Application of wavelet transform to hologram analysis: three-dimensional location of particles

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Abstract

The wavelet analysis provides an efficient tool in numerous signal-processing problems and has been implemented in optical processing techniques, such as in-line holography. When the diffraction pattern recorded on a hologram is analyzed by means of a wavelet transform, the 3-D location of small particles can be determined very accurately. The diffraction process can, in fact, be interpreted as a convolution with a family of wavelet functions, or, merely, as a wavelet transform. The scale parameter of the wavelet family is related to the axial distance z that the wave propagates. The original field is then reconstructed by searching for the optimum value of the scale parameter which produces a maximum of the wavelet transform modulus. The technique proposed is implemented and experimental results are presented.

Introduction

In-line (Gabor) holography is applied to visualize fluid flows seeded with small particles [1], [2]. Such holograms record the far-field diffraction pattern of particles so that a large field is stored. From the reconstructed field, the three-dimensional location, velocity and size of the particles can be determined in order to characterize the fluid flow. However, when the particle density is high, the reconstruction step is tedious because it is necessary to focus on each particle image. Previous papers have dealt with automatic approaches using a vision system [3], [4]. The location of each image can be found by searching for the best focus plane. Such a method is not very reliable and suffers from the noise produced in the medium by the scattering of the nearest particles. Thus, the reconstructed images are not of good contrast. Moreover, the particle depth cannot be estimated with useful accuracy due to the great depth of field in the paraxial viewing of the reconstructed images. Stanton et al. found that, by analyzing the variations of the spot diameter for several non-image planes, particle size and location could be recovered with greater accuracy [5]. However, this procedure is difficult to implement for non-spherical particles.

An alternative solution which does not need focusing is the direct analysis of the diffraction patterns by means of space-frequency operators [6]. Onural [7] proposed to compute the Wigner distribution function to extract the 3-D coordinates of particles. Our aim is to reconstruct successive planes of a sample volume by using the wavelet transform (WT). The one-dimensional case (straight fibres) has already been studied in our laboratory [8], [9]. The proposed application rely on the hypothesis developed by Onural [10]. According to this approach, the diffraction process can be seen as a convolution between the amplitude distribution in the object plane and a family of wavelet functions. The axial distance z between the object and the observation plane is related to the scale parameter a of the wavelet family by the expression: a=λz/π (where λ is the wavelength of the recording laser). The reconstructed original field can, therefore, be obtained from the diffracted field by an inverse wavelet transform.

In this paper, we first describe the principle of the wavelet transform and its application to in-line holography. In Section 2, we present the implementation of the WT for the digital reconstruction of particle holograms. We show that the best estimation of the axial location corresponds to the maximum of the wavelet transform modulus. In Section 3, experimental results are compared with those obtained from a conventional holographic reconstruction. Then an illustration of our method is given in the case of a spray characterization.

Section snippets

Hologram analysis by the use of wavelet transform

If the object field to be studied is not too large, the diffraction pattern of particles can be directly recorded by means of a CCD camera. On the other hand, if the experiment requires the knowledge of the whole of a large volume (engines applications for example), a holographic plate should be used for the recording step. For the reconstruction step, the fringes can still be scanned by a CCD camera using a relay lens. Afterwards, the particle images are localized and visualized by numerical

Preliminary study: calibration of the method

In order to calibrate the method, the diffraction patterns of several particles with known diameters (d⩽60 μm) are directly recorded onto a CCD camera. We have plotted on Fig. 3 the estimated values of the axial distance zest as a function of the recording axial distances zo. This graph indicates that the axial coordinate of particles can be measured with a good agreement.

The mean slopes s40 and s60 of the curves, corresponding to the diameters 40 and 60 μm, are higher than s20 and s30. This

Conclusion

Automatic reconstruction of holograms is difficult when multiple objects are recorded in a strongly perturbed media. We propose a digital method, tested experimentally, which improves the reliability of the reconstruction step on each image. By computing the wavelet transform of the diffraction patterns recorded on a holographic plate, the 3-D location of particles can be recovered with high accuracy (Δz≈50μm for the axial location instead of 0.5 mm by a conventional optical reconstruction). The

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