The Singular Value Decomposition has many very useful applications, including one image compression technique that I’ll discuss in this post. I chose the following image of a baboon to serve as a demo: (click any thumbnail to get the full 298×298 image)

What we’re going to do is experiment with various SVD compressions of this […]

Matrix factorization is a useful technique in process of image fusion. There are numerous applications of image fusion such as robot/machine vision, aerial military affairs, satellite imaging as well as medical imaging in magnetic resonance imaging (MRI) and positron emission tomography (PET). One of the techniques used for image processing is the Non-negative Matrix Factorization […]

As I googled for some sites about the approximation method of least squares, I noticed that this method is a very powerful tool in computer graphics.  I also came across a variation of the method called Moving Least Squares (MLS).  According to Wikipedia, moving least squares is a method of reconstructing continuous functions from a […]

There are many ways to compute the n-th Fibonacci number easily: there is a well-known close formula. Another approach is matrix multiplication.
Let’s take a look at this approach first.
As we all know, Fk+2 = Fk+1 + Fk. Thus, we can expand this equation to a matrix equation:
[1]
Now, it’s easy to see the following:
[2]
Now, […]

What can we do with ODEs? It turns out the ODEs are an integral part of the cloth simulation process.
For simulation purposes cloth is often modeled as a quad mesh whose vertices are connected by stiff springs. In general cloth resists stretching, this property is called extensibility. In order to simulate this, the springs must […]

Many of us have seen Fourier analysis before and may note that it can seem complicated at first glance, but it is comforting to realize that it is well suited to computational solutions. This is the purpose of Discrete Fourier Transforms (DFT), which allow us to reap the virtues of Fourier analysis without the […]

One application of matrices is in the field of encryption and decryption. An encryption and decryption can be modeled using linear algebra methods by using an encrypting matrix A, a message x and an encrypted message b. Using this method allows for easy decryption of the message by multiplying b by the inverse of A.
This […]

Least squares regression is a powerful tool however as we learned it does have its flaws. Mainly this method is generally used under the assumption of a normal distribution. As a result skewed outliers can greatly affect the regression and reduce its accuracy. Even just one large outlier could invalid a regression of hundreds of […]

Today in class we began to talk about the method of Least Squares, a technique used to approximate a function that satisfies some some (noisy) given data. The idea of the algorithm is relatively straightforward: find the best function that minimizes the sum of the squares of the error of each point, given […]

A somewhat dated, but still relevant article published in Nature magazine describes how non-negative matrix factorization (NMF) can be used in object, and specifically, facial recognition.  While many other methods of facial recognition perform comparison analysis on the face as a whole (using, say, the eigenfaces talked about in a previous blog entry), the non-negative […]