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

Digital contours compression MapDigital curves

Format of the input data … … …

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

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

Vector map compression Original curve Restored curve

 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

Product scalar quantizer

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

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

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

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

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

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

Results: test set #1

Results: test set #2

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

The end

Appendix 1: test data #1

Appendix 2: test data #2

Appendix 3: Strong quantization

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

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

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

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