# The Gaussian First things first. You're going to see the Gaussian appear all over this course, and especially in the image processing part. You might as well learn it by heart from the get-go. The Gaussian in one dimension: $$ N(x,\sigma) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{x^2}{e^{2\sigma}}} $$ The Gaussian in two dimensions: $$ N(x,y,\sigma) = \frac{1}{\sigma^2 2\pi} e^{-\frac{x^2 + y^2}{e^{2\sigma}}} $$ We can see from this that the Gaussian is separable, yay! This means that we typically apply two one-dimensional gauss filter operations (one in the x direction and one in the y direction) instead of a two-dimensional gauss directly over the entire image. # Graphics ## Lab exercises The graphics lab exercises/assignments are pretty much lifted directly from [these OpenGL tutorials](http://www.opengl-tutorial.org/beginners-tutorials/). Tutorial 1 covers Lab 1, tutorial 2 covers lab 2 and tutorial 3-4 covers lab 3. You still have to figure out how to answer the questions yourself, but the code presented in the tutorials solve the tasks given in the mentioned labs. ## Rasterization Algorithms Rasterisation (or rasterization) is the task of taking an image described in a vector graphics format (shapes) and converting it into a raster image (pixels or dots) for output on a video display or printer, or for storage in a bitmap file format. ### Bresenham Line Algorithm ### Circle Rasterization Circles possess 8–way symmetry, so it sufficient to calculate one octant and derive the rest. #### Bresenham Circle Algorithm - The radius of the circle is r - The center of the circle is pixel (0, 1) - The algorithm starts with pixel (0, r) - It draws a circular arc in the second octant - Coordinate x is incremented at every step - If the value of the circle function becomes non-negative (pixel not inside the circle), y is decremented ### Point in Polygon Tests - Draw a half line from pixel p in any direction - Count the number of intersections of the line with the polygon P - If #intersections == odd number then p is inside P - Otherwise p is outside P ### Triangle Rasterization Algorithm ### Area Filling Algorithms ### Anti-aliasing Techniques #### Pre-filtering: - extract high frequencies before sampling - treat the pixel as a finite area - compute the % contribution of each primitive in the pixel area #### Post-filtering: - extract high frequencies after sampling - increase sampling frequency - results are averaged down #### Catmull’s Algorithm ### Line clipping #### Cohen – Sutherland (CS) Algorithm - Perform a low-cost test which decides if a line segment is entirely inside or entirely outside the clipping window - For each non-trivial line segment compute its intersection with one of the lines defined by the window boundary - Recursively apply the algorithm to both resultant line segments The low cost test can be done by assigning a 4-bit code to each of the nine sections around a pixel. +------+------+------+ | 1001 | 1000 | 1010 | +------+------+------+ | 0001 | 0000 | 0010 | +------+------+------+ | 0101 | 0100 | 0110 | +------+------+------+ Let the endpoints of a line segment be $c1$ and $c2$. Then we can do the following tests. - If $c1 \vee c2 = 0000$ - Then the line segment is entirely inside - If $c1 \vee c2 \neq 0000$ - Then the line segment is entirely outside #### Skala Algorithm #### Liang - Barsky (LB) Algorithm ### Polygon clipping #### Sutherland - Hodgman (SH) Algorithm #### Greiner - Hormann Algorithm ## 2D and 3D Coordinate Systems and Transformations A point in euclidian space can be defined as 3D vector. Linear transformation is achieved be post multiplying the point to a 3x3 matrix. ### Affine transformations Affine transformations are transformations which preserve important geometric properties of the objects being transformed There are four basic affine transformations: - Translation - Scaling - Rotation - Shearing

To understand what quaternions are, consider real numbers as the 1D number line and complex numbers as the 2D complex plane. Quaternions are "4D numbers". In this course we can use quaternions as 4D vectors with the axes $i, j, k, w$. If you have a $(x, y, z)$ point or vector you can plug these and $w=0$ into the equation $xi + yj + zk + w = 0$ for a quaternion representation. ### Rotation We can use quaternions to rotate points around a vector. For the calculation we need 1. $p$, the point that is to be rotated 2. $\vec{v}$, the vector that $p$ is to be rotated around. - Normalize $\vec{v}$ to the unit vector $\hat{v}$. 3. $\theta$, the arc of the rotation (how many "degrees" to rotate if you will). $$ q = \cos{\frac{\theta}{2}} + \hat{v} \sin{\frac{\theta}{2}} $$ $$ q^{-1} = \cos{\frac{\theta}{2}} - \hat{v} \sin{\frac{\theta}{2}} $$ The rotated point $p'$ is then $$ p' = q * p * q^{-1} $$ Note that quaternion multiplication is non-commutative. # Digital Image Processing ## Typical image processing steps 0. Image aquisition 1. Image enhancement 2. Image restoration 3. Morphological processing 4. Segmentation 5. Representation and description 6. Object recognition ## The Human Eye The human eye has two types of receptors, cones and rods. A typical eye has 6-7 million cones, each connected to a dedicated nerve end. Cones enable color vision. It also has 75-150 million rods, several connected to one nerve end. Rods allows log light sentisivity. ## Sampling and Quantization Sampling and quantization used when converting a stream of continuous data into digital form. Formalized: A continuous function $f(t)$ is to be sampled every $T$ steps of $t$ (which typically represents time). We say that $T$ is the *sampling interval*. The sampled function is then the sequence of values $f_n = f(nT), n \in \mathbb{N}$. Images are typically represented as digitized streams of two dimenstions, x and y. ### Sampling Theorem When a stream contains higher frequencies than the sampling frequency can handle, unwanted artifacts known as aliasing are produced. The Nyquist-Shannon sampling theorem formalizes this: $$f_s \geq 2 \cdot f_{max}$$ This implies that sampling should be performed with a frequency twice as large as the highest frequency that occurs in the signal to avoid aliasing. ## Image enhancement Image enhancement typically aims to do things like: noise removal, highlight interesting details, make the image more visually appealing. There are two main categories of techniques: spatial domain techniques, and transform domain techniques. ### Histograms Histograms of an image provides information about the distribution of intensity levels of an image. Both global (entire image) and local (parts of the image) histograms are useful. ### Spatial domain enhancement techniques Involves direct manipulation of pixels, with or without considering neighboring pixels. Spatial image enhancement techniques that do not consider a pixel's neighborhood are called intensity transformations or point processing operations. Intensity transformations change the value of each pixel based on its intensity alone. Examples include: image negatives, contrast stretching, gamma transform, thesholding/binarization. #### Neighborhood A neighborhood, informally, consists of the pixels close to a given pixel. Formally: $\delta(i,j) = \left\{(k,l) | 0 \lt (k-i)^2 + (l-j)^2 \le c; k, i \in \mathbb{N} \right\}$. #### Spatial Filtering A spatial filter exists of a neighborhood, associated weights for each pixel in the neigborhood, and a predefined operation on the weighted pixels. When the weights sum to 1, the gray value is not changed. ##### Smoothing We can make an averaging spatial filter to smooth an image. Consider an square 8-neighborhood, and the following weights: +-----+-----+-----+ | _1_ | _1_ | _1_ | | 9 | 9 | 9 | +-----+-----+-----+ | _1_ | _1_ | _1_ | | 9 | 9 | 9 | +-----+-----+-----+ | _1_ | _1_ | _1_ | | 9 | 9 | 9 | +-----+-----+-----+ This results in a smoother version of the image, which reduces noise. This averging filter is also known as the box-filter. The averaging spatial filter is a linear filter. An example of a popular non-linear smoothing filter is the median filter. The median filter sets a pixel to median of itself and its neighbors. #### Convolution Convolution is written as $(f*g)(t)$ meaning the function $f$ convolutes $g$ with variable $x$. Convolution can be understood as one wave, or function, travelling through another as shown in the illustration below.  The resulting value (the convolution of f through g) is the area covered by both graphs. Now look at the image and notice that - The convolution $(f*g)(t)$ is zero when the waves are not overlapping. - The convolution $(f*g)(t)$ grows when f moves through g. - The convolution $(f*g)(t)$ is at is maximum when the intersection of the waves is biggest. Remember that we are looking for the area, unlike when colliding water waves where two large wave tops will give us a twice the height. #### Sharpening We use Laplace for this. TODO: write about this. ### Transform domain enhancement techniques Involves transforming the image into a different representation. Examples of transforms include fourier transforms and wavelet transforms. #### Frequency domain Filtering can be done in the frequency domain. We use the discrete fourier transform to enable this. The Discrete Fourier-Transform (DFT) is defined as: $$ F(u) = \sum\limits_{i=1}^{M-1} f(x) e^{-i 2 \pi \frac{ux}{M}}, f(x): x \in [0, M-1] $$ The DFT is reversible, and the Inverse DFT (IDFT) looks like this: $$ F(x) = \frac{1}{M} \sum\limits_{i=1}^{M-1} F(u) e^{+i 2 \pi \frac{ux}{M}} $$ Working on single pixels Neighbourhoods of pixels Filters