1. VOTS VOlume doTS as Point-based Representation of Volumetric Data S. Grimm, S. Bruckner, A. Kanitsar and E. Gröller Institute of Computer Graphics and Algorithms Vienna University of Technology Vienna, Austria

2. Sören GrimmVienna University of Technology Motivation (1/3) Volumetric data: Processing is sampled based Processing is sampled based Given on grid structure, e.g. regular grid Given on grid structure, e.g. regular gridAdvantages: Efficient spatial addressing Efficient spatial addressing Efficient processing, such as rendering, segmentation, etc. Efficient processing, such as rendering, segmentation, etc.

3. Sören GrimmVienna University of Technology Motivation (2/3) Rigid shape of grids is a limitation factor: Sizes are enormously increasing Sizes are enormously increasing Often just parts are of interest Often just parts are of interest Difficult to perform deformations Difficult to perform deformations Difficult to analytically analyze the data Difficult to analytically analyze the data

4. Sören GrimmVienna University of Technology Motivation (3/3) Information dependent storage requirement Information dependent storage requirement Allow to leverage resources Allow to leverage resources Inhomogeneous → grid is efficient Partially inhomogeneous → grid is inefficient

5. Sören GrimmVienna University of Technology What is a VOT (1/2) A VOT is basically a thick Volumetric Point Represents a region by polynomial Represents a region by polynomial It consists of: It consists of:  Coefficients of Taylor series, position, and a validity area 100 samples 7 VOTS

6. Sören GrimmVienna University of Technology What is a VOT (2/2) VOT Evaluation: Evaluation of Taylor series expansion

7. Sören GrimmVienna University of Technology Outline

8. Sören GrimmVienna University of Technology Generation of a VOT Taylor series expansion: For practical reasons: N = 3

9. Sören GrimmVienna University of Technology Cell to VOT conversion Cell

10. Sören GrimmVienna University of Technology Point cloud to VOT conversion (1/4) m scattered data points VOT: Function Fitting?

11. Sören GrimmVienna University of Technology Point cloud to VOT conversion (2/4) Original data values Reconstructed values by Taylor series Linear regression: Minimizing sum-of-squares 20 unknowns, due to symmetry →

12. Sören GrimmVienna University of Technology Taking the partial derivatives with respect to the 20 unknowns Point cloud to VOT conversion (3/4)

13. Sören GrimmVienna University of Technology Point cloud to VOT conversion (4/4) Setting derivatives to zero, leads to a system of linear equations: Inversion of M produces the coefficients

14. Sören GrimmVienna University of Technology Outline

15. Sören GrimmVienna University of Technology Grid to VOTS conversion (1/2) Growing approach: For all sample positions grow a VOT For all sample positions grow a VOT  Size of VOT bounded by specified max error Outcome: VOT candidates Outcome: VOT candidates A small subset of these VOTS is chosen, so that they completely cover the underlying volumetric data A small subset of these VOTS is chosen, so that they completely cover the underlying volumetric data

16. Sören GrimmVienna University of Technology Grid to VOTS conversion (2/2) Goal: small number of large VOTS covering entire volume (small overlap) Goal: small number of large VOTS covering entire volume (small overlap) Sort VOT candidates according to size While space not covered Select largest VOT candidate Update size of remaining candidates Re-sort VOT candidates

17. Sören GrimmVienna University of Technology Outline

18. Sören GrimmVienna University of Technology Maximum Intensity Projection (1/2) Maximum value along a ray: Regular grid→ sample based determined Regular grid→ sample based determined VOTS → analytically determined VOTS → analytically determined

19. Sören GrimmVienna University of Technology Maximum Intensity Projection (2/2) For all VOTS For all VOTS  MIP textures are created  Send to graphics hardware Graphics hardware is used to perform maximum-blending Graphics hardware is used to perform maximum-blending Viewing direction

20. Sören GrimmVienna University of Technology Results  -error 0.01%0.1%1%10% Head: #VOTs 570 K 538 K 333 K 35 K Lobster: #VOTs 114 K 112 K 60 K Head 1.7 M samples Lobster 500K samples VOTS Distribution & Maximum Intensity Projection

21. Sören GrimmVienna University of Technology Conclusion We proposed a new representation of volumetric data: VOTS Intuitive and constructive representation Intuitive and constructive representation Allow user-centric importance sampling Allow user-centric importance sampling Allow to leverage resources Allow to leverage resources Information dependent storage requirement Information dependent storage requirement Allow to analytically process the data Allow to analytically process the data No connectivity for reconstruction is needed No connectivity for reconstruction is needed

22. Sören GrimmVienna University of Technology Future Work Conversion of other grid/data structures Conversion of other grid/data structures Sparse volumes Sparse volumes More sophisticated conversion technique More sophisticated conversion technique Blending between VOTS Blending between VOTS Efficient rendering method Efficient rendering method Exploit derivative information Exploit derivative information New visualization techniques New visualization techniques Investigate filter possibilities Investigate filter possibilities

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