The perception of an image’s content changes little when an increasing function is applied to it. However, if the function is non-increasing, then the perception of the content changes completely. We are sensible to local rather than global changes of contrast. From now on, we are denoting $u \colon \Omega \to \mathbb{R}$ our image, where $\vert \Omega \vert = M \times N$ is a discrete rectangular grid. Also, we are assuming that $u$ takes discrete values $y_0 < \dots < y_{n - 1}$ (usually $0$ to $255$). ...

Our goal in clustering is to group similar data points together. Each group will be called a cluster. Ideally, the intra-cluster distances are minimized and the inter-cluster distances are maximized. Note that this is an unsupervised model, so the following cannot be considered as clustering: Supervised classification; Simple segmentation; Results of a query; Graph partitioning. There are two types of clustering: Partitional clustering: divide data into non-overlapping subsets & each data is in exactly one subset; Hierarchical clustering: A set of nested clusters organized as a hierarchical tree. Types of clusters Well-separated cluster: any point in the cluster is closer to every other point in the cluster than to any point not in the cluster; Center-based cluster: An object in the cluster is closer to its center than to the center of other clusters. The center is usually the centroid or medoid (most representative point); Contiguous cluster: a point in the cluster is closer to one or more other points in the cluster than to any point not in the cluster; Density-based cluster: A cluster is a dense region of points, which is separated by low-density regions, from other regions of high density; Conceptual cluster: Clusters that share some common property or represent a particular concept. K-means clustering Input: A set $S$ of points in the euclidean space and an integer $k > 0$. Output: A parititonal clustering of $S$. ...

Let $A \in \mathbb{R}^{m \times n}$ and $\mathbf{b} \in \mathbb{R}^m$. Our goal is to solve the linear system $A\mathbf{x} = \mathbf{b}$ where $\mathbf{x} \in \mathbb{R}^n$. Info Some facts about matrices: If $A A^\top = A^\top A$, then $A$ is normal; If $A = A^\top$, then $A$ is symmetric; If $A A^\top = A^\top A = I_n$, then $A$ is orthogonal. Norms The following norms are used often: ...

Parametric model A statistical model is parametric if the probability distribution of $X$ belongs to some family of distributions indexed by some parameter $\theta$ of finite dimension. Definition: Parametric model A parametric model is a set of probability distributions $\mathcal{P} = \{P_\theta, \theta \in \Theta\}$ with $\Theta \subset \mathbb{R}^d$ for some finite dimension $d$. Our main goal is to use the observations $X_1, \dots, X_n$ to learn the value of $\theta$. Note that it is possible to do so only if each probability distribution $P_\theta \in \mathcal{P}$ is defined by a unique parameter $\theta$. ...

The pinhole model The first very simple way to acquire images is with the pinhole model. Here, part of the light coming from the object passes through a small aperture $O$ and is projected onto the focal plane. We do this so that each point of the object is represented by a ray. Otherwise, the image would not be formed. The distance $f$ is called the focal length. Let’s suppose we have the following model to describe the pinhole: ...