Interpolation Our goal is to calculate the value of a pixel that is outside the grid of the image. To this end, we will interpolate the image. Depending on our hypothesis, we can arrive to different interpolation methods. Constant by parts: nearest neighbor; Continue: bi-linear; Polynomial of degree 3: bi-cubic; Limited band: Shannon interpolation. Method Initialization Operations per pixel Nearest neighbor $0$ $1$ Bi-linear $0$ $4$ Bi-cubic $4 N ^2$ $16$ Shannon $0$ $N^2$ We will call our discrete image $I_d$ and the continuous image after interpolation $I_c$. ...

Our goal is to characterize efficient estimators for $\theta$ in terms of mean squared error using the notion of Fisher information. Estimator Let $P_\theta$ be a probability distribution where $\theta \in \Theta \subset \mathbb{R}^d$, $d \in \mathbb{N}$. Definition: Estimator An estimator of $\theta$ is any statistic $\hat{\theta}$ taking values in $\Theta$. Bias We want $\hat{\theta}(X)$ to be close to $\theta$. Since the estimator is a random variable, we can calculate its expectation. ...

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: ...