FAST DYNAMIC QUANTIZATION ALGORITHM FOR VECTOR MAP COMPRESSION

Minjie Chen, Mantao Xu and Pasi Fränti

University of Eastern Finland

Vector Compression

Vector data, embrace a number of geographic information or objects such as waypoints, routes and areas. It is represented with a sequence of points in a given coordinate system. In order to save storage cost, compression algorithm for vector data is needed.

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Longitude

La

titu

de

Map of UKGPS traces

Polygonal Approximation

Reduce the number of points in the vector map such that the data is represented in a coarser resolution.(Douglas73’,Perez94’,Schuster 98’, Bhowmick07’)

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Longtitude

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Longtitude

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Number of point is reduced from 10910 to 239

Quantization-based method

Reduce every points’ coding cost. The coordinate value is quantized and differential coordinates is encoded(Shekhar 02’, Akimov 04’)

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Given quantization level l, differential coordinates is quantized as:( ) [ / ] ([ / ] ,[ / ] )i iQ l l x l l y l l v vi i

Coding Q (vi) is equivalent to coding an integer vector q = ([Δxi/l], Δyi/l])

Coding of quantized residual vectors

Integer vector q = ([Δxi/l], Δyi/l]) is encoded by probability distributions of qx and qy:

Codebook itself must be encoded. But a large-sized codebook is intractable in order to achieve a desirable coding efficiency

An intuitive solution is to adopt a single-parameter geometric distribution to model qx and qy:

where px , py can be approximated by maximum likelihood estimation.Other solutions, uniform, negative binomial or Poisson distribution can also be considered

2 2(q) log ([ / ] log ([ / ]r f x l f y l

| || |(| |) (1 ) , (| |) (1 ) yxqq

x x x y y yf q p p f q p p

Coding of quantized residual vectors

Example of using geometric distribution to estimate the probability (allocated coding

bits) of q ,for l = 0.0025

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estimated

real

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estimated

real

For ∆xl For ∆yl

Error Measure (Distortion)

Suppose poly-line {pi,…,pj} is approximated by line segment , the approximation error can be defined as the sum of square distances from vertices pk (i≤k≤j):

r ri jp p

22 ( , ) ( , )

jr r r ri j k i j

k i

e p p d p p p

Poly-line {pi,…, pj} (black line) is approximated by (blue line )with approximating error

2 2 2 22 1 2 3 4( , )r r

i je p p d d d d

12 21

( , )m m

Mr ri i

m

E e p p

The distortion can be calculated by:

This can be calculated in O(1) time by [Perez 94’]

Dynamic Quantization

2min , . . ,E s t R c

The distortion E is minimized under the constraint of bit constraint R:

Dynamic quantization optimizes the cost function:

1 12 21

( ( , ) ( , ))m m m m

Mr r r ri i i i

m

J E R e p p r p p

Combine polygonal approximation and quantization-based method using dynamic programming. [Kolesnikov 05’]:

Dynamic Quantization

The minimization is solved by the shortest path search on a weighted directed acyclic graph (DAG) and dynamic programming. Suppose Ji is the minimum weighting sum from p1 to pi on G, A is an array used for backtracking operation, the recursive equation can be defined by:

2 1{1 1}min ( ( , ) ( , )), 0r r r r

i k k i k ik iJ J e p p r p p J

2{1 1}arg min ( ( , ) ( , ))r r r r

i k k k i k ik i

A J e p p r p p

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DQ

Dynamic Quantization

Two parameters: Lagrangian parameter λ quantization level l

Given one l, different λ → one rate-distortion curve

Existing approach calculates many rate-distortion curves with different l and the best is the lower envelope of the set of curves.

Rate-distortion curve for quantization step qk=0.01/2k, k=0, 1/2,1,…, 5

Time-expensive0 2 4 6 8 10 12

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rate (bits)

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(M

SE

)

Dynamic Quantization – fast solution

Proposed: if ∆x, ∆y follows geometric distribution or uniform distribution, by setting

for each l, one optimal Lagrangian parameter λ is estimated as:

black ‘+’: error balance principle red ‘o’: proposed

/ 0J l

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rate (bits)

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Relationship between λ and l is derived, no need for multiple calculation of rate-distortion curve

21ln 2

6l

Time complexity

Shortest path algorithm on a weighted DAG takes O(N2) time.

Incorporating a stop search criterion in DAG shortest path search

22( , )( , )

( ) ( )i

r rr rA ik i

i

e p pe p p

i k i A

The proposed method can also be applied for bit-rate constraint problem by several iterations using binary search on the quantization level l.

Time complexity reduced as O(N2/M)

Pseudo code

Experiments

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51.55

0.55 0.6 0.65 0.7 0.75

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51.45

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0.55 0.6 0.65 0.7 0.75

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128bits/point, original 10 bits/point

5 bits/point 2 bits/point

Resulting rate-distortion curve

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rate (bits)

dis

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CBCRLDQFDQ

CBC: clustering-based methodRL: reference line methodDQ: Dynamic quantizationFDQ: Fast dynamic quantization

Proof For geometric distribution For uniform distribution

Conclusions

Derivation for optimal Lagrangian multiplier λ for each quantization step l

Fast dynamic quantization algorithm with O(N2/M) time complexity for lossy compression of vector data.

Reference

[Douglas 73’] D. H. Douglas, T. K. Peucker, "Algorithm for the reduction of the number of points required to represent a line or its caricature", The Canadian Cartographer, 10 (2), pp. 112-122, 1973.

[Perez 94’] J. C. Perez, E. Vidal, "Optimum polygonal approximation of digitized curves", Pattern Recognition Letters, 15, 743–750, 1994.

[Schuster 98’] G. M. Schuster and A. K. Katsaggelos, "An optimal polygonal boundary encoding scheme in the rate-distortion sense", IEEE Trans. on Image Processing, vol.7, pp. 13-26, 1998.

[Bhowmick 07’] P. Bhowmick and B. Bhattacharya, "Fast polygonal approximation of digital curves using relaxed straightness properties", IEEE Trans. on PAMI, 29 (9), 1590-1602, 2007.

[Shekhar 02’] S. Shekhar, S. Huang, Y. Djugash, J. Zhou, "Vector map compression: a clustering approach", 10th ACM Int. Symp.Advances in Geographic Inform, pp.74-80, 2002.

[Akimov 04’] A. Akimov, A. Kolesnikov and P. Fränti, "Coordinate quantization in vector map compression", IASTED Conference on Visualization, Imaging and Image Processing (VIIP’04), pp. 748-753, 2004.

[Kolesnikov 05’] A. Kolesnikov, "Optimal encoding of vector data with polygonal approximation and vertex quantization", SCIA’05, LNCS, vol. 3540, 1186–1195. 2005.