Bryan M. Hennelly

Optical image encryption by random shifting in fractional Fourier domains

A number of methods have recently been proposed in the literature for the encryption of two-dimensional information by use of optical systems based on the fractional Fourier transform. Typically, these methods require random phase screen keys for decrypting the data, which must be stored at the receiver and must be carefully aligned with the received encrypted data. A new technique based on a random shifting, or jigsaw, algorithm is proposed. This method does not require the use of phase keys. The image is encrypted by juxtaposition of sections of the image in fractional Fourier domains. The new method has been compared with existing methods and shows comparable or superior robustness to blind decryption. Optical implementation is discussed, and the sensitivity of the various encryption keys to blind decryption is examined.

Fast numerical algorithm for the linear canonical transform.

The linear canonical transform (LCT) describes the effect of any quadratic phase system (QPS) on an input optical wave field. Special cases of the LCT include the fractional Fourier transform (FRT), the Fourier transform (FT), and the Fresnel transform (FST) describing free-space propagation. Currently there are numerous efficient algorithms used (for purposes of numerical simulation in the area of optical signal processing) to calculate the discrete FT, FRT, and FST. All of these algorithms are based on the use of the fast Fourier transform (FFT). In this paper we develop theory for the discrete linear canonical transform (DLCT), which is to the LCT what the discrete Fourier transform (DFT) is to the FT. We then derive the fast linear canonical transform (FLCT), an N log N algorithm for its numerical implementation by an approach similar to that used in deriving the FFT from the DFT. Our algorithm is significantly different from the FFT, is based purely on the properties of the LCT, and can be used for FFT, FRT, and FST calculations and, in the most general case, for the rapid calculation of the effect of any QPS.

Image encryption and the fractional Fourier transform

Summary A number of method have been recently proposed in the literature for the encryption of 2-D information using optical systems based on the fractional Fourier fransform, FRT. In this paper a brief review of the methods proposed to date is presented. A measure of the strength/robustness of the level of encryption of the various techniques is proposed and a comparison is carried out between the methods. Optical implementations are discussed. Robustness of system with respect to misalignment and blind decryption are also discussed.

Reduction of speckle in digital holography by discrete Fourier filtering

We present a digital signal processing technique that reduces the speckle content in reconstructed digital holograms. The method is based on sequential sampling of the discrete Fourier transform of the reconstructed image field. Speckle reduction is achieved at the expense of a reduced intensity and resolution, but this trade-off is shown to be greatly superior to that imposed by the traditional mean and median filtering techniques. In particular, we show that the speckle can be reduced by half with no loss of resolution (according to standard definitions of both metrics).

Resolution limits in practical digital holographic systems

We examine some fundamental theoretical limits on the abil- ity of practical digital holography DH systems to resolve detail in an image. Unlike conventional diffraction-limited imaging systems, where a projected image of the limiting aperture is used to define the system performance, there are at least three major effects that determine the performance of a DH system: i The spacing between adjacent pixels on the CCD, ii an averaging effect introduced by the finite size of these pixels, and iii the finite extent of the camera face itself. Using a theo- retical model, we define a single expression that accounts for all these physical effects. With this model, we explore several different DH record- ing techniques: off-axis and inline, considering both the dc terms, as well as the real and twin images that are features of the holographic record- ing process. Our analysis shows that the imaging operation is shift vari- ant and we demonstrate this using a simple example. We examine how our theoretical model can be used to optimize CCD design for lensless DH capture. We present a series of experimental results to confirm the validity of our theoretical model, demonstrating recovery of super- Nyquist frequencies for the first time.

Extended focused imaging for digital holograms of macroscopic three-dimensional objects

When a digital hologram is reconstructed, only points located at the reconstruction distance are in focus. We have developed a novel technique for creating an in-focus image of the macroscopic objects encoded in a digital hologram. This extended focused image is created by combining numerical reconstructions with depth information extracted by using our depth-from-focus algorithm. To our knowledge, this is the first technique that creates extended focused images of digital holograms encoding macroscopic objects. We present results for digital holograms containing low- and high-contrast macroscopic objects.

Random phase and jigsaw encryption in the Fresnel domain

A number of methods have been recently proposed in the literature for the encryption of 2-D information using optical systems based on linear transforms, i.e., the Fourier (FT), the fractional Fourier (FRT), the Fresnel (FST), and the linear canonical transform (LCT). We propose and examine two encryption schemes using the Fresnel transform (FST) based on the use of random phase screens and random jigsaw transforms (JT). The strength and robustness of the level of encryption of the various techniques are examined with respect to blind decryption. These systems are compared with similar FRT- and LCT-based methods. Optical implementations are also discussed and sampling conditions are investigated in the context of the space-bandwidth product.

Analytical and numerical analysis of linear optical systems

The numerical calculation of the Fresnel transform (FST) presents significant challenges due to the high sampling rate associated with the chirp function in the kernel. The development of an efficient algorithm is further complicated by the fact that the output extent of the FST is dependent on the propagation distance. In this paper, we implement a recently proposed technique for efficiently calculating the FST in which we apply the Wigner distribution function and the space bandwidth product to identify suitable sampling rates. This method is shown to be suitable for all propagation distances. Our method can also be applied to describe the effect of a thin lens modeled as a chirp modulation transform (CMT). Combining our results for the FST and the CMT, we numerically calculate the light distribution at the output of both Cai-Wang and Lohmann Type-I optical fractional Fourier transform (OFRT) systems. Analytic solutions for the OFRT of rectangular window and circular apertures are presented. The analytical solutions are compared to experimental data and to numerical results for equivalent cases. Finally the numerical method is applied to examine the effect that apertured lenses, in the OFRT system, have on the output distribution.

Extended viewing angle holographic display system with tilted SLMs in a circular configuration

This paper presents an extended viewing angle holographic display for reconstruction of real world objects in which the capture and display systems are decoupled. This is achieved by employing multiple tilted spatial light modulators (SLMs) arranged in a circular configuration. In order to prove the proper reconstruction and visual perception of holographic images the Wigner distribution function is employed. We describe both the capture system using a single static camera with a rotating object and a holographic display utilizing six tilted SLMs. The experimental results based on the reconstruction of computer generated and real world scenes are presented. The coherent noise removal procedure is described and implemented. The experiments prove the possibility to view images reconstructed in the display binocularly and with good quality.

Using Commodity Graphics Hardware for Real-Time Digital Hologram View-Reconstruction

View-reconstruction and display is an important part of many applications in digital holography such as computer vision and microscopy. Thus far, this has been an offline procedure for megapixel sized holograms. This paper introduces an implementation of real-time view-reconstruction using programmable graphics hardware. The theory of Fresnel-based view-reconstruction is introduced, after which an implementation using stream programming is presented. Two different fast Fourier transform (FFT)-based reconstruction methods are implemented, as well as two different FFT strategies. The efficiency of the methods is evaluated and compared to a CPU-based implementation, providing over 100 times speedup for a hologram size of 2048 times 2048.