IM201 - Introduction to image processing

In mathematical morphology, the basic structure of an image is a complete lattice. Definition: Complete lattice A complete lattice is a set $K$ equipped with an order relation $\leq$ that satisfies: Reflexivity: $\forall x \in K$, $x \leq x$; Antisymmetry: $\forall x, y \in K$, $x \leq y$ and $y \leq x$ implies $x = y$; Transitivity: $\forall x, y, z \in K$, $x \leq y$ and $y \leq z$ implies $x \leq z$; $\forall x, y \in K$, the supremum and infimum of $x$ and $y$ exist and are denoted $x \lor y$ and $x \land y$ respectively....

Perception Color is the result of 3 things: An illuminant: source of light ($I(\lambda)$); An object: absorbs and reflects light ($R(\lambda)$); An observer: sensor ($S(\lambda) = I(\lambda) R(\lambda)$). Our visual system consists of rods (sensible, imprecise) and cones (not so sensible, very precise). There are three types of cones: S (short) cones: blue; M (medium) cones: green; L (large) cones: red; Let $s(\lambda)$, $m(\lambda)$ and $l(\lambda)$ be the spectral sensibilities of these cones....

Definition: Segmentation To segment an image is to divide it into homogeneous regions, considering one or many attributes (gray level, color, texture, etc.). We call borders the boundaries between these regions. Classical methods of border detection We try to use the local variations of the image to find borders: the 1st and 2nd derivatives. Since we are looking for discontinuities in the image, we need to find the points where the derivative is maximum....

There are inherit defects when an image is captured. Noise The noise is an error of measurement in each pixel. There are essentially 2 sources: Read noise (uniform); Photonic noise (depends on the luminosity): each photon has a probability $p$ to be measured ($p < 1$ to read only the color we want) in a Poisson distribution, then, $\mathbb{E}[Y] = N p (1 - p)$ and $\mathrm{Var}(Y) = N p (1 - p)$....

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 $I_c$....

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 $255$)....

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:...