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. We will focus on the boolean lattice and the function lattice. ...

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. Then, we perceive a spectrum $S(\lambda) = I(\lambda) R(\lambda)$ as three values: ...

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. This is equivalent of looking for places where the second derivative is zero. ...

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 $\mathrm{Var}(Y) = N p (1 - p)$. The relative photonic noise is, however, ...

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 $I_c$. ...