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Optimal Scanning of Gaussian and Fractal
Brownian Images with an Estimation of Correlation
DimensionA.Yu. Parshin, PhD,
Prof. Yu.N. Parshin, Doctor of Technical Sciences
Ryazan State Radio Engineering University , Ryazan, [email protected]
Already known scanning methods
Static scanning methods◦ Usual scanning methods – vertical, horizontal, diagonal.◦ Space-filling curves – Peano-Hilbert curves.
Adaptive scanning methods◦ Choosing scanning method for every image.◦ Calculation of information parameter and considering it as criterion of scanning direction.
Calculation of correlation dimensionUsage of time-delayed values of observable component as values of non-observable components:
Distances between vectors (DE=3):
2332
222
11 mkmkmki txtxtxtxtxtxr
Number of vectors:
EDNN DE – dimension of space of embeddings
Ns – number of signal samples
1,..., EE DnDnn xxx
12 DDE
1 DDE
Dimension of space of embeddings is chosen according to value of object dimension D:
For specific task it is enough to use the following expression:
Calculation of correlation dimension Combined probability density function of independent distances:
Combined probability density function of dependent distances from second vector to each other:
1,0,0
1,0,|1
1r
rrDm
M
m
rDDw
.3
,,...,3,,,0,...,3,,
,
,|
maxmin
maxmin
1min
32
minmax
122
kNm
NkrrrNkrrr
rrrr
D
Dw
kkm
kkm
Dkm
N
Nm kk
rr
Calculation of correlation dimension
Classic interpretation of correlation dimension:
Maximum likelihood algorithm for independent distances:
Likelihood function for dependent distances:
Maximum likelihood algorithm for dependent distances:
M
mm
DDest
r
MDwDE
1
0ln
|maxarg r
.,|||, 1min
32
minmax
11
1
1122112112
Dkm
N
Nm kk
Dm
N
m
rrrr
DrDDwDwDw rrrrr
132
min
1
11 lnln32ˆ
N
Nmkm
N
mm rrrND
rrCD N
nr logloglimlim
0
Correlation function of fractal Brownian surface
The correlation of the pixels series is specified as follows:
Let us define a fractal Brownian surface as an assembly of the random values X(i,j) given on the 2D coordinate axis. A probability measure of the random value increments:
the correlation coefficient of increments equals:
2211221121 ,,,,,, kkkkkkkks jijiRjipjipkkR M
jiXjjiiXjiXjiXX ,,,, 1122
1
1,,,2
21
21
212
212
2211
H
jijjii
jijir
Construction of fractal Brownian surface based on correlation function
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a) H=0,1 b) H=0,5c) H=0,9
Figure 2. Image of 2D Fractal Brownian motion
a) ax=0,01, ay=0,1 b) ax= 0,1, ay= 0,1 c) ax= 0,1, ay= 0,01
Figure 3. Image of 2D Gaussian process
Proposed scanning methods Algorithm 1
Scanning direction is chosen by criterion of correlation coefficient maximum of each subsequent pixel. This method uses image characteristics, averaged by significant ensemble of samples, but it doesn’t consider properties of particular image.
In case of image in the form of fractal Brownian surface correlation of pixels equals:
HxHxx
Hx
Hy
Hyy
Hymn mmnnnmnnR 222222
2
, 4
Proposed scanning methods Algorithm 2
Scanning direction is chosen by criterion of scalar product maximum of previous vector X and following vector Y.
Steps of this algorithm:◦ First vector consists of M image pixels from first line of current frame;◦ One of Q predefined directions from last pixel to end of next vector is considered;◦ standard scalar product is evaluated for pair of vectors previous X and each of Q considered directions Y as
follows:
◦ maximum value of scalar product defines a pair of vectors and thereby the direction of scan trajectory;◦ scan trajectory moves to new pixel, which is last for chosen vector;◦ the algorithm is repeated until majority pixels are covered.
YXYX
YX
,,R
Optimal scanning trajectories
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Figure 4. Optimal scanning trajectory by 1-st method,
image of figure 2c
Figure 5. Optimal scanning trajectory by 2-st method,
image of figure 2c
Figure 6. Optimal scanning trajectory by 1-st method,
image of figure 3a
Figure 7. Optimal scanning trajectory by 2-st method,
image of figure 3a
Correlation dimension estimation
0 0.2 0.4 0.6 0.8 10
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10D
e
H0 0.2 0.4 0.6 0.8 1
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H
a) N=32, K=4, Np=500 b) N=128, K=8, Np=500Figure 1. The dependencies of the correlation dimension estimates on Hurst exponent.
Conclusion◦ Appliance of maximum likelihood algorithm demands provision of maximum
correlations between image pixels in one-dimension sequence. ◦ Scanning by maximum correlation coefficient criterion provides bigger
difference between values of correlation dimension estimates and therefore simplifies solution of object detection problem.
Thanks for Your attention!
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