Cantors Diagonalization
this entry was created on 2008-06-06
In the theory of computation, there is the question of whether or not a person can compute all the programs in existence. Or, another example is, can you run a program with another program as its input and figure out if it would infinite loop. The answer to this question is that it is impossible, and it can be seen through the theory of diagonalization.
In the following table, I claim that all the programs in the world are listed as rows and also as columns.
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Bash Shell Script Image Thumbnail Creation
this entry was created on 2008-05-29
Creating thumbnails automatically
Here is a quick bash script that will automatically generate thumbnails in a series of images in multiple subdirectories...
#!/bin/bash
for img in `find . -name '*.jpg'`
do
convert -resize 15% $img $img._thumb.jpg
x=$(echo $img._thumb.jpg|sed 's/\.jpg.//g')
mv $img._thumb.jpg $x
done
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Linear Algebra part 2 - Inverses and LU
this entry was created on 2008-04-02
Matrix Inverses
As a quick review of the method of matrix multiplication, let's go over what matrices can be multiplied. If you have a matrix of size m x n, where m is the columns and n is the rows, you can multiply this by a matrix of size n x p. The resulting matrix will be of size m x p. The basic equation of matrix multiplication is
(AB)i,j = Σr=1 to nAi,nBn,j
Now to inverses..Not all matrices have inverses. If there is an inverse than,
A-1A = I = AA-1
Where I is equal to the identity matrix. If the matrix does not have an inverse than it is a singular matrix. If this is the case, than a vector other than the zero vector, will take the matrix Ax = 0. This means A is not inveritable. If there does exist an inverse, we can find it through Gauss-Jordan elimination. This is a similar process as the Gauss elimination. Let's go through this with a 2x2 matrix.
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Linear Algebra part 1 - Gauss Jordan Elimination
this entry was created on 2007-10-24
Here are my first set of linear algebra notes. The full videos can be found on MIT's Opencourseware site.
Geometry of Linear Equations
Take for example the equation
2x-y = 0
-x + 2y = 3
The matrix that would be created to represent this system would be,
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Image Rectification
this entry was created on 2007-10-09
Premise
Image rectification is the process of taking a perspective image and altering by ways of a perspective matrix to a rectangle or square. In these notes, I will outline the steps necessary to complete this procedure. For some of the mathematical equations and demonstration purposes, I will be using matlab.
The Image
The image shown above has the points (c,d,f,g) which have the pixel coordinates shown. We can use this information to get the points at infinity. As a rule, the cross product of points give the equations of the lines that connect them. Also the cross product of lines give the points of intersection. Thus we can say,
l1 = c x g and l2 = f x d
intersection of l1 and l2 = l1 x l2
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XMLHttpRequest
this entry was created on 2006-12-20
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