Nibcode Solutions Blog

Systems of linear equations lie at the heart of linear algebra, and they are used to solve practical problems in many fields of study. We can find examples in biology, economics and electronics, to cite only a few examples, where the solution to several problems is reduced to solve a system of linear equations.

So, this week, the problem will be related to this topic, where you should prove you understand the basic concepts when finding the solution of a system of linear equations.

This problem of the week is about the inverse of a square matrix, where you should need to know some basic properties and how to calculate it.

Inverse matrices are useful for a variety of fields, such as:

• Cryptography, to decrypt a message that was encrypted by using an invertible matrix.
• Image processing, where several transformations can be expressed as matrix multiplication, and to reverse those transformation, you just need multiply by the inverse.
• Solving consistent independent systems of linear equations.

In this post, the last one of the series, we'll show an implementation in JavaScript of all the theory we have seen in the previous three posts. We also explain how you can use the HMTL5 canvas object to make your own implementation of the concepts of image processing.

The JavaScript image editor presented in the post, allows you to apply filters and make transformations to any image you choose. Besides, when moving the mouse cursor over the image, you can see, under the image, the 5x5 matrix of pixels surrounding the pixel at the cursor position.

In digital image processing, there is a group of transformations which receive the coordinates of a pixel as input, and return the new coordinates where the pixel should be placed; so that, when computing these transformations for all the pixels of an image, a new image is obtained.

Here, we are going to focus on one kind of these transformations, the affine transformations, which preserve collinearity (points and straight lines, planes, etc.). Affine transformations map points into new points by applying a linear combination of translation, rotation, scaling and skewing operations.

In the previous post we explained how an image can be represented as a matrix of pixels, where each pixel is expressed as a three-dimensional vector, composed by the amount of red, green and blue of the color. In this post, we are going to give some examples about the use of linear algebra in the digital image processing.

One kind of image processing are the filters. Among the most used there are: adjustment of brightness, contrast and colors, grayscale conversion, color inversion (negative), gamma correction, blur and noise reduction.