Mathematical Foundations of Arc Length-Based Aspect Ratio Selection 1 Shandong University 2 Computer Network Information Center 3 University Konstanz Fubo Han 1, Yunhai Wang 1*, Jian Zhang 2, Oliver Deussen 3, Baoquan Chen 1

Outline Aspect ratio selection and banking to 45° Arc-length (AL) based aspect ratio selection Mathematical foundations of AL method Parameterization invariant Connection between AL and banking to 45° Discussions and conclusions

Perceptual Influence of Aspect Ratio X axis: monthly increments Y axis: CO2 measurements aspect ratio=1.15 aspect ratio=0.15 aspect ratio=height/width

Banking to 45° Cleveland’s finding [1] : [1] W. S. Cleveland, M. E. McGill, and R. McGill. The shape parameter of a two-variable graph. Journal of the American Statistical Association, 83(402):289–300, Centering the slopes around 45° minimizes the error in judging slope ratios.

Cleveland’s Methods MS (median absolute slope) AO (absolute orientation) AWO (average weighted orientation) RV (resultant-vector)

Maximizing Orientation Resolution J. Heer and M. Agrawala. Multi-scale banking to 45º. IEEE Trans. Vis. & Comp. Graphics, 12(5):2, GOR (global orientation resolution): LOR (local orientation resolution):

Comparison among Methods J. Heer and M. Agrawala. Multi-scale banking to 45º. IEEE Trans. Vis. & Comp. Graphics, 12(5):2, AWO generates reasonable aspect ratios for most data set LOR produces lower aspect ratio than GOR

Outline Aspect ratio selection and banking to 45° Arc-length (AL) based aspect ratio selection Mathematical foundations of AL method Parameterization invariant Connection between AL and banking to 45° Discussions and conclusions

Arc-Length Method Area-preserving transformation Minimizing total length of line segments J. Talbot, J. Gerth, and P. Hanrahan. Arc length-based aspect ratio selection. IEEE Trans. Vis. & Comp. Graphics, 17(12):2276–2282, ° minimum arc length

AL for 1D Curves [1] J. Talbot, J. Gerth, and P. Hanrahan. Arc length-based aspect ratio selection. IEEE Trans. Vis. & Comp. Graphics, 17(12):2276–2282, Parameterization invariant Results from [1] Good properties Robust ? ×

AL for 2D Contours [1] J. Talbot, J. Gerth, and P. Hanrahan. Arc length-based aspect ratio selection. IEEE Trans. Vis. & Comp. Graphics, 17(12):2276–2282, Results from [1] AL and MS are roughly the same for contour plot. AWO banks circles to ellipses. × ×

Outline Aspect ratio selection and banking to 45° Arc-length (AL) based aspect ratio selection Mathematical foundations of AL method Parameterization invariant Connection between AL and banking to 45° Discussions and conclusions

Parameterization Invariant Line integral representation Uniformly samplingNon-uniformly sampling Same integral result

Line Integral Representation Arc-length (AL) based aspect ratio selection AWO (average weighted orientation) AL is parameterization invariant AWO is parameterization invariant

RV is Parameterization Invariant Resultant vector (RV): RV is parameterization invariant

L1-norm based LOR L1-norm LOR Local orientation resolution (LOR): Maximizing local curvature (MLC) L2-norm LOR

Comparison AL, AWO RV, and MLC are all parameterization invariant.

Outline Aspect ratio selection and banking to 45° Arc-length (AL) based aspect ratio selection Mathematical foundations of AL method Parameterization invariant Connection between AL and banking to 45° Discussions and conclusions

AL and Banking to 45° For single line segment For multiple line segments Bank the line segments with larger absolute slopes to 45°

AL and Banking to 45° Upper and lower bounds of average absolute slopes

1d curves d contours Evaluation close to 1

Outline Aspect ratio selection and banking to 45° Arc-length (AL) based aspect ratio selection Mathematical foundations of AL method Parameterization invariant Connection between AL and banking to 45° Discussions and conclusions

Verification of AL’s Properties Q1: Does AWO behave differently from AL for contours? Q2: Does MS always perform similarly to AL for contours? Q3: Is there any method that has similar performance with AL but is faster and more robust? Q4: Is there any counterexample where all previously methods produce poor results?

Q1: Does AWO behave differently from AL for contours? AL, AWO and RV perform very similarly for both curves and contours 1d curves2d contours 27 1d curves and 26 2d contours MS generates similar results with AL for 2d contours

Q2: Does MS always perform similarly to AL for contours? MS doesn’t always perform similarly to AL for contours MS (median absolute slope): AL ( arc-length ):

Q3: Is there any method that has similar performance with AL but is faster and more robust? RV always sets aspect ratio to 1 for monotonically curve, while AL doesn’t RV is much faster than AL because it has a closed form solution RV is always similar to AL for both 1d and 2d data.

Q4: Is there any counterexample where all previously methods produce poor results? All previously methods produce poor result when most slope values are quite small. AL, AWO, and RV produce overly tall aspect ratios.

Conclusions Rigorous analysis of aspect ratio selection; Unveiling parameterization invariant property using line integral representation; Analyzing mathematical connection between AL and banking to 45° principle; Investigating the properties of AL.

Future Work Exploring perceptual foundations of AL; Extending current methods to 2D scatterplots or line graphs that involve multiple time series. … …

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