i,th pair X(i,), Y(i l ) , and X(i2), Y(i2), i, > 0, and i, > i,. Let arg [ 3 denote phase angle, 6, the rotation phase, and 4s the shifting phase. According to the above properties, we have

4r + i1ds = arg CWAl - arg CX(i,)l = A&, 4, + i, & = arg [Y(i2)l - arg [X(i,)l = ARil

In these two equations, there are two unknowns. We can easily determine

and

also

Now, Y(0) is affected by all kinds of motion, and translation information is only contained in Y(0). With rotation phase, b,, and scaling factor, k,, known, the contribution of rotation and scaling to Y(0) is esliminated so that only the translation com- ponent z,, is left in Y(0). If we call this T(0)

and zo is just

zo = P(0) - X ( 0 )

In the above calculations, we can see that it is assumed that the overall motion consists of an initial translation followed by scaling and rotation. This is a satisfactory assumption because planar motion of a rigid body can always be synthe- tised in this way.

Experimental results: Experiments have been carried out using the ‘Claire’ sequence, from which image segments are extracted for test purposes. The results of the tests can be summed up as follows:

(1) Pure translation (data in pixel distances)

mation error.

(2) Saling and translation (data in pixel distances)

Translation of (2, - 5), (5, - IO), (20, -40), with zero esti-

Actual data Estimated data

Scale factor Translation Scale factor Translation

1.1 0, 0 1.06 0, 0

1.2 IO, - 20 1.19 9.9, -20.1

1.1 0, -10 1.06 -0.3, -10.9 1.2 10, -30 1.19 10.1, -30.2

(3) Pure rotation (data in degrees)

Actual angle 2.0 5.0 10.0 20.0 30.0 -5.0 -10.0 Estimatedanele 2.1 4.7 11.0 21.0 31.0 -7.2 -12.1

(4) General motion (data in pixel distances and degrees)

Actual data Estimated data

Scale Scale factor Rotation Translation factor Rotation Translation

1.2 10.0 5.0, -20.0 1-19 12.2 4.5, -20.0 1.0 5.0 -5, 10.0 1.0 4.7 -5.3, 9.7

From the above results we see that the F D algorithm works well for translation but gives a less accurate result for rotation and scaling by a fractional number. The reason is as follows: as long as we have two sequences which closely describe the two shapes, the technique can tell exactly how motion has occurred from the original position, size, and orientation. The problem is frequently that, for digital images we cannot have an exact description of the shape movement because of the finite display resolution. In the case of rotation, a horizontal straight line becomes a sawtooth after rotation by lo”. Its complex sequence, on which our calculation is based, now no longer describes a straight line but this sawtooth. That is why the estimator can give an exact answer for translation and scaling by an integer factor hut fail to d o so for rotation and scaling by a fractional factor. Choice of the two pairs of F D terms is another important point. The low frequency terms of the FD represent the basic shape and the high frequency terms describe the detail, and it is the detail which rotation and scaling distorts so the two pairs of the F D terms are chosen from low frequency hand. The two pairs of greatest values are chosen to reduce calculation error.

Conclusion: We can see from the above analysis that we need not compute all the terms of the F D but calculate only the low frequency band, for example, the ten lowest terms, which does not require much computational effort. In motion com- pensation, for example, when coding head and shoulder images, the motion parameters obtained by the above tech- nique are good enough for most of the object. Some detailed parts of the object such as eyes and mouth will have further motion of their own. More accurate motion parameters for the detailed parts can be obtained by combining the above technique with intensity matching techniques such as block matching or pel-recursive algorithms.

Q. WANG 10th September 1990 R. J. CLARKE Department of Electrical and Electronic Engineering Heriot Watt University 31-35 Grassmarket Edingburgh EH1 ZHT, United Kingdom

References

ZAHN, C. T., and ROSKIES, R. 2. : ‘Fourier descriptors for plane closed curves’, IEEE Trans., 1972 C-21, (9 pp. 269-281 PERSOON, E., and N, K. s.: ‘Shape discrimination using Fourier descriptors’, IEEE Trans., 1977, SMC-7, (3), pp. 17C179 WALLACE, T. P., and WINTZ, P. A.: ‘An elkient three dimensional aircraft recognition algorithm using normalised Fourier descrip- tors’, Computer Graphics and Image Processing, 1980, 13, pp. 99-126 WALLACE, T. P., and MITCHELL, 0. R.: ‘Analysis of three dimensional movement using Fourier descriptors’, IEEE Trans., 1980, PAMI-2, (6), pp. 583-588

APPLICATION OF HARTLEY TRANSFORM TO TRACKING MOVING OBJECTS IN NOISY ENVIRONMENTS

Indexing terms: Transform, Noise

An application of the fast Hartley transform for tracking moving objects in noisy environments is proposed. Results of simulation of the approach demonstrates that it is promising.

Introduction: Riddle e t al., proposed an algorithm which can detect the tracks of moving objects in noisy environments, where the noise includes white noise, jitter and clutter.’ This algorithm operates by performing a transform in which all pixels with the same two-dimensional velocity map to a peak in a transform space. The fast Hartley transform is emerging as an alternative for the fast Fourier transform for real signals. The use of peak-tracking in transform space has been suc-

1878 ELECTRONICS LETTERS 25th October 1990 Vol. 26 N o . 22

ground (frame numbers = 64, k = 4). Fig. l b is the result of applying the cosine-area transform to the frame sequence of Fig. la. The time axis is scaled so that it corresponds directly to the frame number. Fig. IC is the Fourier transform of Fig. lb, and Fig. Id is the Hartley transform of Fig. lb. From Fig. IC, we obtain fmx = 4 by the detection of the spectral peak. From Fig. Id, we also obtain fmX = 60 by the detection of spectral peak, because of the Hermitian property of DHT, we obtain fmx from eqn. 5 as

f m x = 64 - f m x = 4

The velocity of the moving object is then calculated by

v,,,, = + = 4/4 = 1

2 00

150[

-2001 I I I I I I I I 0 8 16 24 32 40 48 56 64

time a

6 050 I

frequency b m

O O r 0 751

-O 25 t -0501 I J I I I I I I

0 8 16 24 32 4 0 48 56 6 4

c frequency

Fig. 2 Transforms of noisy target representation SNR = 2dB; jitter = 3 pixels a Cosine area transform b Founer transform c Hartley transform

1880

Using the additive noise model stated in Reference 1, we add white noise and jitter into original signal with SNR = 2dB and 3 pixels of jitter, and test this example with the above steps. Fig. 2a is the result of applying cosine-area transform to the frame sequence with noise, and Fig. 26 is the Fourier transform of Fig. 2a. We can also easily obtain f m x = 4 from Fig. 26. Fig. 2c is the Hartley transform of Fig. 2a, using the same step, we get fMx = 4 from eqn. 5. So this application is also promising by the FHT.

Conclusion: We have successfully replaced the FFT by the FHT, and can apply this to the detection of a two- dimensional moving object, by simply extending eqn. 4 to two-dimensional cosine-area transform formula. The result shows that our new approach is a computationally efliciently algorithm which works in relatively robust environments.

Y . 3 . HUANG’ c.-Y. m u

17th August 1990

‘Department oflnformation Engineering Department of Electrical Engineering Tatung Institute of Technology 40 Chung-Shan North Road 3rd Sec, Taipei, 10451 Taiwan, Republic ofChina

References

1 RAJALA, s. A., RIDDLE, A. N., and SNYDER, w. E.: ‘Application of the one-dimensional Fourier transform for tracking moving objects in noisy environments’, Comput. Vision, Graph., Image Process., 1983, 21, pp. 28&293 LB. H. G., and m, K. s: ‘Using the FFT to determine digital straight line chain codes’, Comput. Vision Graph., Image Process., 1982,18, pp. 359-368

3 mu, c.-Y., and LIU, I . - c . : ‘Using FHT to determine digital straight line chain code’, Electron. Lett., 1989, 25, (24), pp. 1629-1631

4 BRACEWELL, R. N.: ‘Discrete Hartley transform transform’, J. Opt. Soc. Am., 1983,73, pp. 1832-1835

2

5 BRACEWELL, R. N.: The fast Hartley transform’, Proc. IEEE, 1984, 72, pp. 101C-1018

4Gbit/s pin/HBT MONOLITHIC PHOTORECEIVER

Indexing terms: Semiconductor devices and materials, Photore- ceivers, Bipolar devices, Photodetectors

Heterojunction bipolar transistors have been monolithically integrated with a pin photodetector to realise a high speed transimpedance photoreceiver. The OEIC, made from MOVPE-growth InP/lnGaAs heterostructures, had a band- width of H G H z with a transimpedance of 750n. It was successfully operated at 4Ghit/s with a sensitivity of -21 dBm at a wavelength of 1 . 5 ~ .

Introduction: Long wavelength opto-electronic integrated cir- cuits (OEICs) are expected to play an important role in multi- gigabit lightwave systems, mainly because of the potential advantages that integration offers. With rapid progress in crystal growth, fabrication, and circuit design, followed by an ever-widening knowledge base, performance of OEICs is con- tinually improving. This has been particularly true in the case of photoreceivers, in which photodetectors are monolithically integrated with preamplifier circuits. Several receivers in the gigabit range have been reported using e.g., InGaAs pin photodiodes followed by preamplifiers made from AlInAs/ InGaAs high electron mobility transistors (HEMTs),’ or InP field effect transistors (FETs).’ In another approach, a metal- semiconductor-metal (MSM) photodetector has been inte- grated with H E M T s . ~ We have previously demonstrated a 1 Gbit/s OEIC receiver using an InGaAs pin photodiode and a preamplifier made from InPflnGaAs heterojunction bipolar transistors (HBTs).~ The HBTs have several advantages when

ELECTRONICS LETTERS 25th October 1990 Vol. 26 No. 22