Visualisierung

Chapter 1

$size_of_effect = \frac{difference}{first_value}$

$lie_factor = \frac{size_of_effect_graph}{size_of_effect_data}$

Change blindness: significant changes are not noticed by the observer

goals: visual exploration, analysis and presentation

pipeline: data acquisition -> filtering -> visualization/mapping -> rendering/interaction

Chapter 2

Sensory memory: Preattentive processing, "instinctive" perception

Short-term memory: seconds to minutes, limited capacity

Long-term memory: days to years, unlimited capacity

Weber's law: linear increase in subjective perception with logarithmic increase in stimulus intensity

Distinguish 200 colors (H), 500 gradations (S), 20 variation (V)

Two stages of visual perception:

1. Preattentive: automatic, parallel processing of visual features
2. Attentive: serial processing, requires focus

Preatentive attributes: < 8 hues, <= 4 orientations, <= 4 sizes, <= 10 else

non-preatentive attributes: parallelism, juncture

Color: use high luminance contrast, avoid red-green combinations, only use max 5 colors

Chapter 3

Best to worst mapping: position -> length -> angle/slope -> area -> luminance/saturation -> volume/curvature

Steven's area judgment scale: $I = \alpha r^{1.4}$

Rule: map most important data to the most accurate visual attribute

Simple line Graphs: direct comparison, fine details BUT clutter, hard to distinguish time series

Stacked Graphs: time series destinguish BUT less vertivcal resolution, hard to compare time series

Small multiples: easy time series destinguish, easy to compare BUT less resolution per time, difficult to compare across time

Horiton Graphs: vertical resolution, quick min/max BUT steep learning curve, hard to compare

Quartils: $Q_1 = (n+1)/4$, $Q_2 = (n+1)/2$, $Q_3 = 3(n+1)/4$

Parallel Coordinates: crossing lines -> negative correlation, parallel lines -> positive correlation

Chapter 4

Derivative to a point $u$: derive $\frac{\delta f(x_0 + t\cdot u)}{\delta t}$

$f(x,y) = \exp(-0.2 \cdot (x^2 + y^2)) \Rightarrow f(x_0 + t\cdot u) = \exp(-0.2 \cdot ((x_0 + t\cdot u_x)^2 + (y_0 + t\cdot u_y)^2))$

On Image:

• forward derivative: $f'\rightarrow_x(x,y)=f(x,y) - f(x-1,y)$
• backward derivative: $f'\leftarrow_x(x,y)=f(x,y) - f(x+1,y)$

Gradient: $\nabla f(x,y) = \left(\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y}\right)$

Derivative to a point $u$: $D_u f = \nabla f\cdot u$

On Image:

• forward derivative: $\nabla f(x,y) = \left(f(x,y) - f(x-1,y), f(x,y) - f(x,y-1)\right)$

Hessian Matrix: First row is derivative to x then x y z, second row is derivative to y then x y z, third row is derivative to z then x y z

Type of critical point ($\lambda$ are eigenvalues of Hessian matrix):

• local minimum: $0 &lt; \lambda_1 \leq \lambda_2 \leq \lambda_3$
• saddle point: $\lambda_1 &lt; 0 &lt; \lambda_3$
• local maximum: $\lambda_1 \leq \lambda_2 \leq \lambda_3 &lt; 0$

Calculate eigenvalues:

• $A - \lambda E$
• $det(A - \lambda E) = 0$
• $det(M_{2x2}) = a_{11}a_{22} - a_{12}a_{21}$
• $det(M_{3x3}) = m_{11}m_{22}m_{33} + m_{12}m_{23}m_{31} + m_{13}m_{21}m_{32} - m_{13}m_{22}m_{31} - m_{12}m_{21}m_{33} - m_{11}m_{23}m_{32}$

Quadrangle Lemma: minimum, saddle, maximum, saddle

Jacobian Matrix: first row is derivative of u to x, y and z, second row is derivative of v to x, y and z and third row is derivative of w to x, y and z

Critical points:

• inflow behavior: $Re(\lambda_i) &lt; 0$
• outflow behavior: $Re(\lambda_i) &gt; 0$
• swirling behavior: $Im(\lambda_1) = -Im(\lambda_2) \neq 0$

Poincaré Index: counterclockwise rotation of a vector, number of counterclockwise rotation of that vector

Divergence: sum of the diagonal elements of the Jacobian matrix

Laplacian: zweite ableitung nach x + zweite ableitung nach y + zweite ableitung nach z

Estemated divergence: $\div f(x,y) = f_x(x,y) - f_x(x-1,y) + f_y(x,y) - f_y(x,y-1)$

Curl: $(w_y - v_z, u_z - w_x, v_x - u_y)$

Chapter 5

Data reconstruction: get continuous data from discrete data

• set constant boxes based on the data
• interpolate the data in the boxes

Radial Basis Function: $RBF(x) = \sum_{i=1}^n f_i \cdot \phi(||x - x_i||)$ with $\phi(x) = \exp(-x^2)$

Better: $RBF(x_i) = \sum_{i=1}^n w_i \cdot \phi(||x - x_i||) = f_i$, that is: solve equation system so that line goes through all points

• drawbacks: every sample point is used, so it is slow, new points need to be recalculated

One solution: $\phi(x) = \frac{1}{x^2} / \sum_{i=1}^N \frac{1}{||x - x_i||^2}$

New formula: $RBF(x) = \sum_{i=1}^n \frac{f_i}{||x_i - x||^2} / \sum_{i=1}^N \frac{1}{||x_i - x||^2}$

• still have to calculate all again for new points

Triangulation: connect points with triangles, then interpolate the data in the triangles

• avoid long, thin triangles -> use Delaunay triangulation
• make circle around triangle, if another triangle has line inside, flip the line

Line-interpolation: $f(y) = x \cdot p_1 + (1-x) \cdot p_2$ with $p_1$ and $p_2$ are the two points of the line

Triangle-interpolation: $f(x) = p_1 \cdot \frac{A_1}{A} + p_2 \cdot \frac{A_2}{A} + p_3 \cdot \frac{A_3}{A}$ with $A$ is the area of the triangle and $A_i$ is the area of the triangle with point $p_i$ and the two other points

Grid-interpolation:

• linear: $f(x,y)$ calculate 1D on x axis for value x, then use the result to calculate 1D on y axis for value y
• bilinear: f(x,y) calculate area of quadrangle $xy$, $(1-x)y$, $x(1-y)$, $(1-x)(1-y)$, then use the areas to calculate the value with values $f_{ij}$ of opposite corners

Chapter 6

Marching cubes: distinguish which way diagonal is cut by calculating middle value $\frac{sum_of_corners}{4}$

Connected Component Analysis: find connected components in a binary image and only keep the largest one

Smoothing:

• Iterative smoothing: average the points with their neighbors

• $x_i \leftarrow x_i + \lambda\sum_j \omega_{ij}(x_j - x_i)$ with $\omega_{ij}$ is the weight of point $j$ to point $i$ and $\lambda$ is the smoothing factor

• Combinatorial: $\omega_{ij} = 1$ if $i$ and $j$ are neighbors, else 0

• Laplace-Smoothing: $\omega_{ij} = \frac{1}{N(i)}$ -> loses volume

• Laplace-Smoothing + HC: Add correction term to Laplace-Smoothing to keep volume

• LowPass Filter: implementation of two laplace smoothing steps, with $\lambda_1 &gt; 0$ and $\lambda_2 &lt; 0$

• Mean value filter: average the normal vector of surfaces in the neighborhood

• Median filter: average the normal vector of surfaces in the neighborhood, but only use the median value

Distance aware smoothing: Smooth more the further away from a certain point -> use distance as scaling factor

Chapter 7

Maximum Intensity Projection: project the maximum value of a volume onto a 2D plane

• Interpolation: use linear interpolation to get the value from inside the voxel
• nearest neighbor: use the value of the nearest voxel
• Nyquist-Shannon Sampling Theorem: to avoid aliasing, sample at least twice the highest frequency of the signal -> $&lt; 0.5$ voxel size

Transfer function: map data values to colors and opacities

Gradients:

• forward difference: $\nabla f(x,y) = \begin{pmatrix} f(x + 1) - f(x) \ f(y+1) - f(y) \end{pmatrix}$
• backward difference: $\nabla f(x,y) = \begin{pmatrix} f(x) - f(x - 1) \ f(y) - f(y - 1) \end{pmatrix}$
• central difference: $\nabla f(x,y) = \frac{1}{2} \begin{pmatrix} f(x + 1) - f(x - 1) \ f(y + 1) - f(y - 1) \end{pmatrix}$

Types of projections:

• First-hit: only the first voxel that is hit by the ray is used
• average: average all voxels that are hit by the ray
• maximum intensity: use the maximum value of all voxels that are hit by the ray
• cvp, threshold: use first voxel that is above a certain threshold
• mida: local maximum of change of intensity along the ray

Chapter 8

Surface normal: $n = -\frac{\nabla f}{||\nabla f||}$

Curvature Information: $\nabla n^\top = -\nabla \frac{\nabla f}{||\nabla f||} = - \frac{1}{|\nabla f^\top|}(I - nn^\top)H$

$P = I - nn^\top$ porjects all points to the plane orthogonal to the normal vector

• $T=trace(G)$ sum of the diagonal elements of the matrix
• $F = |G|F = \sqrt{G{11}^2 + \dots + G_{nn}^2}$
• $\kappa_{1/2} = \frac{T \pm \sqrt{2F^2 - T^2}}{2}$

Isolines on triangles: on side $e_2$: $\frac{\varphi_1}{\varphi_1-\varphi_3}\cdot e_2$

Covariant Derivative: $D_{v(x)}f(x) = &lt;\nabla f(x), v(x)&gt;$

Contours vs crease lines:

• differences:
  • view dependent: contours are view dependent, crease lines are view independent
  • sharp edges: creases show sharp edges
  • contours: crease lines are not contours
• similarities:
  • order 1
  • both no round edges
  • both no bumps
  • both show deformations

Evaluation: Schulze method, pairwise comparison, rank order

Chapter 9

Streamlines: follow the arrows in once specific moment in time

Pathlines: follow the movement of a particle over time

Streaklines: connect particles that have passed through a point at a certain time, that is: connect all partlicles observed over time

Only draw streamlines with a certain distance

Euler Integration: $x_{i+1} = x_i + v(x_i) \cdot \Delta t$

• large errors

Runga Kutta 2nd order: $x_{i+1} = x_i + v(x_i) \cdot \Delta t + \frac{1}{2}v(x_i + v(x_i) \cdot \Delta t) \cdot \Delta t$

1. euler step
2. on way of euler step, go back to halfway point and calculate velocity at halfway point
3. use velocity at halfway point to calculate new position from start position

Runga Kutta 4th order: $x_{i+1} = x_i + \frac{1}{6}(k_1 + 2k_2 + 2k_3 + k_4)$ with

• $k_1 = v(x_i) \cdot \Delta t$
• $k_2 = v(x_i + \frac{1}{2}k_1) \cdot \Delta t$
• $k_3 = v(x_i + \frac{1}{2}k_2) \cdot \Delta t$
• $k_4 = v(x_i + k_3) \cdot \Delta t$

1. euler step
2. on way of euler step, go back to halfway point and calculate velocity at halfway
3. from start position, go with new velocity -> $k_2$
4. at halfway point of $k_2$, calculate velocity and go to new position -> $k_3$
5. at end of $k_3$, calculate velocity and go to new position -> $k_4$
6. use all four velocities (in a certain ratio) to calculate new position

Streamline Integration: Draw streamlines with a minimum distance between them, if dinstace is reached, stop integration

Only draw critical streamlines

How to find critical points:

• compute the gradient of the vector field and set it to zero
• use Jacobian matrix and the eigenvalues at the critical points to determine the type of critical point

Draw fuzzy streamlines

Take noise image, calculate pathlines and then use the noise image to draw the streamlines with a certain opacity

Chapter 10