Reference line approach in vector data compression

Alexander Akimov, Alexander Kolesnikov and Pasi Fränti

UNIVERSITY OF JOENSUUDEPARTMENT OF COMPUTER SCIENCEFINLAND

Digital contours compression

Map Digital curves

Format of the input data

…3615352.4004109581 6925890.76957432833615349.4965740540 6925888.62908302343615344.4561071559 6925889.39925980103615331.3723420152 6925890.78957892023615322.1988572506 6925894.37040082183615306.1865065964 6925915.65528631863615291.9890540070 6925941.02110850723615279.1197768194 6925959.45533969633615261.9772396428 6925983.69090245853615256.3345565684 6925997.44405920523615256.8212792310 6926008.47659149023615262.6289530387 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

…

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 independentExtracting of each level

Low resolution:Result of approximationof high resolution level. Lossless compression

Vector map compression

V e c to r M ap

Tra n s fo rm a t io no f co o rdin a te s

Q u a n t iza t io n En tro py co din g

Original curve Restored curve

xi = xi – Predictor(xi, xi-1)

yi = yi – Predictor(yi, yi-1)

Coordinate transformation: DPCM approach

Y

X

Y

X

Product scalar quantizer

Optimal product scalar quantizer

x y

kjn

M

j

M

k Cknjnnnxy yyxxyxpE

1 1

22

,

])()[(),(

x

jnjyx

M

j Xxjnnx

MM

xxxpx

ME1

2)()(}{

)( {min{min,

}}.))(({min1

2

}{

y

knk

M

k Yyknny yyyp

y

Mean square error E(M) of the 2-D variable =(x,y)

Optimization problem:

MMM yx

The reference line approach

X

Y

X’

Y’

Original coordinates Transformed coordinates

Predictor #1

...

Current point

Predicted point

Point, participatedin prediction

Low resolution level

High resolution level

...

Current point

Predicted point

Point, participatedin prediction

Low resolution level

High resolution level

Predictor #2

Test data

Test data #1: 365 curves170,000 points 5,200 segments LR

Test data #2: 3495 curves221,000 points 13,250 segments

Tested algorithmsDPCM-1: DPCM coordinate transformation for one level

DPCM-2: DPCM coordinate transformation fortwo levels

RL-1: Reference line approach with predictor # 1

RL-2: Reference line approach with predictor #2

Results: test set #1

0,000

0,001

0,002

0,003

0,004

0,005

0,006

4 5 6 7 8 9 10 11 12 13 14 15 16 17

DPCM -1

DPCM-2

RL1

RL2

Results: test set #2

0,0

0,5

1,0

1,5

2,0

2,5

5 6 7 8 9 10 11 12 13 14 15 16

DPCM -1

DPCM -2

RL -1

RL -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 {xi, yi} is separeted into two 1D sets: {xi} and {yi}

Cartesian product quantizer (2)

x y

kjn

M

j

M

k Cknjnnnxy yyxxyxpE

1 1

22

,

])()[(),(

x

jnjyx

M

j Xxjnnx

MM

xxxpx

ME1

2)()(}{

)( {min{min,

}}.))(({min1

2

}{

y

knk

M

k Yyknny yyyp

y

Mean square error E(M) of the 2-D variable =(x,y)

Optimization problem:

MMM yx

Two level resolution vector map compression (1)

1. Two resolution layers2. Low resolution layer is a result of

roughapproximation of high resolution layer

3. Lossy compression of high resolution layer

4. Lossless compression of low resolution layer