CHINESE ASTRONOMY AND ASTROPHYSICS

PERGAMON Chinese Astronomy and Astrophysics 26 (2002) 115-124

Tracking Algorithms for Low-cant rast Extended Objects+ *

RAO Chang-hui’ JIANG Wen-ban’ LING Ningl Jacques M. Beckem ‘Institute of p 0 tics and Electronics, Chinese Academy of Sciences, Chengdu 610209

2 National Solar Observatory, U.S.A.

Abstract We studied the effectiveness of two algorithms, of cross-correlation

coefficient and of absolute difference in the case of extended, low-contrast ob-

jects, such as solar granulation and sunspots. Wave-front overall tilt signals,

acquired via post-processing of sampled solar granulation and sunspot images,

are presented. Furthermore, long-exposure images of solar granulation before

and after the tilt elimination are compared as regards rms contrast.

Key words: contrast-xtended object-cross-correlation coefficient- abso-

lute difference algorithm-atmospheric coherent length

1. INTRODUCTION

As is well-known, due to the influence of atmospheric turbulence, the resolution of optical

images formed by ground-based astronomical telescopes are subject to restriction. The

technology of self-adaptive optics can make real-time measurements and rectify the optical

wave-front disturbed by atmospheric turbulence 11-31. Thus it provides an effective route

for astronomical observations of high resolution. For the high resolution observation of

stellar objects, the self-adaptive optical system provides signals used in the exploration of

wave-front disturbance, and the signals are generally determined by the object itself, by

bright stars in the neighborhood of a very dark object or by a laser guiding star. For high

resolution observations, the signals used in the wave-front exploration can be provided only

by solar surface granulation or sunspots. These objects have a low contrast and large sizes. In general case, the contrast of solar surface granulation is about 4%-g%. We have analyzed

t Supported by National Natural Science Foundation Received 2000-07-03; revised version 2001-l&16

* A translation of Acta Astmn. Sin. Vol. 42, No. 3, pp. 329338, 2001

0275-1062/01/E-see front matter @ 2001 Elsevier Science B. V. All rights reserved. PII: SO275-1062(02)00049-S

116 RAO Chang-hui et al. / Chinese Astronomy and Astrophysics 26 (2UO2) 115-124

the dependence of the observed contrast of solar surface granulation on the aperture of the observing system, on the coherent length of atmospheric turbulence and on the sensitivity of the detector141. In the exploration of the wave-front disturbance of atmospheric turbulence, the accuracy is limited by the contrast and extension of the object. Hence, the method of exploring the wave front of extended, low-contrast objects has become an important subject of research.

According to the literature both in our country and abroad15-Ql, for the exploration of the wave-front disturbance of extended, low-contrast objects, the following two tracking algorithms are generally adopted: the algorithm of cross-correlation coefficient and that of absolute difference. This paper begins with a brief introduction of the basic principles of the two algorithms and the sizes of data computation involved in each. It then evaluates the overall tilt signal of disturbed atmospheric wave fronts from consecutive runs of short- exposure images of solar surface granulation and spots, processed with the two algorithms, and compares the images before and after the elimination of the tilt disturbance.

2. METHODS OF TRACKING

For extended, low-contrast objects, two common algorithms are used: by cross-correlation coefficient and by absolute differences.

2.1 Cross-correlation CoefRcient Algorithm In this algorithm we fist calculate the cross-correlation coefficient between the real

image and a reference image for various position offsets. The position offset corresponding to the maximum coefficient is then taken to be the signal of wave-front error. Let the reference image be IR and its size be N * N and let the real target image be 1~ with size M * M. M < N. Then the correlation is expressed by:

M-l’M-1 M-l M-l M-l M-l

‘UR(%V) = c c IR(i+u,j+‘u)IL(i,j)/[c c Ii(i+%j+‘v)x c c 1;(i,j)]1’2 >

i=O j=O i=O j-0 i=O j=O

(1)

M-l M-l

c c h(i + ,,j + ‘v)h(&j) = F-{F+[IL(5,y)]F+*[IR(Z,Y)]} ’ (2) i=O j=O

The computation in Eq.(l) is made using a direct method, while that in Eq.(2), with Fourier transform.

In order that the wave-front exploration of the system possesses a sub-pixel accuracy, fitting with a quadratic curve may be adopted. The overall tilt signal thus obtained can be expressed as:

RAO Chang-hui et al. / Chinese Astronomy and Astrvphysica 26 (2002) 115-124 117

2.2 Absolute Difference Algorithm

In this algorithm one first of all calculates the sum of the absolute differences between the reference images and the real image for various offsets, and then take the position offset corresponding to the minimum sum as the signal of detecting wave-front error, expressed by

i=o j=o

We may also adopt the quadratic fitting so that the wave-front detection is more ac- curate than one pixel size. Thus the overall tilt signal of wave front may be expressed as:

1 DLR(‘&nin + 1, vmin) - DLR(%in - 1, Vmin)

TX = Umin + 2 DLR(Umin - 1, ~min) + DLR(‘Umin + 1, vmin) - 2DLR(umin, Vmin) ’ (6)

1 DLR(umin, vmin + 1) - DLR(umin, vmin - 1)

TY = vmin + 5 DLR(‘Umin, vmi, - 1) + DLR(Umin, vmin + 1) - 2DLR(umin, vmin) ’ (7)

2.3 Comparison of Two Algorithms Table 1 shows a comparison between the volumes of computation involved in the cross-

correlation coefficient and absolute difference algorithms:

Table 1 Comparison of computing size of the two algorithms

Addition of real numbers Multiplication of real numbers Squares of real numbers cross-correlation

coefficient (direct) MZ(N - M + 1)2 M”(N - M + 1)” (N-M+l)’

cross-correlation coefficient (Fourier)

12N2 log, N 12N2 log2 N (N - M + 1)2

absolute difference algorithm

M2(N - M + 1)2

As may be seen in Table 1, when M and N are large (both larger than 16, say), the amount of computation in the absolute difference algorithm is the smaller. This is because, here, there is only addition of real numbers, and no multiplication or squaring. The amount of computation in the cross-correlation coefficient algorithm is much larger; within it, calculation by Fourier transform involves less computation than by the direct method.

While in respect of the amount of computation, absolute difference is somewhat better than cross-correlation coefficient, this is also the case in respect of system measurement noise. However, in respect of the scintillation caused by atmospheric turbulence, it is the other way round: the algorithm of cross-correlation coefficient is better in this respect.

3. RESULTS OF EXPERIMENTS

On March 30,2000, we carried out observations of solar surface granulation at the vacuum tower telescope of National Solar Observatory of U.S.A. This Observatory is funded by U.S.

118 RAO Chong-hui et al. / Chinese Astronomy and Astrophysics 26 (2002) 115-124

National Science Foundation and operated by Association of Universities for Research in Astronomy. The aperture of the telescope is 760mm. In our experiment, the solar images were formed on the focal surface of a CCD detector via the main optical path as well as the corresponding guiding and imaging light paths, and the images were stored by a real-time

image sampling system. The exposure time of CCD camera is about 8 ms, the A/D number has 16 digits, the readout time of data is about 1.2s, and the sensitivity of the detector

system is O.O9024arcsec/pixel. According to the sampled consecutive images, we applied the two algorithms separately

to the sampled consecutive images to calculate the error signal of the overall wave-front tilt.

We selected a particular sub-region of the first image as the reference image and correlated it

to the sub-region in all subsequent images with different position offsets. We then separately

calculated the sums of cross-correlation coefficients and absolute differences, and obtained

the signal of overall wave-front tilt with a quadratic fit.

Fig. 1 3-D graph of correlation between reference image and real image

-l Time(s) 0 10 20 30 40 50 60

Time(s)

Fig. 2 Time series of tilt signals calculated with the cross-correlation coefficient algorithm

Fig.1 presents a three-dimensional graph of the sums of cross-correlation coefficients and absolute differences, for various position offsets, between one of the real-time images and the reference image. Fig.2 is the time series diagram of 50 consecutive wave-front

RAO Chang-hui et al. / Chinese Astronomy and Astrophysics 26 (2002) 115-124 119

-0.03 1 1,

0 10 20 30 40 50 60

Time(s) (a)

-0.06 ’ I

0 IO 20 30 40 50 60 Time(s)

(b)

Fig. 3 Time series of the difference between the tilts calculated from the two algorithms

tilt signals obtained by the cross-correlation coefficient algorithm. Fig.3 is the time series

diagram of the corresponding differences between the overall tilt signals given by the two

algorithms. It may be seen from this figure that the overall tilt signals yielded by the two

algorithms agree with each other rather well. Besides, the coherent length of atmospheric

turbulence (~0) calculated with the relation between the variance of tilt error signal and

rs is 9.6cm and this agrees rather well with the TO obtained from solar surface granulation

contrast (10 cm)141.

In order to judge the feasibility of the two algorithms, we compare the image qualities of

long-exposure images (consisting of 50 consecutive short-exposure images) before and after

elimination of the overall tilt (similar to the open-loop long-exposure images and tilt-adjusted

long-exposure images in self-adaptive optical systems). As measure of image quality, we use

solar granulation contrast, defined by

Y’ (Arhns x 100% I

7

where (*bms is the root-mean square value of intensity fluctuation of the image and f is

the corresponding mean value.

Next, we take the relative enhancement in the granulation contrast of the tilt-corrected

long-exposure image as a measure of the effectiveness of the tilt correction:

< = -It - To

70

x 100% )

where 70 and -n are, respectively, the contrast of the long-exposure solar granulation image

before and after the elimination of tilt.

Figs.4-8 present the long-exposure images before and after elimination of overall tilt

and, for comparison, the corresponding single-frame short-exposure images. In these figures

(a) is a long-exposure image, (b) and (c) are the long-exposure images after elimination of

tilt, with the overall tilt signals calculated respectively by the cross-correlation coefficient

and absolute difference algorithm, and (d) is a single-frame short-exposure image.

120 RAO Chang-hui e~ al. / Chinese Astronomy and Astrophysics 26 (2002) 115-124

2 4 6 8

o 10 12

~ 14 16 18 20

2 4 6 8

~, 10

14 16 18 20

5 10 15 20 5 10 15 al 'cscc arcsec

(a) open loop. Y =4.18% (b) CC. Y=5.08%

I

5 10 15 20 5 10 15 20 arcsec a rcsgc

(c) AD. Y =5.07% (d) short exposure. 7"=5.67%

20

Fig. 4 Solar granulation image in a 20 arcsecx20 arcsec region. (a) Long-exposure image, (b) and (c), long-exposure images after elimination of tilt, calculated respectively by cross-correlation

coefficient and absolute difference, (d) a single-frame short-exposure image

Fig.4 shows the granulation image in a field of 20.2 arcsec x 20.2 arcsec. The granulation images were used as reference images for extraction of tilt signals. Figs. 5 and 6 refer to two sub-regions of Fig.4(the distance between their centers is about 4.5 arcsec).

Fig.7 presents the long-exposure image, tilt-corrected long-exposure images and single- frame short-exposure image of a sunspot and granulation. Filed size: l l . 7 a r c s e c x 8 .1arcsec .

The sunspot was used as reference signal to extract the tilt signal. Fig.8 refers to the left and upper sub-region of Fig.7, about 11.2 arcsec from the center of the reference region.

Table 2 lists the contrast enhancement factors in Figs. 4,5,6,8. For comparison, the limiting enhancement factor is also given. This limit is defined to be the enhancement factor of single-frame short-exposure solar granulation image relative to the long-exposure image. It expresses the limit of enhancement of contrast after elimination of tilt.

RAO Chang-hui et al. / Chinese Astronomy and Astrophysics 26 (2002) 115-124 121

2 0 03

3

1 2 3 4 arcscc

(a) open loop. )'=4.94%

2 0

4

1 2 3 4 aI'CSCC

(b) cc. r=6.64%

Fig. 5

O O

2 ¢.3 O

4

1 2 3 4 1 2 3 4 arcsec arcscc

(C) A D. Y =6.64% (d) short exposure. Y =7.03%.

Solar granulation image in a 4.7 arcsecx 5.7 arcsec region-the reference region. Meaning of (a)-(d) same as in Fig.4

Table 2 The contrast enhancement factor of long-exposure images of solar granulation

~'o ~(CC) ~(AD) ~(limit) Notes 4.18% 21.53% 21.29% 35.64% Fig.4 4.94% 34.41% 34.41% 42.31% Fig.5 4.04% 32.43% 32.43% 44.06% Fig.6 3.91% 20.72% 20.46% 57.03% Fig.8

From Figs.4-8 and Table 2 the following may be noted: (1) For extended, low-contrast objects, such as solar surface granulation and spots, it is

feasible to use the algori thms of cross-correlation coefficient and absolute difference to carry out correlation tracking.

(2) Elimination of tilt can effectively improve the contrast of open-loop, long-exposure images of solar granulation and enhance the imaging resolution.

(3) The size of the improvement in contrast depends on the size of the viewing field,

122 RAO Chang-hui et al. / Chinese Astronomy and Astrophysics 26 (2002) 115-124

2

4

1 2 3 4 5 1 2 3 4 a r c s e c a r c s ¢ c

(a) open loop. T=4.04% (b) CC. Y =5.35%

¢

e

l

2: g ~3

4

5

1 2 3 4 5 1 2 3 4 5 a.rcgcc a r cscc

(c) AD. Y=5.35% (d) short exposure. T=5.82%

Fig. 6 Solar granulation image in a neighboring region about 4.5 arcsec from the reference region. Meaning of (a)-(d) same as in Fig.4

the larger the field, the smaller the improvement. (Compare Figs. 4 and 5.) (4) The size of the improvement depends also on the size of the reference image. The

larger the reference image, the smaller the improvement in contrast. (Compare Figs. 6 and 8).

(5) The size of the improvement depends, lastly, on the angular distance between the target region and reference region, the larger the separation, the smaller the improvement (compare Figs. 5 and 6). This is caused by the non-isohalation of wave-front tilt.

4. C O N C L U S I O N S A N D D I S C U S S I O N

On the basis of a brief introduction of the principles of two tracking algorithms of extended objects by cross-correlation coefficient and absolute difference, this paper is mainly devoted to an analysis of signals of overall wave-front tilt after processing of the collected images of solar surface granulation and spots and to a comparison of the images before and after the elimination of overall tilt. The results show that by means of correlation tracking with the two algorithms the error signal of overall wave-front tilt may be effectively extracted. The tilt-corrected, long-exposure images of solar granulation have an enhanced contrast. The size of the improvement depends on the size of the target field, on that of the reference

RAO Chang-hui et a L / Chinese Astronomy and Astrophysics 26 (2002) 115-124 123

Fig. 7

5 10 a,rcscc a rcscc

(a) open loop. (b) CC.

15 20

2 4 6 8

o° 10 ~ ~2

14 16 18 20

2 4 6 8

14 16 18 21)

5 I0 15 20 5 I0 15 arcsec arcsec

(c) AD. (d) short exposure.

20

Sunspot and granulation image in a 11.7 arsec field. Tracking was on the sunspot. Meaning of (a)-(d) same as in Fig.4

image and on the angular distance between the current and reference region: an increase in any of these three factors decreases the improvement in contrast. We should point out here that in this work we have not made a thorough comparison of the merits and shortcomings of the two tracking algorithms (of cross-correlation coefficient and absolute difference) for extended, low-contrast objects. We also have not performed a deep study of the influence of the contrast and extension of the object under investigation on the detecting accuracy of these two algorithms. These will be our next topic of research.

A C K N O W L E D G E M E N T In this work much assistance has been offered by colleagues JIANG Ling-tao, XU Bing, JIANG Ping, REN Guo-qiang, ZHOU Wei-chao and TAO Hui- min of Institute of Optics and Electronics of Chinese Academy of Sciences, FANG Cheng, DING Ming-de and GAO Xiu-fa of Department of Astronomy of Nanjing University, as well as Steven Hegwer and Tim Henry of U.S. National Solar Observatory. To all them, the authors would like to express sincere thanks.

124 RAO Chang-hui et al. / Chinese Astronomy and Astrophysics 26 (2002) 115-124

Fig. 8

0.5 1

1.5 2

2.5 ou

3 3.5

4

4.5 5

0.5 I

1.5

2 2.5

r~

3 3.5

4

4.5 5

1 2 3 4 5 1 2 3 4 a l ' c s e c a l ' c s e c

(a) open loop. Y=3.91% (b) CC. 2'=4.72%

0.5 1

1.5 2

8 2.5 t~ ~ 3

3.5 4

4.5 5

0.

I.

~2. r~

3.

4°

1 2 3 4 5 1 2 3 4 5 a r c s e c a l ' c sec

(c) AD. )'--4.71% (b) short exposure. Y=6.14%

Granula t ion image in the upper-left corner region of Fig.7, about 11.2 axcsec away from

the reference region. Meaning of (a)-(d) same as in Fig.4

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