CIE L*a*b* color model

Transcription

1 CIE L*a*b* color model To further strengthen the correlation between the color model and human perception, we apply the following non-linear transformation: with where (X n,y n,z n ) are the tristimulus values of the reference white point : GenICT II 135

2 CIE L*a*b* color model L* represents a luminance channel (relative to reference white point). a* and b* represent chromatic channels : GenICT II 136

3 CIE L*a*b* color model Chromaticity diagram for various luminances: : GenICT II 137

4 CIE L*a*b* color model Perceptually uniform: The CIE L*a*b* color model is perceptually uniform, i.e., a change of the same amount in a color value should produce a change of about the same visual importance. In other words, Euclidean distance in the color space is propotional to human perception : GenICT II 138

5 L*a*b* & RGB L*a*b* chromaticity diagram reduced to the colors that can be represented in RGB color space : GenICT II 139

6 6.6 Summary : GenICT II 140

7 Summary We have introduced additive (RGB) and subtractive (CMY) color schemes, where the channels represent colors as in a tristimulus. HSV color scheme is based on channels reflecting visual perception properties. CIE XYZ/xyY follow the human tristimulus. CIE L*a*b* is perceptionally uniform : GenICT II 141

8 7. Color Mapping

9 Color map construction Example 1: Luminance map By assigning to each of the RGB color channels the same values one exclusively obtains greyscale colors. A linear mapping of intensity values of f to greyscale values generates a luminance map : GenICT II 143

10 Color map construction Example 2: Linear color transition One can steadily decrease one color channel and increase another to produce color transitions : GenICT II 144

11 Color map construction Example 3: Rainbow color map : GenICT II 145

12 Color map construction More intuitively, the rainbow color map can be generated using the HSV color space: : GenICT II 146

13 Luminance vs. rainbow color map Color order shall be based on human perception : GenICT II 147

14 Rainbow vs. luminance map Luminance provides better high-frequency separation than hue : GenICT II 148

15 Using luminance and hue : GenICT II 149

16 Comparing maps with changing luminance and hue : GenICT II 150

17 Saturation vs. luminance variation Saturation change exhibits low-frequency changes better Luminance change exhibits high-frequency changes better : GenICT II 151

18 Color bands The transitions do not need to use the full color spectrum. Color reduction leads to a banding effect : GenICT II 152

19 Number of bands Only for low-frequency changes, additional bands can help : GenICT II 153

20 Highlighting color map : GenICT II 154

21 Summary Numerical data: continuous color transition consider perception (luminance, saturation) Ordered data: banded color transition Categorical data: distinct colors : GenICT II 155

22 8. Multidimensional Data

23 Multidimensional data In the following, we are looking at all attributes of all samples simultaneously. The number of attributes is typically higher than the dimensionality of a visual system (2D or 3D). Sometimes the dimensionality can be reduced by removing redundant information (= dependencies between attributes). Still, one often needs to look into a larger number of attributes simultaneously : GenICT II 157

24 Multidimensional data Multidimensional data can be stored in a matrix of size n x d with n samples and d attributes: : GenICT II 158

25 8.1 Scatterplot Matrices

26 Scatterplots We had seen that for sets of samples with two numerical attributes, scatterplots are very common : GenICT II 160

27 Scatterplot matrices Scatterplots can be drawn for any combination of 2 attributes. These plots can be arranged in a matrix, the so-called scatterplot matrix : GenICT II 161

28 Animated Scatterplots One can traverse a scatterplot matrix by exchanging only 1 attribute. An animation can show the transition : GenICT II 162

29 Animated Scatterplots : GenICT II 163

30 Animated Scatterplots : GenICT II 164

31 8.2 Multidimensional Scaling

32 Linear projections Consider a linear projection from a high-dimensional attribute space into a 2D (or 3D) visual space. Try to find an optimal projection with respect to some metric. They give the best 2D (or 3D) view on the data. It is a slice through the high-dimensional space. The visual encoding is that of a scatterplot : GenICT II 166

33 Principal Component Analysis (PCA) PCA tries to find the most relevant directions (principal components) in the data. The first two principal components span the projection space. Maximize variance: Compute covariance matrix X T X where X has entries x ij. The principal components are the eigenvectors of the largest eigenvalues of the covariance matrix : GenICT II 167

34 Non-linear projections Linear projections cannot detect low-dimensional curved features (e.g., manifolds) in a high-dimensional space. Non-linear projections can be used to reproduce the nonlinear high-dimensional features in lowerdimensional spaces : GenICT II 168

35 Multidimensional Scaling Metric Multidimensional Scaling (MDS) tries to minimize the difference between distances in the original space and the projected space: E = [ d( k, l) d ( k, l)] 2 M k l This serves as an objective function that needs to be minimized : GenICT II 169

36 Isomap Uses geodesic instead of Euclidean distance: This geodesic distance is approximated by calculating the neighbourhood of the each point, building a graph that connects the neighborhood with distances as edge weights, and computing distances on that graph (Dijkstra s shortest path algorithm). Then, perform an MDS with these geodesic distances : GenICT II 170

37 Self-Organizing Maps Algorithm that performs clustering and non-linear projection onto lower dimension at the same time Finds and orders a set of reference vectors located on a discrete lattice Learning rule: m ( t + 1) = m ( t) + h ( t)[ x( t) m ( t)] i i ci i Objective function uses a neighborhood kernel: E = h x m SOM k i ci k i : GenICT II 171

38 Comparison MDS tries to preserve the metric (ordering relations) of the original space, long distances dominate over the shorter ones SOM tries to preserve the topology (local neighbourhood relations), items projected to nearby locations are similar : GenICT II 172

39 Example: PCA vs SOM : GenICT II 173

40 Design goals Distance preservation Neighborhood preservation Cluster segregation Clutter avoidance (using screen space) : GenICT II 174

41 8.3 Star Coordinates

42 Motivation Looking at all configurations in a scatterplot matrix is a tedious process. Is there no way that we can depict d-dimensional points in a 2D or 3D visual system? : GenICT II 176

43 Star coordinates The main idea of a star coordinate system is to place d-dimensional point in a d-dimensional coordinate system that is drawn in a 2D visual space. Hence, the d axes of the star coordinate system are linearly dependent. The axes emerge from an origin o and have distinct directions. Then, a d-dimensional point p i is mapped to : GenICT II 177

44 Star coordinates Placing a d-dimensional point in star coordinates: : GenICT II 178

45 Star coordinate layout The layout of the axes can be interactively changed or automatically generated : GenICT II 179

46 Star coordinates versus projections Star coordinate plots represent a linear projection. Interacting with the axis of the plot changes the projection : GenICT II 180

47 3D star coordinates The concept can be generalized to a 3D visual system (points are clustered and cluster boundaries rendered): : GenICT II 181