1. Reference line approach in vector data compression Alexander Akimov, Alexander Kolesnikov and Pasi Fränti UNIVERSITY OF JOENSUU DEPARTMENT OF COMPUTER SCIENCE FINLAND

2. Digital contours compression MapDigital curves

3. Format of the input data … 3615352.4004109581 6925890.7695743283 3615349.4965740540 6925888.6290830234 3615344.4561071559 6925889.3992598010 3615331.3723420152 6925890.7895789202 3615322.1988572506 6925894.3704008218 3615306.1865065964 6925915.6552863186 3615291.9890540070 6925941.0211085072 3615279.1197768194 6925959.4553396963 3615261.9772396428 6925983.6909024585 3615256.3345565684 6925997.4440592052 3615256.8212792310 6926008.4765914902 3615262.6289530387 6926012.7475718036 … 151.540252685547 -24.045833587646 151.531372070313 -24.058473587036 151.537963867188 -24.086318969727 151.565521240234 -24.096805572510 151.614135742188 -24.052780151367 151.616073608398 -23.998264312744 151.639846801758 -23.977359771729 151.683868408203 -23.988887786865 151.788024902344 -24.098888397217 151.880798339844 -24.181110382080 151.906097412109 -24.194442749023 151.933441162109 -24.217914581299 …

4. Multiresolution vector map compression Low resolution High resolution...... Average resolution Compressed file Choose decoding accurancy... Layer 1 Layer K Layer N............

5. Two level resolution vector map compression += High resolution: Original data, Lossy compression Two level resolution: Data is stored separetely, with ability of independent Extracting of each level Low resolution: Result of approximation of high resolution level. Lossless compression

6. Vector map compression Original curve Restored curve

7.  x i = x i – Predictor(x i, x i-1 )  y i = y i – Predictor(y i, y i-1 ) Coordinate transformation: DPCM approach Y X YY XX

8. Product scalar quantizer

9. Optimal product scalar quantizer Mean square error E(M) of the 2-D variable  =(x,y) Optimization problem:

10. The reference line approach X Y X’ Y’ Original coordinatesTransformed coordinates

11. Predictor #1... Current point Predicted point Point, participated in prediction Low resolution level High resolution level

12. ... Current point Predicted point Point, participated in prediction Low resolution level High resolution level Predictor #2

13. Test data Test data #1: 365 curves 170,000 points 5,200 segments LR Test data #2: 3495 curves 221,000 points 13,250 segments

14. Tested algorithms DPCM-1: DPCM coordinate transformation for one level DPCM-2: DPCM coordinate transformation for two levels RL-1: Reference line approach with predictor # 1 RL-2: Reference line approach with predictor #2

15. Results: test set #1

16. Results: test set #2

17. Conclusions The reference line approach allows to reduce distortion in lossy compression of two levels vector map The necessarity of independent storage of different resolution levels lead us to increasing of compressed file size

18. The end

19. Appendix 1: test data #1

20. Appendix 2: test data #2

21. Appendix 3: Strong quantization

22. Scalar quantization Relatively fast optimal algorithm: O(MN) Low storage space requirements

23. Cartesian product quantizer (1) 2D data {x i, y i } is separeted into two 1D sets: {x i } and {y i }

24. Cartesian product quantizer (2) Mean square error E(M) of the 2-D variable  =(x,y) Optimization problem:

25. Two level resolution vector map compression (1) 1.Two resolution layers 2.Low resolution layer is a result of rough approximation of high resolution layer 3.Lossy compression of high resolution layer 4.Lossless compression of low resolution layer