Table of Contents 1. Motivation 2. Properties of the Legendre polynomials 3. Numerical integration via Gaussian Quadrature 4. Conclusion 5. References All the code used to generate the animations in this article is available at the following public repository: Auxiliary repository 1. Motivation There are many scenarios in which we need to compute the integral of a function over a given interval. For instance, in physics, we may want to calculate the work done by a force over a certain distance....
Table of Contents 1. Motivation 2. Orthogonalization via Gram-Schmidt 3. Legendre Polynomials 4. Conclusion 5. References All the code used to generate the animations in this article is available at the following public repository: Auxiliary repository 1. Motivation Orthogonal bases are a powerful tool in linear algebra. They allow us to decompose vectors into simpler components, and they are at the core of many numerical algorithms. For instance, imagine we are given an arbitrary function....
Table of Contents 1. Derivative: The slope of the tangent line 2. Integral: The area under the curve 3. Differentiating the area under the curve 4. Integrating the slope of the tangent line 5. Conclusion 6. References Back in high school, I remember being introduced to the key concepts in calculus. First, the derivative was presented as the slope of the tangent line to a curve at a given point....
Table of Contents 1. Data compression 2. Compressed sensing 2.1. Problem formulation 2.2. When is it supposed to work? 2.3. The Restricted Isometry Property (RIP) 3. Why does the L1- norm work? 4. Conclusion 5. References In this article we will focus on the topic of Compressed Sensing. We will start by motivating the interest in this recent field. Sparse signals are ubiquitous in nature, and the ability to recover them from a small number of measurements has a wide range of applications....
Table of Contents 1. Continuous Training 2. Data collection 2.1. Frontend 2.2. Backend 3. Training Pipeline 3.1. Training 3.2. Evaluation 3.3. Model Deployment 3.4. Proxy Deployment 4. Putting it all together 5. References In this series we will build from scratch a Continuous Training System. However, to keep things simple, we will use a toy example and run the system locally. In a real-world scenario, the system would be deployed in a cloud environment and the data would be stored in a distributed storage system....
Table of Contents 1. Introduction 2. Via geometric features 2.1. From points/lines 2.1.1. Problem formulation 2.1.2. Noise amplification: the horizon line 2.1.3. Solution: least squares estimator 2.2. From conics 2.3. From multiple features 3. Via a pair images 4. Via ML model 5. Summary 6. References 1. Introduction So far, we have figured out how to: Mathematically characterize the transform between the 3D world and a 2D image of it Map different types of objects between the two domains However, one might wonder: how do we actually compute the homography matrix that fully describes that transform in the first place?...
Table of Contents 1. Motivation: sports analytics 2. Points 3. Lines 4. Conics 4.1. Projection 4.2. Distortion: Objects behind the camera plane 5. Images 5.1. Projection via object decomposition 6. References 1. Motivation: sports analytics At this point, we know how to mathematically characterise the mapping between the 3D world and a 2D image capturing it. So it seems natural to wonder: what can we do with it? In this post, I will focus on a use case that I happen to be familiar with, but there are many others you can think of....
Table of Contents 1. Pinhole camera model 2. Intrinsic matrix 2.1. Setup 2.2. Homogeneous coordinates 2.3. Accounting for distortions 2.3.1. Digital images 2.3.2. Rephotographing Images 3. Extrinsic matrix 4. Homography matrix 5. References 1. Pinhole camera model When we capture something on camera, there is an interesting phenomenon going on: compression. We are taking a photograph of a 3D world, and capturing it in a 2D image. This 3D→2D space mapping inevitably leads to information loss....