h rn 71 v

2 30 3 M TI Fr,

10

0

0 1 2 3 4

MSM dc-bias (V)

(a)

50

40

30

20

10

0

f

0 1 2 3 4

LO Voltage (V)

(b)

Figure 5 (a) Variation of the noise figure of the mixer with the dc bias voltage. l-VLo = 0.5 V, 2-VLo = 1 V, 3-V~o = 1.5 V. (b) Variation of the noise figure of the mixer with the LO voltage. 1 A!& = 3 V, 2--Vd, = 0.5 V

REFERENCES 1. Q. Z. Liu and R. 1. MacDonald, “Controlled Nonlinearity Mono-

lithic Integrated Optoelectronic Mixing Receiver,” IEEE Photon. Technol. Lett., Vol. 5, 1993, pp. 1403-1406.

2. Q. Z. Liu, R. Davies, and R. I. MacDonald, “Fiber-optic Mi- crowave Link with Monolithic OptoelectronIc Up-Converters.” IEEE Photon. Technol. Lett., Vol. PTL-7, 1995, pp. 567-569.

3. T. E. Darcie, S. O’Brein, G. Raybon, and C. A. Burrus, “Subcar- rier Multiplexing for Multiple-Access Lightwave Network,” J. Lightwave Techno[., Vol. LT-5, 1987, pp. 1103-1 110.

4. Q. Z. Liu, R. Davies, and R. I. MacDonald, “Characterization of Fiber-optic Microwave Link with Optoelectronic Upconverter

Using Impulse Response Identification System,” Microwaae Opt. Technol. Lett.. Vol. 10, 1995, pp. 135-138.

5. K. R. Prasad. Q. Z. Liu, and R. I. MacDonald, “Analysis of an Optoelectronic Mixer Based on Metal-Semiconductor-Metal Photodetector,” Microwarie Opt. Technol. Lett., Vol. 10, 1995,

6 . M. T. Abuelma’atti, “A Simple Algorithm for Fitting Measured Data to Fourier-Series Models,” Int. J. Math. Educat. Sci. Technol., Vol. 24, 1993, pp. 107-112.

pp. 210-215.

Received 2-15-96: recised 5-2-96

Microwave and Optical Technology Letters, 13/4, 193-197 0 1996 John Wiley & Sons, Inc. CCC 0895-2477/96

IMAGE INTERPOLATION WITH

GAUSSIAN RADIAL BASIS FUNCTIONS ADAPTIVE RECEPTIVE FIELD-BASED

Farid Ahmed Behrend College School of Engineering and Engineering Technology Pennsylvania State University Erie, Pennsylvania 16563

Steven C. Gustafson and Mohammad A. Karim Center for Electro-Optics University of Dayton 300 College Park Avenue Dayton, Ohio 45469

KEY TERMS Interpolation, radial basis function, receptive field width, Gaussian jiinc- tion, power spectra

ABSTRACT Image intelpolation with radial basis function (RBF) neural networks is accomplished. A n RBF nehvork is first trained with an image so as to satisfv the gray-r>alue constraints at each pixel. Each pixel is then divided into subpiwels, and the subpirel gray ualues are calculated with the trained network. Two-dimensional Gaussian basis functions are used in the neurons of the hidden layer. A distortion measure of interpolation is provided. It may be used to determine an acceptable range of receptii,e field width. This range cun then be employed in an adaptuie scheme for the detemiination of receptiue field width of the Gaussian functions that renders a high-fidelity interpolation with the presercation of edge details. 0 1996 John Wiley & Sons, Inc.

INTRODUCTION Image interpolation deals with the task of producing a high- resolution image from a lower-resolution image. Conven- tional interpolation techniques include linear, bilinear, spline, and the nearest-neighbor interpolations [ 1-41. All of these methods aim at reducing the mean-squared error, but they often perform poorly near edges and corners. Also, perfor- mance degrades for large magnification ratios. In this work we propose a radial basis function (RBF) neural nehvork [5, 61 for high-fidelity image interpolation. The goal is to achieve high-quality (subjectively) images even at large magnification ratios. We particularly emphasize the effect of the receptive

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 13, No. 4, November 1996 197

field widths (standard deviations) of the Gaussian functions on image fidelity, which should be small enough to ensure the preservation of edges and corners. On the other hand, a larger receptive field width ( (T ) is necessary for accomplishing desired smoothness. The effect of (T is investigated. and an adaptive scheme for the determination of u is proposed. This scheme preserves edge information while retaining smooth- ness. Results are compared for the zero-order-hold [8], bilin- ear interpolation. spline interpolation, and RBF interpolation with the same variance throughout.

INTERPOLATION AND RBF NEURAL NETWORK

The problem of interpolation can be stated as follows [51: Given N different input vectors, x,, x , E R". 1 = 1,2. . . . , N, and the corresponding A' real output vectors, y,, y, E R. I = 1 , 2 . . . . , N. find a function F . F : R,, .+ R. satisfying the interpolation conditions

The solution to the above problem is a linear combination of N functions:

where { h ( l / x - s,ll) I i = 1.2, . . . . hi) is a set of N arbitrary functions known as radial basis functions, and 1 1 . I / denotes a norm that is usually taken to be Euclidean.

Poggio and Girosi [7 ] solved the approximation problem with the use of the regularization theory. by finding the function F ( x ) that minimizes the following metric H

where y r is the desired output, F ( s ) is the approximation of the unknown function, and A is a positive real number called the regularization parameter. P is a constraint operator. which embeds a priori information about the solution, and usually rcflects some degree of smoothness in the unknown function. Thercfore, H [ F ( x ) ] expresses the compromise be- tween the error of the approximation and the degree of the smoothness of the unknown function, and more precisely expresses the trade-off between interpolation and generaliza- tion. The solution to the optimization problem of Eq. 3, as suggested by Poggio and Girosi [7]. is a linear combination of the generalized radial basis functions (GRBF), given by

where CA?; x , ) is the Green's function of the differential operator PP. with P being the adjoint operator of P. I f P is an operator with radial symmetry, the Green's function G is radial and the approximating function becomes

where G(x-: x , ) , in the case of a Gaussian function with

Output Layer

Xl

x1

van

Figure 1 Radial basis function neural network

K-dimensional input, is given by

This equation suggests the network structure shown in Figure 1 as a method for its implementation. This is the architecture for an RBF neural network. The RBF neural network is a feed-forward network having an input layer, a hidden layer, and an output layer. The weights of the input-to-hidden-layer neurons are assigned the values of an input training vector. The number of neurons in the hidden layer is equal to the total number of training vectors. In other words, each train- ing vector is represented by one neuron. These neurons compute K-dimensional Gaussian functions.

Now let us suppose h,, represents the Gaussian function output for the ith hidden unit due to the j th input vector:

where CT, is the standard deviation that defines the receptive field width of Gaussian function G, .

In our application, we center a Gaussian function at each pixel of the given image. For a total of N training pixels, let H be the N X N matrix consisting of all such hidden layer outputs. T = [ f , f 2 ... t,]' is the target vector constructed from the gray values of the training pixels. The amplitudes of the Gaussian functions represented by the output layer weight vector W = [ w I w 2 ... w , ] ~ is then obtained from the solu- tion of the simultaneous linear equations

where h , , = h(llx, - .x,ll)

198 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol 13, No 4, November 1996

ADAPTIVE SCHEME FOR u

The receptive field width, known as standard deviation a , determines contribution of each Gaussian function in its neighborhood. The larger the value of u, the greater the smoothing effect. So a small u is desired near edges and corners. Let the neighborhood of a pixel p be defined as

The proposed formulation for u at pixel p is given by

1 U =

a + b [ x 2 + xq + xs + Xs + l / & ( X , + xg + x-/ + x9)1 '

(9)

where, the parameters a and b determine the maximum and minimum values of u and the x i s are edge values in a neighborhood defined by

(10) x i = { 1, for an edge, 0, otherwise.

SIMULATION The input image shown in Figure 2 is a 20 X 20 pixel segment of the eye region of a human face. The location of pixels in the image is represented as an input or training vector so that there are 400 two-element input vectors corresponding to 400 pixel locations. Thus the number of neurons in the hidden layer of the RBF neural network is 400. Each hidden-layer neuron is provided a standard deviation ri, and the center location of each pixel is given by = IQI, 4 1 , which is basically the coordinate of the ith training pixel. These two parameters (a, and U , ) describe the Gaussian function at hidden-layer neuron i.

The network is trained so that the integral of the sum of all the Gaussian functions over each pixel is equal to the gray value of that pixel, as described by Eq. (8). The gray values of all the pixels comprise the target vector T , which is found from the output neuron. Here, W is a 400 X 1 matrix, H is a 400 X 400 matrix, and the result is a 400 X 1 target vector T. The weight matrix W contains the amplitudes of the 400 Gaussian functions and is evaluated by computing H - ' T . For high values of a , the matrix H becomes ill conditioned so that singular-value decomposition or similar methods must be invoked [9]. One of the authors 1101 has shown that the maximum value of u is constrained by the diagonal domi- nance property of the matrix H.

The trained network is then used for the interpolation of the image. A magnification ratio of 2, for example, is investi- gated so that the new interpolated image is of size 40 x 40, which yields 1600 pixel locations. These pixels are used as input to the trained network to obtain 1600 outputs that correspond to the gray values of all the subpixels. The simula- tion was performed for zero-order hold (replication), bilinear interpolation, bicubic interpolation, and RBF interpolation.

Figure 3 shows the corresponding interpolated images. Histograms of the power spectra of these images are dis- played in Figure 4 for the sake of comparison. Note that for replication there is a high-low frequency component due to smooth gray values from zero-order hold: the subpixels of a

Figure 2 Original segment of the eye image

particular pixel of the zero-order hold image have the same gray values. Bilinear and bicubic interpolation reduced this smoothness to yield more detailed information from the image. Figure 4(d) shows that RBF interpolation performs well in terms of keeping both the smooth and the detailed information. In particular, note that it has larger number of pixels with power spectra smaller than 0.1 and larger than 0.6, when compared with bilinear and bicubic interpolation.

In the above simulation the standard deviation a was taken to be 1.0. Figure 5 shows the histogram plots of power spectra for values of a equal to 0.2, 0.5, 1.0, and 1.5, respectively. For smaller values of a , the subpixels are influ- enced mostly by the corresponding pixel and the contribution from neighboring pixels is reduced. As a result, the gray values of the subpixels are close to the gray values of the original pixel, as in the zero-order hold case. On the other hand. higher values of a result in greater contributions from neighboring pixels and a smearing effect. Specifically, note the poor performance with higher values of a : There are more contouring effects with less detailed and smooth infor- mation. The interpolated image corresponding to a = 1.0 is a trade-off between these two cases. Here the power spectrum is wider than that for higher a , so contrast is improved.

Interpolation was also performed with the adaptive r described by Eq. (9). Figure 6 shows the corresponding power spectrum plot. Note particularly how the smooth and detailed information of the image are maintained more effectively, compared to the previous cases. Note that the choice of the parameter values of a and b in Eq. (9) was influenced by the numerical intractability resulting from higher and lower val- ues of u. We chose a = l/amdx and b is then obtained from Eq. (9). For an objective evaluation of the interpolation performance, we calculated the distortion measure d of the interpolated images as follows:

where I is the interpolated image in consideration and I , is the image obtained from replication of the original image. Distortion is calculated with respect to the replicated image, so that the dimensionality of the images is the same. Figure 7 shows the variation of distortion with respect to the standard-deviation values. The plot is shown on an arbitrary

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 13, No 4, November 1996 199

Figure 3 Images interpolated by a magnification factor of 2 with (a ) replication, (b) bilinear interpolation, (c) bicubic interpolation, and (d) RBF interpolation

=-n- SH14 -10

I ' . 1 . ' ' I ' ' ' . 1 I ' I ' . ' 111 ' , . .

Figure 4 Histogram plots of power spectra corresponding to (a ) replication. (b) bilinear interpolation. ( c ) bicubic interpolation, and ( d ) RBF interpolation

200 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS Vol 13 No 4, November 1996

Figure 5 Histogram plots of power spectra for RBF interpolation with different values of the standard deviation: (a) 0.2, (b) 0.5, (c) 1.0, and (d) 1.5

Adadwe standard deviation

Figure 6 Histogram plot of power spectrum for adaptive receptive field width

scale for clarity. The minimum value of distortion was ob- tained at u = 0.9. This value of 0.13 is found to be less than the distortion of 0.154 for the case of bilinear interpolation. An acceptable range of u values can also be obtained with this measure, as shown in Figure 7. For the adaptive case, the distortion value is found to be 0.162, which is larger than the minimum obtainable with CT = 0.9. The increased distortion here accounts for the improved edge preservation with adap- tive u.

CONCLUSION The interpolation of images with radial basis functions is investigated. It is shown that the choice of the values of the receptive field widths of the hidden-layer Gaussian functions in RBF neural networks is very important for image enhance- ment and fidelity. An adaptive scheme for u selection that

I 0.5 1 1.5

0.1 1 -dud dsvi.tbn

Figure 7 Distortion measure of the interpolated images

allows smaller widths near edge locations and larger widths for isolated points is found to result in an enhanced image that preserves edges.

REFERENCES 1. G. Chen, D. Figueiredo, and J. P. Rui, “Unified Approach to

Optimal Image Interpolation Problems Based on Linear Partial Differential Equation Models,” IEEE Trans. Image Proc., Vol. 2,

2. De Natale, G. B. Francesco, S. G. Desoli, D. Giusto, and V. Gianni, “Spline-Like Scheme for Least-Squares Bilinear In- terpolation of Images,” in ?’roc. IEEE Int. Conf Accoust., Speech, Signal Processing, Vol. 5, 1993, pp. 141-144.

3. S. W. Lee and K. J. Paik, “Image Interpolation Using Adaptive Fast B-Spline Filtering,” IEEE Int. ConJ Accoust., Speech, Signal Processing, Vol. 5, 1993, pp. 177-180.

4. Robert H. Bamberger, “Method for Image Interpolation Based on a Novel Multirate Filter Bank Structure and Properties of the Human Visual System,” Proc. SPIE, Vol. 1657,1992, pp. 351-362.

NO. 1, 1993, pp. 41-49.

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 13, No. 4, November 1996 201

5 . T. Poggio and F. Girosi, "Networks for Approximation and Learning, Proc. ZEEE. Vol. 78. 1990. pp. 1481-1495.

6. P. D. Wasserman, .4di anced Methods irr Neural Cornpicring. Van Nostrand Reinhold, New Yixk. 1993.

7. T. Poggio and F. Ciirosi. "Regularization Algorithms for Learn- ing that are Equivalent tu Multilayer Networks." Science. 1990. pp. 978-982.

8. A. K. Jain, Fundnm~nruls of Digifal I ~ i a g e Processing. Prentice Hall, Englewood Cliffs. YJ. 1989.

9. W. Cheney and D. Kincaid. Niitnen'cal ,Mathematics and Comput- ing, Brooks-Cole, Pacific Grove. CA. 1985.

10. Steven C. Gustafson. Gordon R. Little. John S. Loomis, and Todd S. Puterbaugh. "Optimal Reconstruction of Mising-Pixel Imagcs. Appl. Opr.. Vol. 31. 1992. pp. 6879-6830.

Receri eci 5-27-96

COMPARATIVE PROPAGATION STUDY OF PULSE-MODULATED MICROWAVES Thad G. Rolling and T. Koryu lshii Department of Electrical and Computer Engineering Marquet:e University Milwaukee Wisconsin 53233

I. INTRODUCTION

In previous works, it was shown that the signal velocity of pulse-modulated microwaves is superluniinal under certain conditions [l-31. However. there also have been reports to the contrary [4, 51. The purpose of this article is to confirm the superluminal propagation speed in a waveguide by an alternate experimental approach. The leading edge of the pulse-modulated microwaves under investigation is consid- ered the part of the pulse that exists just above the noise threshold level of the system. The signal velocity [6, 71 in this article is defined as the speed of a point on the detected pulse that exists just at the noise threshold level of the system. The experiment shown here will show by direct com- parison that the signal velocity inside a waveguide is greater than the signal velocity in open air. known to be approxi- mately equal to 3 x I ( ) * m/s.

II. EXPERIMENTAL SETUP AND PROCEDURE

The experimental setup shown in Figure 1 was constructed to compare the speed of propagation or pulse-modulated mi- crowaves in open air to the speed of propagation in a wave- guide at the noise threshold. The experimental block diagram is given as Figure 2. External modulation of the reflex klystron

was accomplished using a BK Precision 3011B function gen- erator to modulate the repeller voltage of the HP715A power supply. This was accomplished by connecting the inner con- ductor of the coaxial output jack of the function generator to the repeller jack of the HP71SA with a 10-nF capacitor in series. The function generator provided the trigger input to the Tektronix 46SB oscilloscope. The generated pulses have a pulse repetition rate of 100 kHz and a pulse width of 1.88 ps. The pulse rise time to the double noise threshold level [S] was measured to be 3 ns.

An X-band ferrite isolator, Micro-Radionics 09941-001-02, was used to protect the klystron from reflections. A variable attenuator. HPX382A, and cavity frequency meter, HPX5328, were used to attenuate the pulse-modulated microwaves and measure the carrier frequency, respectively. Two HPX880A E-H tuners were used to match the system.

A slotted line section was used to position the first detcc- tor. which measured the departure time of the signal. This detector was a microwave tunable probe, model FXR B200A, with a 1N23C microwave detector crystal.

The two waveguide switches, HPX614A, were used to switch the system between air-waveguide composite propaga- tion or waveguide-only propagation. With the switches in position A in Figure 2, the signal propagated through 1.105 m o f open air with two pyramidal horn antennas of aperture 6.4 x 5.3 cm. length 12.3 em, and 0.900 m of waveguide for a total distance of propagation between the detectors of 2.005 m. With the switches in position B in Figure 2, the path consisted of 2.005 m of WR-90 waveguide propagation only hetween the detectors.

The second waveguide switch was connected to the second HPXXXOA E-H tuner. The E-H tuner was connected to a waveguide detector mount and a shorting plunger, HPX485B with a 1N23C microwave detector crystal. The arrival time of the pulse was measured with the use of the waveguidc detec- tor mount.

With this experimental setup, the speed of propagation of the leading edge of pulse-modulated microwaves was mea- sured. In order to observe the leading edge of the pulse, at the noise threshold level an important condition of 100% modulation must be insured. This condition is imperative because if microwave energy exists between the pulses, the leading edge of the pulse will not be truly observable. A horizontal time scale of 2 ns per division and a vertical

Figure 1 Photograph o f experimental setup

202 MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol 13 No 4, November 1996