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FAST DYNAMIC QUANTIZATION ALGORITHM FOR VECTOR MAP COMPRESSION Minjie Chen, Mantao Xu and Pasi...
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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
Latit
ude
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Longtitude
Latit
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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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Longtitude
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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
-0.1 -0.05 0 0.05 0.10
0.02
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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
-5.75 -5.7 -5.65 -5.6 -5.55 -5.5 -5.45 -5.4 -5.35 -5.3 -5.2550
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original
PA
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
10-8
10-7
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10-4
rate (bits)
dis
tort
ion
(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
0 5 1010
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10-6
10-5
10-4
rate (bits)
dis
tort
ion
(M
SE
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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
0.55 0.6 0.65 0.7 0.75
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51.55
0.55 0.6 0.65 0.7 0.75
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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
2 4 6 8 10 1210
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10-7
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10-5
10-4
rate (bits)
dis
tort
ion
(M
SE
)
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.