On the Theory of Optical Hilbert transform for incoherent objects

The Hilbert transform as been investigated abundantly in coherent imaging. To the best of our knowledge, it is for the first time investigated in the context of incoherent imaging. We present a two-pupil optical heterodyne scanning system and analyze mathematically the design of its two pupils such that the optical system can perform the Hilbert transform on incoherent objects. Computer simulations of the idea clarify the theoretical results. ©2007 Optical Society of America OCIS codes (090.0090) Holography; (090.1760) Computer holography ; 110.4850 Optical transfer functions; 100.0100 Image processing References and links 1. A. Sagan, S. Nowicki, R. Buczynshi, M. Kowalczyk, and T. Szoplik, “Imaging phase objects with squareroot, Foucault, and Hoffman real filters: a comparison,” Appl. Opt. 42, 5816-5824 (2003). 2. A. W. Lohmann, E. Tepichin, and J. G. Ramirez,” Optical implementation of the fractional Hilbert transform for two-dimensional objects,” Appl. Opt. 36, 6620-6626 (1997). 3. S. Lowenthal and Y. Belvaux, “Observation of phase objects by optically processed Hilbert transform,” Appl. Phys. Lett. 11, 49-51 (1967). 4. R. Gale Wilson, "Wavefront-error evaluation by mathematical analysis of experimental Foucault-test data,” Appl. Opt. 14, 2286-2297 (1975). 5. T.-C. Poon and A. Korpel, “Optical transfer function of an acousto-optic heterodyning image processor," Opt. Lett. 4, 317-319 (1979). 6. T.-C. Poon, “Scanning holography and two-dimensional image processing by acousto-optic two-pupil synthesis," J. Opt. Soc. Am. 2, 521-527 (1985). 7. T.-C. Poon and T. Kim, Engineering Optics with MATLAB, World Scientific (2006). 8. T.-C. Poon and T. Kim, "Optical image recognition of three-dimensional objects," Appl. Opt. 38, 370381(1999).


Introduction
The Hilbert transform has drawn some attentions recently due to its ability for phase-retrieval as well as for image processing [1,2].The classical Hilbert transform can be implemented coherently by π -phase shifting in the Fourier plane of an optical system [3].It is also wellknown that by half-plane filtering in the Fourier plane, we can obtain the Hilbert transform of the original complex light field [4].Unfortunately, the Hilbert transform of the information is superposed coherently with the original light field.Unless the two fields can be separated by some phase extraction methods, we cannot obtain the Hilbert transform of the original light field.For incoherent image processing, up till now, no optical systems have been able to extract the Hilbert transform information from incoherent objects, i.e., the quantity, such as intensity, to be Hilbert transformed is real and non-negative.In this paper, we propose a quarter-plane filtering in the optical transfer function (OTF) domain to extract the Hilbert transform of the incoherent object.The notion of quarter-plane filtering is for 2-D transformation as half-plane filtering only leads to 1-D transformation.The proposed filtering can be performed with a two-pupil optical heterodyne scanning system originally developed by Poon and Korpel [5], and subsequently analyzed by Poon, using an optical transfer function approach [6].In section 2, we first review and formulate the definition of an analytic signal of a function and from which we can obtain the Hilbert transform of the function.For brevity, one-dimension formalism is developed and extension to two dimensions is trivial.In section 3, we briefly discuss the two-pupil system and summarize the relevant results that are useful for our present development of the Hilbert transform.In section 4, we analyze the design of pupils so as to obtain the Hilbert transform and show some 1-D simulations to clarify the idea.Finally, in section 5, we make some concluding remarks.

Analytic signal and the Hilbert transform
Consider a real function ) (x g with its Fourier transform defined by where x and x k are the Fourier transform variables.The integral is known as the analytic signal associated with g(x).
) (x g a + can be written in terms of the inverse Fourier transform as (4) The analytic signal given by Eq. ( 2) is a complex function and can be written as follows: Hence we see that from the definition given by Eq. ( 5), we can extract the original function and its Hilbert transform by simply taking the real part of and the imaginary part of Using Eq. ( 3), we can re-write Eq. ( 7) as We can also create an analytic signal by including negative frequencies only.Similar to the definition given by Eq. ( 2), we then have and hence, for this case, we have We shall design the pupils in the two-pupil optical heterodyne scanning processor so that Eq. ( 10) can be realized.

Two-pupil optical heterodyne scanning processor
Figure 1 shows a typical two-pupil optical heterodyne scanning image processor, which was originally developed and analyzed by Poon  k being the wave number of the light.Note that in Eq. ( 11), only the intensity of the object is processed and hence the system is an incoherent optical system.

Pupil function designs
The processing OTF, given by Eq. ( 12), can be altered by properly designing the pupil functions.It is interesting to point out that Eq. ( 11) has the same functional form as that of Eq. ( 10) by recognizing that the spectrum of , and Ω OTF needs to be designed such that it is given by ) ( ) ( Note the obtained OTF is masked by ) ( ) ( and hence the notion of quarter-plane filtering.With this OTF, Eqs.(11a) and (11b) now can be written, respectively, as , the real-time scanned and processed outputs given by Eq. ( 14a) and (14b) represent the recovery of the original incoherent object, , and its Hilbert transform, respectively by comparing the equations with Eq. ( 10).In a general case, for 0 ≠ z , Eqs. ( 14a) and (14b), after some manipulations, can be worked out, in the spatial domain, to be and they are shown in Figs.2(c) and 2(d), respectively.We clearly observe the reconstruction of the original rectangular object and its Hilbert transform.

Concluding remarks
We have analyzed mathematically a two-pupil optical heterodyne scanning image processor by designing the pupils for the extraction of the Hilbert transform of incoherent objects.We believe this is the first time the Hilbert transform of incoherent objects has been addressed and the result of it could be useful for image processing applications as well as real-time holographic recording of the Hilbert transform of a 3-D object.Experimental investigations are currently underway.

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Figure1shows a typical two-pupil optical heterodyne scanning image processor, which was originally developed and analyzed byPoon [6].We shall briefly describe the principle of operation and then summarize some of the important results, which are relevant to our pursuit of the Hilbert transformation.Beamsplitters BS and BS1, and mirrors M and M1 form the Mach-Zehnder interferometer.The pupil, ) , ( 1 y x p , is illuminated by a collimated laser at