.
And the U matrix that we have has a particular property. U matrix, can be thought of columns, and each columns with U, has a property that take a half part between themself equal to one, meaning that each column, the U, is a norm of one. Furthermore, if I take two columns U, and take that part out of them, the answer is zero, meaning the U's are constructed, such that all the columns, are perpendicular to each other. Second perpendicular to first, third perpendicular to both first, and second. Such you can think of the U's, essentially basis function, spanning the same space, as the columns of A.
Similarly, the draws of the V transpose, also had the following form that takes any draws of the V transpose will have perpendicular vectors. Or mathematically, one could write, U transposed times U equal to I and V transposed V equal to I. So that's, the first fact of
the singular value decomposition. That a matrix A can decompose into a set of column vectors U perpendicular to each other, a diagonal matrix D, and a set of rogue perpendicular vectors V transposed. And multiplying the three matrices, reconstruct the matrix A exactly. And we can always do this. In the math lab, this is a function called SVD.
The second, a fact of this decomposition is that the singular value matrix is diagonal forms. Meaning the elements are zero, and the non-zero elements sits, on the diagonal. And one can always sort it, from large to small. So sigma one, larger than sigma two, all the way bigger than zero. So all the values on diagonals are positive, or zero. We will see cases when the last element is zero, or several of the elements are zero. The U matrix is going to be the size of same, as A and by n. The diagonal matrix singular value matrix is size, m by m, and the V transposed matrix is size m by m.
We can think of the U matrix, as set of basis. For example, if A is made of curve, and since this space where every column of A is a curve, we can think of the U matrix enclose the basis function, where the first base is, for example, is a horizontal curve, and the second basis function is a curve going down. And the third basis encode a curve going up, and down. Then we can think of the matrix V together with the singular value D, provides a coefficient for every column A. So, the coefficient of the beta1, beta2, beta3. So through a scaling of a beta1, beta2, beta3, it provides a reconstruction in column of A, which is represented by a curve here. As such we can think of U is basis, the columns of V transpose provide us the addresses. Multiply to the scale factor, give us the beta1 to beta n.
So, the first factor we can think of is SVD, as transforming matrix A, into it's basis function, and the transformed addresses. So let's just think of this painting by Mondrian, what is the SVD of this painting? Well first of all, what is this painting? The painting in our world is in fact a matrix. Where rows, and columns are pixels. And Mondrian, paintings are very good illustration of this basis, because all the Mondrian paintings are made of rows, and columns. So let's take this Mondrian painting, and first convert a colored image to a gray skull image, shown on the top left. Right here use the colormap jet in the matlab illustration, where red indicated bright pixels, blue means dark pixels. So this is a Mondrian in the black, and white world, but illustrated in the colormap jet. And we were take SVD of this image, which is made of thousands rows, many thousand columns, we see that we can construct SVD of this matrix. And to reconstruct from it a U matrix. And here, I illustrated only the first four columns of the U to highlight what it looks like. So first note, the first columns are almost uniforming it's values. The second column we can see, has some positive value which indicated in orange, and some negative values indicated in blue. As such, you can see that this perpendicular to the first column. In the image, what it's capturing, it's saying that Mondrian paintings are made of two halves.On the top half, which are one color, and the bottom part of the image is another color. Move out to the third column of U, we see, again, is made of some value, positive mean indicated in red, blue indicating the negative values. Again, this vector is perpendicular, to the first two columns. And again if you look at this picture, what it is indicating is telling us the Mondrian paintings for this picture has several rows have the same color, hence we have a basis vector with uniform color in it. Now just take in the four columns of SVD of this picture I can reconstruct this picture by taking the first four columns of U. The first four by four matrix of a singular value matrix D, and the four rows of V. And reconstructing the matrix through the matrix multiplication is shown in the bottom right. As you can see, this is not exactly the same but it's a very close approximation than where we started. As such, we can think of SVD is a process of simplifying what's in the matrix, very large matrix and then using a set of basis functions, illustrating U and using a set of addresses in V.
If I were to take more rows and more columns view matrix but not all of them. For example, I only take the first 20 columns of the U matrix, I can reconstruct the Mondrian painting fairly closely. In the center at what I've shown here is the log of the singular values. As you can see, the log of a single values drop off pretty rapidly, indicating that many dimensions in the basis function has a scale factor near 0. And we're going to use this factor for many different cases. And we're going to use this for using SVD as a clean up process of cleaning up a matrix. Or we can use this factor to approximate a large dimensional matrix with a few basis functions. So now let's look at Gauss himself in rows and columns. So Gauss is not painted by Mondrian so he has many orientations edges horizontal, not lines with horizontal vertical directions, but nonetheless we can take Gauss' image as a matrix and reconstruct itself. So on the middle we have a Gauss image reconstructed with more basis functions and the right we see a Gauss image reconstruct with a few vectors. And this is shown in grayscale. As you can see, it still captured the structure of his face fairly reasonably with on a few basis functions. And this is because many rows and many columns of this image are redundant, they are not exactly the same but it can be represented by a few basis vectors.
So this is a look at a picture as a matrix where there are thousands rows and thousands of columns. One can also look at a picture as a vector by taking all of the rows, all of the columns and stacking them into one, long vector. So if I have a 1000 by 1000 image, that would be a vector of a million dimension long. This is illustrating here images not as a matrix, but image as a vector. As such that I can take many, many pictures, many, many pictures of people's face for example. Each one forms a vector and I can stack the vector together each one, 1 million direction long, 1 dimensional long, and then maybe thousand peoples, we have a million by thousand matrix. I can take a single value decomposition with that matrix that gives me a set of basic function U, and it reshape the use back to the image space. And this is what happens when we do that. On the left we have a set of images of the face. Each one stretch into vector space, placed column-wise next to each other. We computed SVD of this matrix, and we take the first few columns of U, and reshape back to the image space. And this is called eigenfaces, as you can see those eigenfaces capture different characteristics of the face. For example some capture the hair, some capture the mouth. And from those basis functions, one can re-express our faces as addresses in those spaces.
Here is a set of images captured by an artist for people in Berlin. On the left we have the first basis function for the man. On the right we have a first basis picture of the woman. Taking a view of eigenvectors as well as addresses, we can see here are three faces reconstructed by the same eigenvector or use different coefficients on this eigenvector. So, here we show different ethnic group it will show you by this eigenvector, address from one point address space to another point address space. Here we have a different eigenvector corresponding to the age changes. Again, as we change the address on this eigenvectors faces transform itself. So now we are seeing eigenvectors, singular value vectors, as our basis functions and as such that we can think of it as our basis functions and their addresses and together they reconstructed the matrix or in this case image for me. Once we have decomposed a matrix A into a singular value decomposition, U times D times V transpose, we can also understood the concept of rank and row space. A rank of matrix A is simply defined as the number of the non-zero singular values in the matrix D. So we call that r, rank of A equal to r. The number of elements non-zero on the diagonal. And r is going to be less than equal to the dimension of m or n. The null space is corresponding to exactly those elements associated with singular values equal to zero. So if the rank is r, then the columns in V or rows in V transpose associated with those dimensions where single value is 0, are the null space. So we take the columns, from r plus one all the way to n. If r is the rank, r plus one to n columns of V is the null space of A. And we have the following property that A Times the nullspace V, r plus one to n, equal to 0.