Optical correlation

The general aim of optical correlation is to make use of the unique spatial filtering properties of optical systems for the analog computation of the correlation product of two images. This mathematical operation is a basic tool for many problems in image processing and pattern recognition. When I first met with the subject in 1992, at the beginning of my doctoral research, the basic principles were known and demonstrated since quite a long time, but the recent availability of reliable spatial light modulators (SLM's) with good resolution shed a new light on optical correlation. For the first time, programmable and numerically interfaced optical correlators could be achieved. The remaining problems to be solved were mainly a better signal processing formulation of correlation filters, their optical implementation on SLM's, and better overall performances for optical correlators.

Theoretical studies (signal processing)

With Philippe Réfrégier, we investigated the regularization of linear correlation filters [3], that is the stability of the output of a correlator as a function of variations in input images, and especially of noise properties. Significantly, we established that optimal trade-off filters, previously introduced by Philippe Réfrégier, could be interpreted as regularized versions of the inverse filter. Then, with François Goudail, we studied the influence of non overlapping noise on linear correlation filters [4]. Later, I proposed an approach based on Bayesian detection theory that generalizes and encompasses many previous heuristic approaches [8].

During a collaboration with Philippe Réfrégier and Barham Javidi (university of Connecticut), we gave a backing for non linear correlation based on multi-criteria optimization[1, 3], thus formalizing previous heuristic approaches that had shown that non linear correlation is adaptive and robust to variations in input images, and then performs better than linear correlation when the operating conditions are not precisely known.

Optical implementation of correlation filters

During my doctoral work, I have studied the problem of the optimal implementation of correlation filters on SLM's. This question is of utmost importance for the realization of efficient optical correlators that perform at the limits prescribed by signal processing principles. In the literature, several solutions had been given for particular coding domains, fro instance pure phase or binary phase. I proposed an original and efficient method of constrained multi-criteria optimization that can be used for any coding domain [2]. This method can be seen as an optical implementation algorithm of optimal trade-off filters that preserves the optimality of the solution (Fig. 1). In the frame of the doctoral work of Jérôme Colin, whose goal was the realization of a high-speed nonlinear photorefractive optical correlator, we have extended the constrained multi-criteria optimization method to nonlinear correlation [10].

With Anders Grunnet-Jepsen and Sylvie Tonda, we proposed two practical approaches to the implementation of adaptive filters in a linear optical correlator. We first showed that optimal trade-off filters could be adequately approached by the convolution of a reference image with some small kernel, typically less than 11 by 11 pixels. This dramatically reduces the size of the filter bank that has to be constructed for every reference object [5]. We then proposed an adaptive estimation method for the power spectral density of the input scene image [9], that chooses in a filter bank the best suited filter according to the situation.

Experimental demonstrations

During my doctoral research, I studied a correlation architecture based on the shadow casting principle. The goal was was to try and equal coherent optical correlators (Vander Lugt and joint transform correlators mainly) in terms of performances, while retaining the small cost and robustness of incoherent optics. I studied in detail the optical architecture and demonstrated the existence of a trade-off between photometric properties and the resolution loss caused by diffraction [6, 7]. I adapted my multi-criteria constrained optimization method, used in this case following a bipolar scheme, and demonstrated experimentally the efficiency of this approach [6] (Fig. 2). This incoherent correlator is patented.

References

1. Ph. Réfrégier, B. Javidi, and V. Laude, ``Non linear joint Fourier transform correlation: an optimal solution for adaptive image discrimination and input noise robustness,'' Opt. Lett. 19, 405-407 (1994).
2. V. Laude and Ph. Réfrégier, ``Multicriteria characterization of optimal Fourier spatial light modulator filters,'' Appl. Opt. 33, 4465-4471 (1994).
3. Ph. Réfrégier, V. Laude, and B. Javidi, ``Basic properties of nonlinear global filtering techniques and optimal discriminant solutions,'' Appl. Opt. 34, 3915-3923 (1995).
4. F. Goudail, V. Laude, and Ph. Réfrégier, ``Influence of nonoverlapping noise on regularized linear filters for pattern recognition,'' Opt. Lett. 20, 2237-2239 (1995).
5. A. Grunnet-Jepsen, S. Tonda, and V. Laude, ``Convolution-kernel-based optimal trade-off filters for optical pattern recognition,'' Appl. Opt. 35, 3874-3879 (1996).
6. V. Laude, P. Chavel, and Ph. Réfrégier, ``Implementation of arbitrary real-valued correlation filters for the shadow-casting incoherent correlator,'' Appl. Opt. 35, 5267-5274 (1996).
7. V. Laude, ``Diffraction analysis of pixelated incoherent shadow casting,'' Opt. Commun. 138, 394-402 (1997).
8. V. Laude and S. Formont, ``Bayesian target location in images,'' Opt. Eng. 36, 2649-2659 (1997).
9. V. Laude, A. Grunnet-Jepsen, and S. Tonda, ``Input image spectral density estimation for real-time adaption of correlation filters,'' Opt. Eng. 38, 672-676 (1999).
10. J. Colin, N. Landru, V. Laude, S. Breugnot, H. Rajbenbach, and J.-P. Huignard, ``High-speed photorefractive joint-transform correlator using optimized nonlinear filters,'' JEOS A 1, 283-285 (1999).