# 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 The Bresenham line algorithm is used to determine which points in a raster should be plotted to form a close approximation to a straight line between two points. In the picture we have a line going from starting point (x0, y0) to end point (x1, y1). The grey squares are the pixels used for drawing the approximation of the line.  function BresenhamLineAlgorithm(x0, x1, y0, y1) int ∆x = x1 - x0 int ∆y = y1 - y0 double error = 0 double slope = abs(∆y/∆x) int y = y0 for x from x0 to x1 plot(x,y) error = error + slope if error ≥ 0.5 then y = y + 1 error = error - 1 In this example we choose the _slope_ to always be between 0 and 1. This means we always increment _x_, and sometimes increment _y_. We decide if we should increment _y_ by looking at the _error_. The _error_ is the distance between the actual point on the line and our current approximation. If the _error_ is greater than 0.5 we increment _y_ by 1, and reduce the _error_ by 1. Since the _slope_ is always between 0 and 1 in this algorithm it will only work in the first octant. To extend the algorithm to work in every octant we can do as follows: function switchToOctantZeroFrom(octant, x, y) switch(octant) case 0: return (x,y) case 1: return (y,x) case 2: return (-y, x) case 3: return (-x, y) case 4: return (-x, -y) case 5: return (-y, -x) case 6: return (y, -x) case 7: return (x, -y) Octants: \2|1/ 3\|/0 ---+--- 4/|\7 /5|6\ ### 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 ray 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

- 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 by 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 ### Transformation Matrices Quick note: a _hyperplane_ refers to a substance of one dimensionality less than its ambient space. I.e. in a 3-dimensional space, the $xy$-plane is a hyperplane. In a 2-dimensional space, the $x$ and $y$ axis are hyperplanes. #### Translation Translation defines movement by a certain distance in a certain direction. Translation of a point __p__ by a vector $\vec{\textbf{d}}$ results in a new point $\textbf{p'} = \textbf{p} + \vec{\textbf{d}}$. All we do is add the vector to the point. The translation transformation matrix is an instantiation of the general affine transformation, $\Phi(p) = A \cdot p + \vec{t}$ with $A = I$ and $\vec{t} = \vec{d}$. The 3D homogenous translation matrix is: $$ T(\vec{d}) = \begin{bmatrix} 1 & 0 & 0 & d_x \\ 0 & 1 & 0 & d_y \\ 0 & 0 & 1 & d_z \\ 0 & 0 & 0 & 1 \end{bmatrix} $$ The $n$-dimensional translation matrix is derived by instantiating the general affine transformation with $A = I(n)$. ($I(n)$ is the $n$-dimensional identity matrix) If you need the homogenous variant, use $I(n+1)$. The inverse translation of $T(\vec{d})$ is $T^{-1}(\vec{d}) = T(-\vec{d})$. #### Scaling We have $n$ scaling factors $s_{i}$, $i = 1, 2, 3, ..., n$ corresponding to the dimensionality of the space. If a scaling factor $s_{i} < 1$ the object's size is reduced in the respective dimension, whileas if $s_{i} > 1$ it is increased. $s_{i} = 1$ has no effect. $s_{i} = 0$ reduces the object's size in the respective dimension to zero. If $s_{i} = 0$, $\forall i$ the object disappears. _Mirroring_ about a major hyperplane can be done by using $s_{i} = -1$ as the factor for the hyperplane which isn't involved in the mirroring. I.e. to perform 2D mirroring about the $x$ axis, use $S(1, -1)$. To perform 3D mirroring about the $xy$-plane, use $S(1, 1, -1)$. The 3D homogenous scaling matrix is: $$ S(s_x, s_y, s_z) = \begin{bmatrix} s_x & 0 & 0 & 0 \\ 0 & s_y & 0 & 0 \\ 0 & 0 & s_z & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$ For 2D homogenous scaling, remove the third row and third column. For 3D non-homogenous scaling, remove the fourth row and fourth column. For 2D non-homogenous scaling, remove the third and fourth rows and columns. The inverse of the scaling matrix $S(s_x, s_y, s_z, ...)$ is $S^{-1}(s_x, s_y, s_z, ...) = S(\frac{1}{s_x}, \frac{1}{s_y}, \frac{1}{s_z}, ...)$ #### Rotation The book defines _positive rotation_ about an axis $a$ as one which is in the _counterclockwise_ direction when looking from the positive part of $a$ towards the origin. The three dimensional rotation transformation is defined as $R_{a}(\theta)$, where $a$ is the axis to rotate about and $\theta$ is the measure of how much you want to rotate it. The 3D homogenous rotation matrices are: $$ R_{x} (\theta) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & cos(\theta) & -sin(\theta) & 0 \\ 0 & sin(\theta) & cos(\theta) & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$ $$ R_{y} (\theta) = \begin{bmatrix} cos(\theta) & 0 & -sin(\theta) & 0 \\ 0 & 1 & 0 & 0 \\ -sin(\theta) & 0 & cos(\theta) & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$ $$ R_{z} (\theta) = \begin{bmatrix} cos(\theta) & -sin(\theta) & 0 & 0 \\ sin(\theta) & cos(\theta) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{bmatrix} $$ For the inverse 3D rotation transformation, use $-\theta$. In 2 dimensions we rotate about a point. The standard rotation transformations specify rotation about the origin. The 2D homogenous rotation matrix is: $$ R(\theta) = \begin{bmatrix} cos(\theta) & -sin(\theta) & 0 \\ sin(\theta) & cos(\theta) & 0 \\ 0 & 0 & 1 \end{bmatrix} $$ #### Shearing Shearing increases $n-1$ of an $n$-dimensional object's coordinates by an amount equal to the $n$th coordinate times a shearing factor (specified seperately for each $n-1$ dimensions). That is, $SH_{xy} (a, b) \cdot p$ increases $p_x$ by $p_z * a$ and $p_y$ by $p_z * b$. The 2D homogenous shearing matrices are: $$ SH_y (a) = \begin{bmatrix} 1 & a & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$ $$ SH_x (b) = \begin{bmatrix} 1 & 0 & 0 \\ b & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix} $$ The 3D homogenous shearing matrices are: $$ SH_{xy} (a, b) = \begin{bmatrix} 1 & 0 & a & 0 \\ 0 & 1 & b & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$ $$ SH_{xz} (a, b) = \begin{bmatrix} 1 & a & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & b & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$ $$ SH_{yz} (a, b) = \begin{bmatrix} 1 & 0 & 0 & 0 \\ a & 1 & 0 & 0 \\ b & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$ The inverse of a shear is obtained by negating the shear factors. ## Quaternions 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 logarithmic light sensitivity. ## 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 is moving through g, before reaching the edge of g. - The convolution $(f*g)(t)$ is at is maximum when the intersection of the waves is biggest. - The convolution $(f*g)(t)$ shrinks when the front of f has reached the end of g and has begun "leaving" it. 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