L Denis
email: loic.denis@cpe.fr

C Fournier

T Fournel

C Ducottet

Numerical suppression of the twin-image in in-line holography of a volume of micro-objects

Keywords: numbers: 07.05.P, 42.40 digital holography, in-line holography, deconvolution, twin-image

We address the twin-image problem that arises in holography due to the lack of phase information in intensity measurements. This problem is of great importance in in-line holography where spatial elimination of the twin-image cannot be carried out as in off-axis holography.

A unifying description of existing digital suppression methods is given in the light of deconvolution techniques. Holograms of objects spread in 3D cannot be processed through available approaches. We suggest an iterative algorithm and demonstrate its efficacy on both simulated and real data. This method is suitable to enhance the reconstructed images from a digital hologram of small objects. * Note that we do not consider here the effects due to sampling (pixel integration) and finite sensor size. In practice these affect the system response (i.e. perform a low-pass filtering).

Introduction

Twin (i.e. virtual) images were considered as a limitation of holography since its very beginnings [START_REF] Leith | Reconstructed wavefronts and communication theory[END_REF]. They are defocused images of the holographed objects that superimpose on the real images of the objects in the optical or numerical reconstructions. Their origin is tied to the principle of in-line holography and is recalled at the end of this introductory section.

The twin-image problem has received considerable attention since Gabor's pioneering work on holography. It is still a concern today as the in-line holographic setup is widely used in optical holography, soft X-rays holography and electronic microscopy. Basically, two classes of approaches have been proposed to tackle this problem: a first class which involves a change in the hologram-recording setup, and a second class that attempt to numerically suppress the twin-image. The best known method in the first category is off-axis holography, first suggested by Leith and Upatnieks [START_REF] Leith | Reconstructed wavefronts and communication theory[END_REF]. An offaxis hologram leads to spatially separated real and twin images at the reconstruction step. The real image can then be selected by spatially filtering the reconstructed planes [START_REF] Cuche | Spatial filtering for zero-order and twin image elimination in digital off-axis holography[END_REF]. The digital off-axis setup however proves less relevant for the study of a collection of small objects due to sampling constraints [START_REF] Xu | Properties of digital holography based on in-line configuration[END_REF] and practical considerations (robustness to vibrations). In-line digital holography is therefore preferred in this context and the twin-image problem has still to be addressed. By combining different holograms of the same objects recorded at distances with λ/4 shifts, the phase shifting technique [START_REF] Yamaguchi | Phase-shifting digital holography[END_REF] performs phase reconstruction and therefore achieves twin-image suppression. Optical scanning holography [START_REF] Poon | Twinimage elimination experiments for three-dimensional images in optical scanning holography[END_REF] also reconstructs the phase thanks to an additional modulation/demodulation step. These two techniques are however limited to non-moving objects since the shifts or scans must be performed while the objects remain fixed. They are therefore irrelevant for the study of objects in motion. In this paper, we address the techniques adapted to the study of micro-objects in a volume. We consider only the in-line Gabor setup in the following analysis.

Numerical techniques (i.e. methods of the second aforementioned category) attempt to improve the system response R of the holographic imaging technique (hologram recording + reconstruction) by changing the reconstruction step. Many techniques have been described in the literature as following a different approach. We provide in this paper, a unifying description of some of them in the light of deconvolution techniques in section 2. We show that the case of objects spread in a volume requires the development of a specific algorithm. Section 3 describes the suggested algorithm as an extension of existing iterative techniques. We show results on simulated and real data and discuss the applications and limits of the algorithm in section 4. Before describing twin-image suppression methods, we recall the origin of the twin-image phenomenon and give a quantitative description of twin-images.

Twin-image origin and characteristics: Hologram formation corresponds to the recording of the intensity pattern of the free propagated waves diffracted by the objects.

Hologram reconstruction however does not exactly solves the inverse propagation problem but rather propagates further the intensity pattern. The complete system response R of a point object in the focus plane therefore is not punctual but is the sum of a (real) punctual response and a (twin) defocused pattern [START_REF] Onural | Digital decoding of in-line holograms[END_REF]

* : R(x, y) = δ(x, y) + 2 h 2z (x, y) , (1)
with δ(x, y) denoting Dirac's distribution, the real part, and z the distance separating the point object and the hologram. The symbol h z is Fresnel function of parameter z and is the convolution kernel that models, under Fresnel's approximation, the diffraction at distance z. It is defined by:

h z (x, y) = 1 jλz exp jπ x 2 + y 2 λz . (2)
The symbols x and y are the spatial coordinates in transverse planes (i.e. orthogonal to the optical axis) and z is the depth coordinate (along the optical axis). The symbol λ is the radiation wavelength and j = √ -1. By convention, we underline complex-valued variables in this paper.

It is worth noting that a point object is recorded, in the hologram plane, as an intensity pattern which is classically approximated by the difference: 1 -2 (h z ) [START_REF] Onural | Digital decoding of in-line holograms[END_REF]. This difference can be expanded into three terms: 1 -2 (h z ) = 1 -h z -h -z due to the conjugation property conj(h z ) = h -z . The hologram intensity can then be seen as the diffraction amplitude created by two symmetrical objects (resp. real and virtual) located on both sides of the hologram, hence the twin-image can be interpreted as the diffraction pattern of a virtual object.

In a focus plane, the amplitude of the twin-image depends on that of the real image, on the size of the holographed object and on the recording distance. Let ϑ be the object's aperture. In its focus plane, the reconstructed amplitude is given under a linear system approximation by the convolution with the system response:

[ϑ * R] (x, y) = ϑ(x, y) + 2 ϑ * (h 2z ) (x, y). (3)
We derive in Appendix A an approximation of the twin-image for small and symmetrical objects (i.e. whose largest dimension d verifies the condition d √ 2λz):

[ϑ * R] (x, y) ≈ ϑ(x, y) + 2F[ϑ i ]( x 2λz , y 2λz ) h 2z (x, y) . ( 4
)
This approximation is useful to quantify the amplitude of the twin-image with respect to the real image. The twin-image approximate expression given in equation 4 involves the real part of Fresnel's function with parameter 2z and the Fourier transform of the object's aperture scaled by a factor 1/(2λz). The recording distance plays the role of a scaling factor: the twin-image is dilated when the recording distance increases. Then, the larger the recording distance, the larger the spatial separation between the real image of the object and its superimposed twin-image.

The signal to noise ratio between the real image of an object and its corresponding twin image depends both on the recording distance and the object's size. For a given recording distance z, a dilation of the object by factor a leads -according to the 2D Fourier transform scaling property -to a multiplication of the twin-image's amplitude by a 2 . The wider the object, the higher the corresponding twin-image amplitude. Conversely, for a given object, the twin-image amplitude is proportional to the inverse of the recording distance z through Fresnel's function. Hence, the larger the recording distance, the smaller the amplitude of the twin-image.

Numerical suppression of the twin-image in the light of deconvolution techniques

The many numerical techniques introduced in literature for twin-image suppression can be classified into distinct families of approaches. We will first briefly sum up these approaches and then show that there are strong connections between physicallybased, filtering and iterative approaches. Image deconvolution provides an interesting framework to give a unified description of these methods. On the basis of this description, we derive in section 3 an algorithm adapted to holograms of micro-objects spread in a volume.

Classes of existing approaches

Physically-based techniques: These techniques date back to the origins of holography. Bragg and Rogers [START_REF] Bragg | Elimination of the unwanted image in diffraction microscopy[END_REF] suggested the use of a second (auxiliary) hologram recorded at twice the distance of the main hologram to subtract the twin image. A similar approach can be conducted by subtracting from the reconstructed plane an estimate of the twinimage obtained by reconstructing an out-of-focus plane located at three times the infocus distance. The twin-image is then pushed back at twice its original distance and its amplitude reduced accordingly. This approach has been followed by authors using one [START_REF] Onural | Digital decoding of in-line holograms[END_REF][START_REF] Yang | Reconstruction of near-field in-line hologram[END_REF] or two [START_REF] Zhang | Reconstruction of a complex object from two in-line holograms[END_REF][START_REF] Zhang | Reconstruction of in-line digital holograms from two intensity measurements[END_REF][START_REF] Denis | Twin-image noise reduction by phase retrieval in in-line digital holography[END_REF] holograms.

Inverse filtering approaches: Some authors [START_REF] Maleki | Noniterative reconstruction of complex-valued objects from two intensity measurements[END_REF][START_REF] Nugent | Twin-image elimination in Gabor holography[END_REF] try to invert the system response R by designing an inverse filter in the Fourier domain. Different strategies are proposed to circumvent the filter singularities problem.

Iterative approaches: Iterative techniques have been designed [START_REF] Lannes | Correction des effets de l'image jumelle et du terme quadratique en holographie de gabor[END_REF][START_REF] Koren | Twin-image elimination in in-line holography of finite-support complex objects[END_REF][START_REF] Koren | Iterative algorithms for twin-image elimination in in-line holography using finite-support constraints[END_REF] to improve the reconstructed amplitude, using a finite support constrain.

Phase retrieval techniques:

With two or more holograms, more general phase reconstruction algorithms may also be used. Gerchberg-Saxton algorithm [START_REF] Gerchberg | A practical algorithm for the determination of phase from image and diffraction plane pictures[END_REF][START_REF] Fienup | Phase retrieval algorithms : a comparison[END_REF] has been applied to digital holography in the work of Liu and Scott [START_REF] Liu | Phase retrieval and twin-image elimination for in-line Fresnel holograms[END_REF], and more recently in the work of Zhang et al. [START_REF] Zhang | Whole optical wave field reconstruction from double or multi in-line holograms by phase retrieval algorithm[END_REF]. It iteratively reconstructs the phase in each hologram plane by performing back and forth propagation and enforcing at each step a constraint based on the measured hologram intensities. An improved version [START_REF] Yang | Gerchberg-Saxton and Yang-Gu algorithms for phase retrieval in a nonunitary transform system : a comparison[END_REF] of Gerchberg-Saxton algorithm has been suggested to account for the numerical errors (h z * h -z = δ due to windowing effects).

Maximum a posteriori approach: Sotthivirat and Fessler suggested [START_REF] Sotthivirat | Penalized-likelihood image reconstruction for digital holography[END_REF] a reconstruction of the image of a plane object using a maximum a posteriori approach (i.e. regularized image restoration). In the case of the study of small objects spread in a plane, the definition of an appropriate prior model is however difficult.

Other approaches: A quite different approach has been followed in [START_REF] Lai | Off-axis reconstruction of in-line holograms for twin-image elimination[END_REF]: the hologram is decomposed into blocks and reconstructed as a combination of off-axis sub-holograms. Tiller et al. suggest using two close holograms to deduce the phase on the basis of the connection between intensity deviation between two close planes and phase value.

The deconvolution point of view

Twin-image suppression has already been identified as a deconvolution problem by other authors (see [START_REF] Onural | Digital decoding of in-line holograms[END_REF]). We show here that the deconvolution provides a framework that reveals previously unnoticed connections between various approaches.

Let us consider a collection of N objects defined by their apertures (ϑ i ) i=1..N . In a plane located at distance z from the hologram, the complex amplitude due to the objects is

a z = N i=1 ϑ i * δ x i ,y i * h z i -z
, where (x i , y i ) is the transversal location of object i and z i is the distance between object i and the hologram plane. In the case of a single real-amplitude object, a z = ϑ as is the case for objects spread in a unique plane located at the recording distance z. Note that phase objects such as micro-lenses or biological structures may also be considered, the aperture ϑ then is complex-valued.

The intensity recorded in the hologram plane (after suppression of the continuous background constant) is given by [START_REF] Onural | Digital decoding of in-line holograms[END_REF]:

I H = -2 a z * h z = -a z * h z -conj(a z ) * h -z .
(

) 5
The numerically reconstructed amplitude in the plane z is obtained [START_REF] Kreis | Methods of digital holography : A comparison[END_REF] by convolving the hologram with Fresnel's kernel of parameter -z:

A R = I H * h -z = -a z -conj(a z ) * h -2z (6)
The common principle of all methods for numerical suppression of the twin-image is expressed either: (i) in the hologram plane (Eq. 5), or (ii) in the reconstructed plane (Eq. 6).

Objects within a plane

In this section, we consider that all objects are located at the same distance z (the recording distance) from the hologram and that their transmittance 1 -ϑ i is real-valued. The amplitude a z therefore is real and equations 5 and 6 become:

I H = -2a z * h -z , and (7)
A R = -a z * δ + h -2z . (8)
The twin-image suppression problem then corresponds to a deconvolution problem in both cases: (i) recover a z from I H ; and (ii) recover a z from A R . We describe in the following the main twin-image suppression techniques and underline the connections between each of them using the common framework of deconvolution techniques.

Fourier-quotient method: The basic technique to perform deconvolution is to design a filter in the Fourier domain that inverses the convolution kernel. The transfer function of the inverse filters corresponding to approaches (i) and (ii) are:

F[F (i) I ](u, v) = 1 cos [λzπ(u 2 + v 2 )]
, and

F[F (ii) I ](u, v) = 1 1 + exp -2jπλz(u 2 + v 2 ) . (9)
These filters have a singularity for each zero of their denominator. Different solutions have been suggested:

• the transfer functions are set to one whenever a singularity is reached [START_REF] Maleki | Noniterative reconstruction of complex-valued objects from two intensity measurements[END_REF],

• a positive constant is added to the denominator [START_REF] Nugent | Twin-image elimination in Gabor holography[END_REF],

• the filter is expanded into a series and only the first terms of the series are kept [START_REF] Onural | Digital decoding of in-line holograms[END_REF][START_REF] Yang | Reconstruction of near-field in-line hologram[END_REF].

We give here the series expansion used in [START_REF] Yang | Reconstruction of near-field in-line hologram[END_REF] as it will provide a direct connection with iterative approaches. It is based on the classical expansion 1/(1

+ x) = 1 -x + • • • + (-1) n x n : F[F (ii) I ](u, v) = 1 -exp -2jπλz(u 2 + v 2 ) + exp -4jπλz(u 2 + v 2 ) -• • • +(-1) n exp -2njπλz(u 2 + v 2 ) , (11)
and can be physically interpreted as pushing back the twin-image at distance (2n + 2)z:

-A R * F (ii) I = -A R * δ -h -2z + h -4z -• • • + (-1) n h -2nz = a z + (-1) n a z * h -(2n+2)z . (12)
Iterative approaches: The inverse filter F (ii) I obtained by series expansion can be put into the following recursive form:

a 0 = A R a n+1 = A R -a n * h -2z (13)
This corresponds exactly to the iterative algorithm of van Cittert [START_REF] Van Cittert | Zum Einfluss der Spaltbreite auf die Intensitätverteilung in Spektrallinien[END_REF][START_REF] Hill | Convergence of the van Cittert iterative method of deconvolution[END_REF] applied to deconvolve equation 8.

In the case of small objects, an a priori spatial constraint can be added if the support of the objects is known. Let us note M as the binary function indicating whether point (x, y) is in an object (M (x, y) = 1) or not (M (x, y) = 0). This spatial constraint can be enforced at each iteration (i.e. by projection on the sub-space of solutions that satisfy the constraint):

a 0 = M • A R a n+1 = M • A R -a n * h -2z (14)
This algorithm corresponds to the one introduced by Lannes [START_REF] Lannes | Correction des effets de l'image jumelle et du terme quadratique en holographie de gabor[END_REF]. Koren et al. suggested [START_REF] Koren | Twin-image elimination in in-line holography of finite-support complex objects[END_REF][START_REF] Koren | Iterative algorithms for twin-image elimination in in-line holography using finite-support constraints[END_REF] another iterative algorithm also using a spatial constraint:

     a 0 (twin) = (1 -M ) • A R + M • ĀR , with ĀR the mean value of A R a n+1 (twin) = (1 -M ) • a 0 (twin) + M • (a n * h -2z ) a n = γ(1 -M ) • (a n (twin) * h 2z ) + M • (a n (twin) * h 2z ), with γ ∈ [0, 1]. ( 15
)
This algorithm goes back and forth between two planes corresponding respectively to the twin image a (twin) and the real one a. The underlying idea of this algorithm is that the real and virtual images carry the same information and can be interchanged through a convolution with h 2z and h -2z . Getting a real image without a spurious twin-image is indeed equivalent to getting a twin-image with no associated real image. The twin-image is well known outside the support of the real images. Inside this support, however, the twin and real images are superimposed. The algorithm therefore reconstructs iteratively the twin-image: at each step, the twin-image is updated inside the support M and the real image is reduced outside M . This algorithm requires a good knowledge of the support M .

Objects in a volume

When the studied objects are spread in a volume, the twinimage suppression problem expressed in equations 5 and 6 is no longer a deconvolution problem. However, if two holograms of the same objects -recorded at different distances or with a different wavelength -are available, they can be combined to obtain the classical deconvolution formulation. Let I 1 and I 2 be two holograms of the same objects respectively recorded at distances z and z +d. Then, according to equation 5, they are related to the complex amplitude of the objects through:

-I 1 + (I 2 * h d ) = a z * h z * (δ -h 2d ). ( 16
)
Several authors [START_REF] Maleki | Noniterative reconstruction of complex-valued objects from two intensity measurements[END_REF][START_REF] Zhang | Reconstruction of a complex object from two in-line holograms[END_REF][START_REF] Zhang | Reconstruction of in-line digital holograms from two intensity measurements[END_REF][START_REF] Denis | Twin-image noise reduction by phase retrieval in in-line digital holography[END_REF] applied a deconvolution to equation 16 using similar approaches as those described as the Fourier-quotient methods in the previous paragraph.

Multi-hologram approaches (either deconvolution-based or performing phaseretrieval) are, however, limited in practice (except some specific setup like multiwavelength holography [START_REF] Javidi | Threedimensional image fusion by use of multiwavelength digital holography[END_REF] or semi-transparent holograms in X-ray holography [START_REF] Xiao | Digital image decoding for in-line x-ray holography using two holograms[END_REF]) to the study of still objects as they require a recording of two holograms at different distances. We therefore suggest a new algorithm in section 3 that extends iterative approaches with a support constraint to the case of a single hologram of 3D objects.

Algorithm for twin-image suppression in the case of 3D objects

The use of a single hologram to reconstruct a twin-image free volume of objects prevents us formulating the problem directly as a deconvolution problem (see section 2.2.2). An iterative approach with finite-support constraint is however possible. It can then be interpreted in the context of wavelet denoising. Classical reconstruction gives images corrupted by the twin-images. Twin-image noise can be spatially localized by using a Fresnel transform with opposite parameter. The Fresnel transform is a redundant scaling transform [START_REF] Onural | Family of scaling chirp functions, diffraction, and holography[END_REF] that separates, at some scales, signal (i.e. real images) from noise (i.e. twin-images). We can therefore apply the technique of wavelet coefficient thresholding [START_REF] Krim | On denoising and best signal representation[END_REF].

In the following paragraphs we the algorithm that we suggest for twin-image suppression. Section 4 then illustrates some results on both simulated and real holograms.

Principle

The aim of the algorithm is to spatially separate the real and virtual images by masking. Figure 1 illustrates the twin-image phenomenon. Small diffracting objects were used to simulate the complex amplitude in a recording plane. The intensity of the field has been stored and two reconstructions have been performed. Image (a) has been computed from the simulated complex amplitude. It is therefore free of the twin-image phenomenon. Only two of the three objects are in focus. The third one is slightly out-of-focus. Image (b) is the result of the simulated hologram reconstruction. The lack of phase information results in the presence of the twin-image. Low frequency fringes appear in image (b) that could not be seen in image (a): this is the so-called "twin-image". In-focus images can easily be spatially separated from their twin-image. Out-of focus objects, on the other hand, cannot be separated. Twin-image suppression, in the case of objects in a volume, must be performed in 3D. In the following, we consider a set of planes (z 1 , . . . , z p ) that coarsely sample the volume. The hard-thresholding wavelet coefficient algorithm can then be written:

     a H 0 = I H a n (twin) (z i ) = (1 -M (z i )) • a n H * h z a n+1 H = 1 p p i=1 a n (twin) (z i ) * h -z i , (17)
with M being a 3D binary mask that indicates the support of the objects, and with arg(a) being the phase of complex number a. We describe its construction in the next paragraph. The algorithm iteratively reconstructs the complex amplitude a H in the hologram plane I H . The noise is suppressed in each plane a (twin) (z i ) of the twin halfspace.

The total redundancy of Fresnel transform is a drawback of the algorithm efficacy. If we consider the sampling of the volume into p planes and that the twin-image of a given object is suppressed only in one of the p planes by mask M , then the amplitude of this twin-image is reduced by factor (p -1)/p after summing back all filtered planes. Direct use of the wavelet denoising procedure, therefore, is not usable. Instead of denoising the transformed planes, it is more efficient to suppress the phase corresponding to the twin-object in the hologram plane. This leads to the modified version of the algorithm illustrated on figure 2, for i = 1 to p:

                     a 0 H = I H    a n (0) H = a n H a n (i) H = a n (i-1) H -M (z i ) • a n (i-1) H * h z i * h -z i a n+1 H = a n (p) H . ( 18
)
This time, the complex amplitude in the hologram plane a H is modified p times per iteration. The complex amplitude of the twin objects that are in focus in plane z i :

M (z i ) • a n (i-1) H * h z i * h -z i is subtracted from the hologram amplitude a H .
The subtraction takes place before processing the next plane z i+1 which avoids multiple suppression of a virtual object.

Iterative cleaning

The mask M is obtained by rough segmentation of the real-objects half-space. In the case of amplitude-only objects, thresholding is sufficient to perform an approximate segmentation. Phase objects on the other hand might be segmented based on a focusing criteria such as that suggested by Dubois et al. [START_REF] Dubois | Focus plane detection criteria in digital holography microscopy by amplitude analysis[END_REF] and on the phase in the reconstructed planes. Note however that we have not checked in this study the efficacy of the method on holograms in this latter case. The mask can be either computed once, before algorithm 18, or updated during the algorithm execution. In case of under-segmentation (i.e. non-detection of some objects) the twin-image suppression will be less efficient since the twin-images corresponding to the non-detected objects will remain.

To soften the constraints on the quality of the segmentation M and to reduce the z sampling steps, we suggest modifying the binary mask obtained by segmentation by: (i) dilating it with a disk a few pixels wide, (ii) low-pass filtering the result (11 × 11 Gaussian kernel with 2 pixels standard deviation). It is then possible to suppress a twinimage even if the considered plane is not exactly in-focus. The low-pass filter is aimed at reducing sharp boundaries that could create ripples after Fresnel transform. The process of cleaning a twin-object from a slightly out-of-focus plane is illustrated in figure 3(a). A hologram of a disk-shaped object of 90µm diameter, recorded at 120mm from the object, has been simulated. The cleaning plane is located at 2mm from the focus-plane in order to prove that the z sampling may be loose. The virtual object is efficiently suppressed in few iterations. The evolution of the relative energy of the virtual object is depicted in figure 3(b). The rapid decrease in the first iterations proves the efficiency of the cleaning algorithm. We found in practice that the twin-image suppression was satisfying after less than 10 iterations in the examples shown in section 4.

Results

We give some illustrations of the algorithm described in section 3 in three different cases: a simulated hologram of a large depth volume with some small objects; a simulated hologram of a uniform collection of particles; and an experimental hologram of water droplets.

Synthetic hologram of a simple three dimensional distribution of small objects The objects studied here are spread over three parallel planes each located at a 20 mm distance. The simulated hologram is recorded at 100mm from the closest plane for a 1024 × 1024 sensor with 6.7 × 6.7µm 2 pixels (8 bit quantization depth). gives the signal to noise ratio in each 5 plane before (lowest curve) and after (highest curve) twin-image reduction. The signal to noise ratio have been computed by using the complex amplitude in the hologram plane as it is available in numerical simulations.

Let us write I (th)

Rz the theoretical intensity (i.e. the twin-image free computed with the complex amplitude) and I

with N the number of pixels. In this case, the twin-image reduction algorithm has increased the SNR by 10dB.

Experimental hologram of a water spray A cloud of water droplets was produced by a water spray. A hologram was recorded on a PCO Sensicam camera (1280 × 1024 6.7×6.7µm 2 pixels with 12 bit quantization depth) using a YAG pulsed laser (lambda = 532nm, pulse duration= 7ns). The droplets were located within a bounding volume of 60mm depth. Their size is roughly 90µm.

Conclusion

The twin image occurs because the traditionally used reconstruction does not invert the recording step. Numerical techniques for twin-image suppression try to improve the system response. We have shown that identifying the twin-image suppression problem as a deconvolution problem provides a unifying framework. We have set bridges between different methods previously described in the literature as being distinct approaches. Existing algorithms cannot be applied to the case of small objects spread in a volume. We therefore derived an iterative algorithm that is both an evolution of finite- support techniques for plane objects and is inspired from wavelet denoising. The first results on simulated and real holograms are encouraging as the twin-images are efficiently cleaned. More generally, further experimental evaluation in various conditions should be completed to clarify the potential of the method.

It seems that the method could also be applied to holograms of phase-objects. An experimental validation on such holograms should however be considered.

The twin-image suppression technique we described might be usefull in particle holography for fluid dynamics. Its application for correlation-based measurements of particle holograms would suppress the indetermination due to the twin-image [START_REF] Ooms | Exctracting 3d particle field displacement from on-axis digital holographic piv measurements[END_REF]. The algorithm could also be extended to twin and real objects suppression and possibly provide an alternative to Monnom et al. technique for out-of-focus object influence reduction [START_REF] Monnom | Improvement in visibility of an in-focus reconstructed image in digital holography by reduction of the influence of out-of-focus objects[END_REF]. It could then be used in applications such as interrogation cell correlation for velocity measurements [START_REF] Denis | Cleaning digital holograms to investigate 3d particle fields[END_REF] or multi-plane velocimetry [START_REF] Cöetmellec | Application of in-line digital holography to multiple plane velocimetry[END_REF]. Note that the described twin-image cleaning algorithm is limited to low or medium concentrated holograms as it relies on the spatial separation between real images and the surrounding twin-images. Furthermore, objects close to the hologram boundaries get blurred by the incorrect twin-image cleaning due to border effects. This problem requires further work and can be reduced by windowing [START_REF] Cuche | Aperture apodization using cubic spline interpolation: application in digital holographic microscopy[END_REF].

Denoising and segmentation appear as tightly linked problems. For efficient twinimage suppression, denoising must be performed jointly with volume segmentation. The study of small objects in a volume (i.e. finite support objects surrounded by empty regions) suggests the use of adapted image processing techniques. Enforcing a finite support constraint as we do in this paper is a possible approach. In the case of well defined objects, such as spherical particles, a parametric estimation approach in an inverse problems framework proves to result in superior performance [START_REF] Soulez | Inverse-problem approach for particle digital holography: accurate location based on local optimization[END_REF][START_REF] Soulez | Inverse problem approach in particle digital holography: out-of-field particle detection made possible[END_REF].

Appendix A. Approximate expression of the twin-image of a small object

We derive here an approximate expression of the twin-image in the focus plane of the real image. We consider small objects, a situation common to particle holography and three dimensional imaging of small elements such as micro-organisms in water [START_REF] Malkiel | The three-dimensional flow field generated by a feeding calanoid copepod measured using digital holography[END_REF]. The twin-image amplitude is given (eq. 3) by: ϑ * (h 2z ) (x, y) = ϑ * (h 2z ) (x, y) For symmetrical objects, the aperture ϑ is even and the Fourier transform is real. The twin-image is then given by: ϑ * (h 2z ) (x, y) ≈ h 2z (x, y) F[ϑ i ]( x 2λz , y 2λz ), (A.4)

and appears clearly as a modulated signal in two ways:

• the frequency is linearly modulated with rate r/(2λz), with r the distance to the object center,

• the amplitude is modulated according to the Fourier transform of the object aperture.

Figure 1 .

 1 Figure 1. Twin-image appearing at the reconstruction step: (a) "twin-image free" reconstruction using the phase; (b) classical reconstruction from the hologram, showing the twin-image

Figure 2 .

 2 Figure 2. Twin-image cleaning scheme

Figure 3 .

 3 Figure 3. Illustration of the iterative suppression of a virtual object: (a) evolution of the twin-object in its in-focus plane (first row) and in the cleaning plane (second row) (please note that the contrast of the figure has been increased for visualization purposes); (b) relative energy of the twin-object at each iteration.

  Figure 4(a) gives a diagram representing the object. The corresponding computed hologram is displayed in figure 4(b). Figures 4(c) to (e) show the three reconstructed planes using the classical reconstruction algorithm. The reconstructed planes after twin-image cleaning are given in figures 4(f) to (h). The twin-image reduction is very efficient on this simple experiment.Simulated hologram of opaque particles One hundred opaque spheres (diameter 90µm) were spread in a 6 × 6 × 6 × mm 3 volume. A hologram was numerically simulated at a distance z = 120mm for a 1024 × 1024 sensor with 6.7 × 6.7µm 2 pixels (8 bit quantization depth). Figure5(a) displays the hologram. Classically reconstructed and twin-image cleaned planes are displayed in figures 5(b) and 5(c) respectively. Five "cleaning" planes were used in the algorithm (p = 5 in algorithm 18). Figure5(d)

  Rzthe intensity obtained with a reconstruction algorithm (i.e. classical reconstruction, or twin-image suppression using algorithm 18 prior to reconstruction). The signal to noise ratio shown on figure5(d) is:

Figure 6 (

 6 a) shows the experimental hologram. The classical reconstruction is given in figure6(b). The reconstruction after twin-image reduction is displayed in figure6(c). The twin-image has been strongly attenuated. The algorithm seems robust against noise due to hologram acquisition (background noise due to spurious interference patterns).

Figure 4 .Figure 5 .

 45 Figure 4. Twin-image cleaning applied on a simple synthetic hologram

Figure 6 .

 6 Figure 6. Twin-image cleaning applied on an experimental hologram of a water spray: (a) experimental hologram; (b) a reconstructed plane; (c) the same plane after twin-image suppression

  , η) exp j π 2λz [(x -ξ) 2 + (y -η) 2 ] dξdη , (A.1)as we consider objects described by a real-valued aperture ϑ. This expression can be approximated by:ϑ * (h 2z ) (x, y) ≈ 1 2jλz exp j π 2λz [x 2 + y 2 ] • ϑ i (ξ, η) exp -jπ λz [xξ + yη] dξdη (A.2)for small objects such that 1 2λz [ξ 2 + η 2 ] max 1, i.e. whose largest dimension d verifies the condition d √ 2λz. The integration corresponds to the Fourier transform of ϑ up to a spatial scaling transform with a scale factor of 1/(2λz): ϑ * (h 2z ) (x, y) ≈ h 2z (x, y)F[ϑ i ]( x 2λz , y 2λz ) . (A.3)


C Mougin

R Scalla

F Cabanne

OCCURRENCE OF CYTOCHROMB P 450 MONO-OXYGENASES IN THE METABOLISM OF CHLOROTOLURON BT WHEAT MICROSOMES

Wheat cell suspension cultures metabolise the herbicide chlorotoluron by ring-methyl hydroxylation, N-demethylation and sugar conjugation (Cole and Owen, 1987;Canivenc etal., 1989). Cell suspension cultures were used to study oxidative transformation of chlorotoluron in vitro. Achlorophyllous cells of wheat cv. Koga II were cultivated as previously described (Canivenc el a/., 1989).

Microsomes were isolated by differential centrifugation from cells treated with cyometrinil.

Aerobic incubations of microsomes with chlorotoluron and NADPH led to the formation of ring-methyl hydroxylated and N-monodemethylated chlorotoluron. Metabolites were analysed in subcellular fractions after differential centrifugation. They were mainly found in the microsomal fraction in mixture with NADPH-cytochrome P 450 (cytochrome c) reductase activity. Microsomes metabolised the herbicide at rates ranging from 600 to 650 and 200 to 250 pmoles mg" 1 30 min" 1 for the ring-methyl hydroxylase and N-demethylase, respectively.

Microsomal pellets contained substantial levels of cytochrome P 450 (300 pmol mg protein" 1 ).

In cells grown in the presence of cyometrinil, chlorotoluron ring-methyl hydroxylase, N-demethylase, and lauric acid hydroxylase activities were stimulated respectively 1.6-, 3.3-and 1.4-fold compared to controls. The cytochrome P 450 content of microsomes was increased 1.8^fold by cyometrinil.

Ring-methyl hydroxylase and N-demethylase activities exhibited their highest rates with NADPH. Other reductants such as NADH, dithionite or ascorbate were ineffective. Molecular oxygen was also necessary for the NADPH-dependent activities. Herbicide hydroxylase and N-demethylase were drastically reduced by inhibitors of the P 450 system, namely ABT, />-CMB, menadione and cytochrome c.

Carbon monoxide (CO) inhibited the hydroxylase activity with partial reversion by light. In contrast, N-demethylase activity was apparently insensitive to CO. Only the monodemethylated metabolite could also be formed in the presence of cumene hydroperoxide as oxygen donor.

These data allow us to postulate that chlorotoluron ring-methyl hydroxylase and N-demethylase activities probably belong to the family of cytochrome P 450 mono-oxygenases. The fact that the N-demethylation can consume cumene hydroperoxide leads to a possible role of another type of mono-oxygenase, for example a peroxygenase-type enzyme, in addition to cytochrome P 4 so•


C Benjamin
email: cbenjami@roazhon.inra.fr

Rural Realities : Trends and Choices

Diversification activities, on-farm or off-farm, can allow farmers not to rely exclusively on farming for the household's income. In many rural areas, on-farm tourism and off-farm employment have became an important and steady source of income for farm households. The most significant change over the last ten years has been the substantial growth in off-farm labour participation by farm spouses. This paper based on the neoclassical farm-household is an attempt to characterize the variables that affect the decisions to work off-farm for the French farm households and the decision whether or not to hire labour on farm. Furthermore the impact of the Common Agricultural Policy (CAP) on the participation in the off-labour market is studied. In a simplified model where male and female labour are perfect substitutes we can show that the reduction in the cereals price and the set-aside would induce a decrease in the reservation wage and hence an increase in the probability of off-farm participation. On the other hand the receipt of compensatory payment would have a negative impact on the probability of off-farm work participation. Empirical results show that while the CAP reform will increase wives' off-farm participation, education and age have a greater positive effect. Furthermore, simulations show that the impact of the CAP reform on off-farm labour market participation is higher for wives who do not have a high school diploma than for those who do. 1

Introduction

The growing importance of diversification activities for farm households has been identified as a new feature of Western agriculture. Diversification activities, on farm or off farm, can allow farmers not to rely exclusively on farming for the household's income. Indeed, in many rural areas, on-farm tourism and off-farm employment have became an important and steady source of income for farm households. In France, nonagricultural income represents an increasing percentage of farm household income: rising from 15% in 1956 to more than 42% in 1988.

Severa! factors have contributed to this increase. The number of farms involved in tourism activities has increased by roughly 20% since 1978, although only a fraction of farms are engaging in tourism activity. The most significant change over the last ten years has been the substantial growth in off-farm labour participation by farm spouses: the percentage of young wives under 35 years old working off farm rose from 26 % in 1979 to 52% in 1988. The prevalence and magnitude of off-farm employment income in European countries have been described which are, to our knowledge, mainly descriptive [START_REF] Gasson | Farm Diversification and Rural Development[END_REF]The Arkleton Trust, 1991 ). Empirical studies of off-farm work participation choices have been, in most cases, applied to American and Canadian agriculture [START_REF] Lass | Factors Affecting the Suppl y of Off-Farrn Labor: A Review of Empirical Evidence[END_REF][START_REF] Huffman | Off-Farm Work Decisions of Husbands and Wives: Joint Decision Making[END_REF].

Growth in part-time fanning and in diversification of agricultural activities may be affected by the 1992 Comrnon Agricultural Reform (CAP) reform. This recent shift in support policy implies a major change in the nature of support provided to the cereals sector. This consists of support price cuts and a mandatory acreage set-aside in retum for compensatory payments. We might expect that this reform would affect off-farm labour decisions of French agricultural households. To date, research on off-farm labour supply did not provide information on the reactions of farm households to changing economic conditions, such as output price variations (Lass et al.). The purposes of this paper are i) to identify the factors that lead French farmers to supply off-farm labour ii) to find out how the changes in the nature of support given to the European agricultural sector will affect the off-farm labour decisions and iii) to quantify the impacts of the variations of output price, land used, and the amount of subsidies on the probability of off-farm work for French grain farmers.

Our first step is to present a model of household behaviour to identify the variables that affect the labour decisions. The neo-classical firm-household model Nakajima (1969) emphasizes the interdependence between the firm's profit maximisation and household's ut1lity maximisation behaviour. Previous studies have modelled within this framework off-farm work decisions of farm operators and their spouses (see [START_REF] Lass | Factors Affecting the Suppl y of Off-Farrn Labor: A Review of Empirical Evidence[END_REF] for a recent survey). Our analysis is along the same lines but differs in that it explicitly includes hired labour decision as a decision variable. The decision of the household to work off farrn and the decision to use hire labour on the farm are joint and our model takes into account their simultaneity. The farm household mode! allows us to characterize a rule for off-farm work participation: the household defined as an entity, will not work off farm when its marginal value of time from not participating in the offfarm labour market (its reservation wage) is greater than the wage rate offered off-farrn.

Furtherrnore, the household will use hired labour when the marginal value of hired labour equals its cost. The comparative static of the reservation wage and the marginal value of hired labour with respect to the instruments used in the new CAP regime allow the formation of hypotheses regarding how CAP will affect labour decisions.

The plan of the paper as is follows. Section 2 details the analytical model. In the third section, the theoretical model is applied to a sample of 1518 grain farms. In the empirical analysis we attempt to characterize the variables which affect the decisions to work off-farm for the French farm households and the decision whether or not to hire labour on farrn. In section 4 our attention is directed to the quantitative impact of the CAP reform on the probability of off-farrn work. Finally, we conclude with some observations regarding policy analysis and further research in this area.

The theoretical framework

A simp/ified mode/

The farrn household is defined as an economic entity 1 and is assumed to maximise its utility function subject to income, time and non-negativity constraints :

max U ( LF, LO, C; E , A) LF.LO.C.LH subject to PcC ~ 1t (p, v, LF, LH,Z; E, D) + w 0 LO+ B-wLH LF+ LO+ LEI ~ T LF ?.0
1 We assume that male and female family labour are perfect substitutes.

[1]

[1.a] [1.b] [1.c] LO~0 c~o LH~0 [1.d] [I.e] [l.f]
The utility function (U (.)) depends on on-farm family labour (LF), on off-farm family labour (LO), and on the vector of consumption goods (C). The utility fonction depends also on human capital variables (E) and on household characteristics (A). The associated price of consumption goods is (pc). The restricted conditional profit fonction is denoted 7t. lt depends on the product price vector (p), the price vector of variable input (v), on-farm family work, hired labour level LH, fixed factors (Z), human capital variables and other exogenous shifters of supply (D). The wage rate for hired labour is noted w. Other variables used include the off-farm wage rate ( w 0 ) 2 , exogenous household income (B), leisure time of the household ( LEI ) and total time available to the household (1).

The utility fonction is well-defined and continuous from above, quasi-concave in LF , LO and C, nonincreasing in LF and LO and nondecreasing in C [START_REF] Diewert | Applications of duality theory[END_REF]. The restricted profit fonction is nonnegative, linearly homogeneous, continuous and convex in prices, nondecreasing, continuous and concave in quantities [START_REF] Lopez | Estimating Labor Supply and Production Decisions of Self-Employed Farrn Producers[END_REF]. Furthermore, on-farm and off-farm family labour are assumed to be perfect substitutes in the utility fonction as well as family labour and hired labour in the production function despite their differential opportunity costs.

Constraint (1.a] is the budget constraint : the value of consumption must be lower than or equal to total income (M) which is defined as the sum of farm profits, off-farm wage and exogenous income. Constraint (1.b] is the time constraint: the household can allocate his available time to on-farm work, off-farm work and to leisure. Constraints [I.e] to [1.f] are positivity constraints. We suppose that the budget and the time constraints are bounded.

Furthermore, the price of the consumption good is used as the numeraire.

Assuming interior solutions for on-farm family labour and for the consumption, the associated Kuhn-Tucker conditions for a maximum are:

-(au / 8LF )/(8U/ 8M) = ôn /8LF [2]
where t and µ are the Lagrangean multipliers associated with the non-negativity constraint [l.d) on off-farm family labour and the non-negativity constraint [1.f] on hired labour.

Equation [2] states that the marginal rate of substitution between on-farrn family labour and income is equal to the marginal value of farm labour. Equation [3] provides the criterion for off-farm participation. Off-farm work will be zero if the marginal rate of substitution between off-farm family labour and income is greater than the off-farrn wage rate. An interior solution occurs for off-farm family labour when the marginal rate of substitution is equal to the off-farm wage rate. Equation [START_REF] Benjamin | L'affectation du travail dans les exploitations agricoles : approche microéconomique et application sur données françaises[END_REF] defines the criterion for employing hired labour on farm. Hired labour is used on farrn when its marginal value product is equal to its price.

Hence, the participation rule for the off-farm labour market for the household is defined as

LO>0 if w,, >-(8U/8LO/8U/8M~w=o LO=0 if w,, ~-(8U/8LO/àU/ àM~w=o [5] [6)
Since on-farm work and off-farm work are perfect substitutes we can rewrite the participation rule for the off-farm labour market as:

ro > o if w 0 > an /oLF 1 w=o LO= 0 if w 0 ~ an /8LF 1 w=o [7a] [7b)
The right-hand side of these inequalities gives the marginal value of time when the household allocates all of its work time to on-farm labour, i.e. it defines the reservation wage.

The criterion of participation in the off-farm labour market is based on a comparison of the offfarm wage rate and the reservation wage. Factors which influence the reservation wage and the off-farm wage rate will affect the criterion of participation. For instance, variables that raise (or decrease) the off-farm wage rate increase (or decrease) the probability of off-farm work while variables that raise ( or decrease) the reservation wage rate decrease ( or increase) the probability of off-farm work.

Along the same lines, the rule for using hired labour on farm is defined ~:

LH > o if an /8L1fi 1 .H= o > w [8]
The left-hand side of these inequalities defines «the reservation wage of hired labour».

Variables that rai se ( or decrease) this value increase ( or decrease) the probability of using hired labour.

Both labour decisions rules [7] and [8] are likely to be affected with the CAP reform through the variations in exogenous variables implied by the reform.

Effect of the CAP reform on the participation in the off-farm labour market

The new CAP regime marks a major change in the nature of support given to EC cereal farmers. Central to the reform is a phased reduction in institutional prices for cereals. To offset the income effects of the reduction in support prices, farmers will receive compensation payments if they comply with set-aside requirements (they have to set aside 15% of the base area 3 ). This reform will induce variations in output price, in land area and in exogenous income.

By assumption, the off-farm wage rate depends only on the human capital variables and on the location parameters. lt is not affected by the CAP reform. On the other hand, the reservation wage is a function of exogenous shifters of supply and demand: output and input prices, fixed assets, other factors that shift the production level, human capital variables, household characteristics, exogenous income and local labour market variables. The reform will affect the household's reservation wage and therefore the participation rule in the off-farrn labour market. The total effect of the CAP reform on the reservation wage can be written as dwr = ôw,/ôpdp +ôwr/ôZdZ +ôw,/ôBdB [9] where w, is the household's reservation wage

We have to consider the comparative static of the household's reservation wage with respect to product price, to land area and to non labour income. We can prove that ôw, /ôp > 0, aw,/ôZ > 0 and aw,/ôB > 0 4 • lt follows that the reduction in cereals price and the set-aside of land would induce a decrease in the reservation wage of the members of the household, hence an increase in the probability of off-farrn participation. On the other hand, the receipt of the compensatory payment for reduced price support would have a negative impact on the probability 3 Small producers (i.e. those who have an insufficient land area to produce 92 tonnes of cereal at reference yîélds) are exempt from this requirement. 4 Formai analysis ofthese cases and the comparative static results presented in this paper are outlined in [START_REF] Benjamin | L'affectation du travail dans les exploitations agricoles : approche microéconomique et application sur données françaises[END_REF] of off-farm participation. Hence, the total effect on the reservation wage is indeterminate because there are opposite effects. However, under the compensation payments planned in the new CAP regime we can predict the sign of the total effect.

The sign of the effect on the reservation wage depends eventually on the farm yield. Indeed, when the individual yield is greater than or equal to the reference yield5 , th~ total effect on the reservation wage is negative. As a conclusion we expect that the effect of a decrease in product price, set-aside and receipt of compensatory payments will induce a decrease in the reservation wage and hence an increase in the probability of off-farm participation.

Empirical results

We turn to French data to characterize the variables which affect the variations of the probability of off-farm participation and of the probability of using hired labour. In the empirical analysis, we distinguish the wife's behaviour from the husband's behaviour because both spouses might be not influenced by the same factors.

The theoretical model developed in section two is applied to a sample of French field crops farms . Two considerations explain this choice: i) the importance of the off-farm participation labour rate in this specialisation and ii) the focus of the CAP reform on the cereals sector. We, first, define the econometric specification. Next, we present the data base. Finally, the empirical results are discussed.

Econometric specification

The empirical model includes three participation decision rules based on the comparison of the reservation wage and the corresponding market wage. The model for estimating off-farm participation and hired labour use probabilities may be written as:

i=f, h [10)
where f and m represent female and male respectively. X; and Y define the vectors of exogenous variables, (3; and a are the parameters to estimate and E and µ; are stochastic errors.

We define three binary variables, D; (i=f,h) which equals one if the individual i (the former or his wife) works off-farrn and zero otherwise, DLH which equals one if the farrn uses hired labour and zero otherwise. The decision rules are finally expressed in terms of observable exogenous variables, unknown parameters and stochastic error terms.

Prob(LO;

> 0) = Prob(D; = 1) = Prob(w,; < w~) = Prob(µ; < w~ -~ ;X;) Pr ob( LH > 0) = Pr ob(DLH = 1) = Pr ob(ôrr./8LH < w) = Prob(r, < w-aY) [11] [12]
Since the participation decisions of the husband and the wife in off-farm work and the decision for hiring labour on farm are simultaneous, a joint estimation procedure should increase the statistical efficiency of the parameter estimates. Assuming that stochastic errors have a normal distribution, the appropriate statistical model is the multinomial probit model.

Unfortunately, there does not exist analytical solution in that case and the multinomial probit mode! must be solved by numerical methods which are difficult to implement. Consequently a simplified procedure has been preferred, three univariate probit models are estimated.

Data

The model is estimated using a random sample of French farm households drawn from the 1988 General Census of French Agriculture. A total of 1518 households were retained after eliminating observations with missing data and the farmers who did not grow grain crops. In that subsample, 331 wives (about 24 %) report off farm activities, 157 husbands (about 10 %) work A dummy variable represents the general education level of each individual ( equal to 1 if the individual attained the baccalaureat or more, 0 otherwise). Agricultural education is likewise represented by a dummy variable set to 1 if the individual has an agricultural high school diploma, 0 otherwise. Finally, participation in recent agricultural courses is defined by a dummy variable. Household characteristics include: the number of children under 6, between six and twelve, child older than 12 and the total number of family members living on-farrn. Farm characteristics include: the fann size and durnmy variables which describe farms involved in farm tourism, fanns taking on-farm diversification activities (which include mainly on-farm agricultural products sales), farms benefiting from a development plan and farms using a computer and a tractor of more than 80 HP. A last dummy variable describes the status of the farm (which equals I if the farm is a G.A.E.C. 6 ) . Location characteristics are defined by the average condition in the French department where the farm is located. lt includes the population density of the department. Labour market conditions are measured for females (for males) by the proportion of farmers' wives (farmers) who work off-farm in the department. These regional characteristics might show opportunities to work off-farm. Data definitions with means and units of measurement are provided in Annex 1.

Participation equations

Results of the estimation are presented in tables 1 and 2. 6 G.A.E.C. is a fonn of partnership arrangement, in these farms farmers organise their business as companies and receive a salary for their work. * For the continuous variable the figure indicates the value of the quasi-elasticity i.e. the percentage change of the probability upon a one percent increase of the variable considered. At the values used to compute the derivatives and the quasi-elasticities the wife's probability ofoff-fann participation equals 13.07%.

For each explanatory variable we calculate the partial derivative. They are evaluated at the continuous variables' sample mean and the zero value for the dummy variable. For the price variable, we assume that the product price is equal to one hundred and ten francs per quintal (the average in the grain farms' sample). Each derivative gives the increase in the probability of being an off-farm worker for an unit increase in the variable's the sample mean. For husbands and wives, the probability of off-farm participation is affected significantly by age. The life-cycle effect is quadratic. At young ages the probability of participation increases and then decreases for aider ages. The maximum positive effect of age occurs at 34 years old for the wives (at 54 years old for the males), it becomes negative at 67 for the wives (at 108 years old (!) for the males). In both cases, the level of general education has a positive impact on participation decision. For the wives, the effect is stronger, the marginal effect of having an high school diploma is equal to 0.17 (for the males the effect is equal to 0.12). However, the individual who has more agricultural schooling has a lower probability of wage work b_}lt the effect is not significant in each equation. Greater farming experience of the farm operator has a negative effect on the individual' s part1c1pation. An additional year of farming experience decreases the probability of working off farm by around 0.04 for the wives (by around 0.01 for the husbands).

Finally the probability of participation is reduced significantly for wives and husbands who have Farm size has a negative and significant effect on wives' and husbands' off-farm labour participation. The effect is stronger for the husbands. A decrease of 10 % in farm size increases the probability of the husband' s off-farm labour participation by 1.17%. For the wives, the probability of off-farm participation is affected significantly by the price. The product price has a negative and significant effect on the probability of off-farm work participation. The probability of off-farm labour participation goes up by 1.03 % when the product price decreases by 10%, other things being equal. For the husbands, the price effect is positive but is not significant. The probability of the wife' s participation in the off-farm labour market increases significantly when the farm is in partnership. Thus, when the farm is a G.A.E.C. this probability increases by 0.09.

However, this variable has no significant effect on the husband's behaviour. Other dumrny variables which characterize the equipment of the farm (benefit of a development plan, use of a micro-computer) decrease the probability of off-farm labour by wives and husbands participation but these effects are not significant. On-farm tourism activities have a positive and significant effect on the wife's participation and the husband's participation in off-farm labour. The effect is stronger for males. This effect can look somewhat puzzling. We might expect an opposite sign of this variable. In fact, this impact can be explained as follows. On farm tourism activities are parttime or seasonal activities. The existence reveals a need of another source of income than farm income. On the other hand, diversification activities which can take place all the year have a negative effect on the wife's participation. It is worth noting that this variable is not significant for the husband. This result suggests that these activities are realized by the wives. Hence, diversification activities and off-farm labour may be substitutes i.e. alternative means to diversify household income.

Subsidies have a negative and significant effect on off-farm labour participation for both individuals. The effect is four times as strong for the husbands as for the wives. An increase of 10% in subsidies decreases the husbands' probability of off-farm participation lfy 0.50%.

Regional variables reflect differences in off-farm employment opportunities. The effect of the density is somewhat puzzling. An increase in population density has a negative impact on the wife's participation in off-farm labour. In fact, we expected a positive effect since a high population could induce better off-farm employment opportunities. On the other hand, the proportion of wives (husbands) who are working off-farm has a positive and significant effect on the wife's (husband's) participation in off-farm labour market.

Hired labour probit equation

Table 3 presents the estimation of the probability of employing hired labour on farm. -

We pay attention to the variables which have coefficients significantly different from zero.

Among the human capital variables, the woman's general education has a positive impact on the probability of hiring labour. The marginal effect of achieving formai education is equal to 0.098. This variable has the strongest positive effect of the probability of the wife working offfarm this result among the human capital and household characteristics. This result suggests that hired labour is a substitute for the woman's labour (for the women who have a general education level).

The husband's participation in recent agricultural courses increases the probability of using hired labour by 0.083. This result indicates that hired labour is a complement for the farm labour of the man.

The presence of children increases the probability of usi~~ hired labour on farm. The effect is stronger for the number of young children. An additional year of one child under 6 increases the probability of using hired labour by around 0.09. This effect can be explained by the substitution of hired labour for female farm labour because of child care. The number of family members living on farm has a negative effect on the probability of hiring labour.

Farm size has a positive effect on the probability of hiring labour on farm. The effect is stronger for the husbands: a decrease of 10 % in farm size decreases the probability of hiring labour by 1.99%. If the farm is engaged in tourism or diversification activities the probability of hiring labour incrcascs. This result is due to labour needs on farm.

Subsidies have a positive effect: an increase of 10% in the subsidies mcreases the probability by 0.37%.

Sensitivity analysis of off-farm labour decisions

To further illustrate the results, we use the estimates of the pro bit models to conduct some simulations. The probabilities of the individuals' off-farm labour participation are calculated under various levels of exogenous variables. First, these simulations illustrate the impacts of variations in the structural variables (human capital characteristics) and variations in instruments of policy (subsidies). Next the sensitivity analysis is conducted to quantify the impact of the new CAP regime.

For the simulations, we consider an average cereal farm which receives no subsidies. We assume that the product price is equal to one hundred francs per quintal. The other continuous variables are taken to the sample mean and the dummy variables are equal to zero.

Change in the human capital variables

Figure 1 shows the variations of the probability of wife's off-fann participation with respect to age. Two scenarios are considered. In the first one (lower curve) the wife has no highschool general education. In the second case (upper curve) we suppose that the wife has at least a high school diploma.

I \ / \ , \ / \ / \ 0,4 -/ \ 0,3 0,2 0,1 \ \ \ \ \ \ \ 1 1 1 1 1 \ \ \ \ \ \ \ \ \ \ \ ' \ ' ' 0 -+----------1-~1 --1--1-~-, 20

Agef

The shape of both curves is explained by the quadratic life cycle effect outlined earlier: for w1ves older than 34, an additional year of age has a positive but diminishing effect on the probability of off-farm labour participation. Figure 1 proves the strong positive effect of the wife's general education. Whatever the age, with at least a high school diploma the probability of the wife's participation in off-farm labour is more than twice as high as the probability of wives who have no general education. For instance, if we assume that the wife is 30 years old and has no general education, the probability of off-farm participation equals 21.82%. If the wife has a general education diploma the probability is equal to 50.90%: the education induces about a 133% increase in the wife's off-farm participation.

The same analysis is conducted to study the effect of the husband's age on his probability of off-farm participation. Figure 2 shows how the probability changes as the husband's age mcreases.

Ageh

The highest value for the probability appears for an older age than for the wife. As in the first case, general education causes an outward shift of the curve. However this shift is not so important as it was for the wife. For instance if we assume that the husband is 30 years old and has no general education, the probability of off-farm participation equals 11.90%. If he has a general education diploma the probability is in this case 24.85%. The education induces about a 110% increase of the husband' s probability of off-farm participation.

Change in nonfarm income

We consider that the change in nonfarm income is due to an increase in subsidies received by the farm. Increase in subsidies can be a means to maintain activity on farm and to avoid depopulation of the country. Indeed, aids to farms can be intended to promote forms of diversification activities on farm or to compensate for difficult farming conditions in less favoured areas. We calculate the amount of subsidies that the farm household should receive in the order that the wife's probability of off farm participation does not increase. The amount depends in particular on the wife's age and on the department where the farm is located.

Link between nonfarm income and the wife's age

The link between the wife's age and the amount of nonfarm income is illustrated with the figure 3. The concave curve, drawn with respect to the wife's age, defines the amount of subsidies that the farm should receive so that the wife's probability of off-farm participation would be equal to 0.5. Above the curve, the probability is lower than 0.5 (the probability is decreasing with the level of subsidies). Below the curve, the probability is greater than 0.5. Since we admit that the event of off-farm participation is predicted to occur if the probability of off-farm participation is greater than or equal to 0.5 (Judge 1988 p. 794 ), above (below) the curve and the wife is predicted not to work (to work) off farm.

The horizontal line (A,B) measures the amount of the compensatory payment for the reduced price that the farm would receive under the CAP reform 7 . We assume that before the CAP reform the farm has no subsidies. Hence, the wives who are between 23 and 46 years old are predicted to work off farm. The intersecting points with the horizontal line and the curve define the ages for which the wives' behaviour is not affected by the receipt of subsidies: they still are working off farm. Indeed for wives who are between 26 and 42 years old, the amount planned in the CAP reform is not sufficient to keep them off the off-farm labour market. For instance if the wife is 36 years old compensatory payments would need to be twice as high as those given under the CAP.

For the wives who are under 23 years old, the amount of compensatory payments planned in the CAP reform is sufficient to keep them off the off-farm labour market. lt is worth noting that the wives' behaviour who are older than 46 does not vary with or without compensation.

They do not participate in the off-farm labour market.

Link between nonfarm income and the location ofthefarm

One potential policy to reduce the population decline in some rural areas might be to raise subsidies for such areas. The amount of aids necessary to keep the wives off the off-farm labour market vary by the department i.e. depends on the location of the farm (figure 4). On figure 4, the downward slope line defines the amount of subsidies that the farm should receive in order that the wife's probability of off-farm participation will be equal to 0.5 (we assume that the wife is 30 years old). Above (below) the line the probability is less (greater) than 0.5 and the wife is predicted not to work off farm (to work). As for the figure 5, the horizontal line gives the amount of the compensatory payment for reduced price that the farm will receive under CAP reform. For the departments, which have a density greater than 190, the amount of compensatory payments planned in the CAP reform is sufficient to keep the wives off the offfarm labour market.

Effect of the CAP reform on wife's participation

The predicted effect of each instrument of the CAP reform is confirmed in the wife's participation equation8 .

We consider a farm under CAP reform i.e. which faces a 30% decrease of the product price and sets aside 15% its acreage and in turn receives compensation. We study the effect of the new levels of the three variables on the decision to work off-farm. For the other continuous variables, we use the sample mean and the dummy variables are equal to zero. We conduct the analysis in the case where the wife has no general education and in the case where the wife has at least a high school diploma. The impact on the probability of the wife's participation in the offfarm labour market is shown by the figure 5. As we predicted from the theoretical analysis, the effects of the price cuts, set-aside and the compensatory payments is positive on the probability of off-farm labour participation. In both cases (with or without general education) there is a vertical shift of the curve which describes the variations of the probability of off-farm participation. However, the probability increases by less than the shift induced by the variation in the general education. For instance, we assume that the wife is 30 years old. When she has no general education, the probability of off-farm participation increases from 21.82% to 23.93% (around a 10% increase). If the wife has a general education diploma, the CAP reform induces about a 5% increase in the wife's off-farm participation (from 50.90% to 53.67%).

The difference between these figures and the previous results indicates that while the CAP reform will increase wives' off-farm participation, education and age have a greater positive effect. Furthermore, these simulations show that the impact of the CAP reform on off-farm labour participation is higher for wives who do not have a high school diploma than for those who do.

Conclusion

This paper analyses the off-farm labour supply and hired labour demand of farm households in a theoretical context of a household mode! behaviour; Furthermore, the paper has attempted to address the empirical effect of the CAP reform on labour decisions of French agricultural households. Indeed, changes caused by this support shift may induce a reallocation between work activities for the members of the household. First, the firm-household mode! has been used to derive the rule concerning the off-farm labour market participation and the decision of employing hired labour on farm. Comparative static analysis in a simplified mode! allows us to predict how the probability of off-farm participation. The study focuses mainly on the decision to work off farm. Both the decrease in the product price and the new set aside requirements induce a decrease in the reservation wage and hence an increase in the probability of off-farm labour participation. On the other hand, receipt of compensatory payments has a positive effect on the reservation wage, i.e. a negative effect on the probability of the off-farm labour participation.

Probit models allow to search for the factors affecting the probability of working off farm for the wife and the husband and the probability of using hired labour on the farm. Empirical results provide information on relationship between family labour and hired labour and support the interest of a joint analysis of labour decisions of the agricultural households. Furthermore, parameters are used to quantify the effects of the variation in the product price, the farm size and subsidies. For the wives, the effect of the three instruments are significant (unlike the husband) and positive on the probability of off-farm labour participation. The findings suggest that increases in wives' human capital (for instance, in general education and in age) will have the greatest effect on the probability on off-farm labour participation.

Severa! aspects of the analysis presented here deserve further consideration. First, the empirical analysis allows an examination of interrelationships between family labour and hired

  off farm and 574 farrns (about 38%) use hired labour. Four classes of variables have been distinguished corresponding to: i) individual characteristics, ii) family characteristics, iii) farm characteristics and iv) location characteristics. Individual characteristics are age and several variables measuring human capital: general education, agricultural education, agricultural training courses and agricultural work experience.

  participated in agricultural training since 1980. The probability of the wife' s off-farm participation decreases with the number of children regardless the children's age. But the effect of children on the wife' s probability of participation decreases with children' s age. The probability of the husband' s off-farm participation does not depend on family characteristics.

Figure 1 .

 1 Figure 1. Impact of the wife' s age on her probability of off-farm labour market participation 0,6

Figure 2 .

 2 Figure 2. Impact of the husband's age on his probability of off-fann labour market participation 0,6

Figure 3 .

 3 Figure 3. Frontier of participation in off-farm labour: the age effect Thousands Francs25 -

Figure 4 .

 4 Figure 4. Frontier of participation in off-farm labour : the location effect 30 Thousands francs

Figure 5 .

 5 Figure 5. The CAP and the education effect on the wife's probability of off-fann labour participation 0,6

Table 1 .

 1 Probit analysis of the wives' off-fann labour market participation

	lndependent variables	Coefficients	t-student	Partial derivative
	lntercept	-1.291	-1.53
	Human capital
	Agef	0.115	2.97	-0.0094
	Agef"'agef	-0.0017	-3.86
	Dumeducf	0.801	7.07	0.17
	Dumeduch	0.078	0.49	0.0167
	Dumagrf	-0.l 14	-0.31	-0.0241
	Dumagrh	0.059	0.38	0.0126
	Exp	-0.036	-5.36	-0.0357
	Dumpennf	-0.652	-3.62	-0.1 389
	Dumpennh	0.189	1.59	0.0402
	Household characteristics
	Less6	-0.17	-1.67	-0.0361
	Kids612	-0. 136	-1.75	-0.036 1
	Oldl 2	-0.081	-1.24	-0.0172
	Nbind	-0.06 1	-1.36	-0.0129
	Farm characteristics
	Size	-0.003	-3.33	* -0.0515
	Price	-0.004	-1.77	* -0.1027
	Dumgaec	0.441	2.58	0.0935
	Dummicro	-0.088	-0.66	-0.0186
	Dumdivers	-0.254	-2.28	-0.0539
	Dumtourism	0.68	2.11	0.1444
	Dumdev	-0.034	-0.25	-0.0072
	Dumtract	-0.01	-0.66	-0.0022
	Exogenous income
	Subven	-I.00E-05	-1 .46	* -0.0167
	location characteristics
	Density Averoff	-0.001 3.065	-2.49 4.07	* -0.124

* 0.1248

Table 2 .

 2 Probit analysis of the husbands' off-fann labour market participation

	lndependent variables	Coefficients	t-student	Partial derivative
	lntercept	-3.962	-3.27
	Human capital
	Ageh	0.1737	3.44	0.0406
	Ageh*ageh	-0.0019	-3.76
	Dumeducf	0.199	1.26	0.0533
	Dumeduch	0.51	2.61	0.117
	Dumagrf	-0.505	-0.85	-0.1085
	Dumagrh	-0.215	-0.86	-0.058
	Exp	-0.035	-4.4	-0.0082
	Dumpennf	0.262	1.07	0.0604
	Dumpennh	-0.494	-2.6	-0.1218
	Household characteristics
	Less6	-0.165	-0.97	-0.0386
	Kids612	-0.076	-0.69	-0.0178
	Oldl2	0.044	0.57	0.0102
	Nbind	-0.022	-0.45	-0.005
	Farm characteristics
	Size	-0.008	-3.87	* -0.1458
	Price	0.004	1.50	* 0.0963
	Dumgaec	0.287	1.29	0.067
	Dummicro	-0.062	-0.38	-0.0144
	Oum divers	-0.129	-1.01	-0.03
	Dumtourism	1.093	3.19	0.2553
	Dumdev	-0.395	-1.69	-0.0922
	Dumtract	-0.488	-3.46	-0.1139
	Exogenous incume
	Subven	-3.00E-05	-2.12	* -0.0507
	Location characteristics
	Density	-0.0003	-0.62	* -0.0087
	Averoff	4.072	5.54	* 0.9510

* For the continuous variable the figure indicates the value of the quasi-elasticity i.e. the percentage change of the probability upon a one percent increase of the variable considered. At the values used to compute the derivatives and the quasi-elasticities the husband's probability of off-fann participation equals 15.04%.

Table 3 .

 3 Pro bit analysis of the use of hired labour For the continuous variable the figure indicates the value of the quasi-elasticity i.e. the percentage change of the probability upon a one percent increase of the variable considered. At the values used to compute the derivatives and the quasi-elasticities the probability of hiring labour equals 31.06%.

	Independent variables	Coefficients	t-student	partial derivative
	Intercept	-1.714	-2.79
	Human capital
	Agef	0.031	1.26	0.0013
	Agef'l'agef	-0.00029	-1.17
	Dumeducf	0.28	2.61	0.0988
	Dumeduch	0.012	0.08	0.0041
	Dumagrf	-0.105	-0.3	-0.0371
	Dumagrh	0.226	1.55	0.0796
	Exp	0.007	1.29	0.0024
	Dumpennf	-0.00008	-0.0 1	-0.0001
	Dumpennh	0.236	2.15	0.0833
	Household characteristics
	Less6	0.155	3.67	0.0361
	Kids612	0.256	1.59	0.0289
	Old12	0.129	3.67	0.0172
	Nbind	-0.086	2.44	-0.0129
	Farm characteristics
	Size	0.007	8.4	* 0.1991
	Price	-0.003	-1.33	* -0.1028
	Dumgaec	-0.026	-0.17	-0.0936
	Dummicro	0.209	1.84	0.0186
	Dumdivers	0.216	2.52	0.054
	Dumtourism	0.292	1.04	0.1444
	Dumdev	-0.002	-0.02	-0.0072
	Dumtract	-0.109	-1.19	-0.0022
	Exogenous incarne
	Subven	l.50E-05	2.38	* -0.0371

*

This off-farm wage rate is assumed to be independent from hours worked but to be a function of human capital variables and location paramcters.

A reference yield is used to calculate the rate of compensation per hectare.

In the simulations of the efTect of the CAP reform we do not consider the husband's behaviour and the decision to use hired labour since in both probit equations the coefficient o f the product price is not significant.

labour. In particular, hired labour was shown to be a substitute for female labour. This result was derived under the assumption that we can define a consistent aggregate of hired labour by assimilating seasonal and regular hired labour. However previous studies [START_REF] Gunter | Dynamic labour demand schedules for US agriculture[END_REF] found that substitution relationships between family labour and these two types of hired labour are different. Hence the disaggregation of hired labour is a possible improvement. Furthermore, we might expect that variations in exogenous variables would induce different effects on these both types of labours. Second, an alternative empirical specification would be to consider explicitly in a simultaneous model (as a logit multinomial) the labour regimes (self sufficient, sellers, employers) which can appear for a farm household. This estimation procedure which is heavier in the interpretation of the results, will allow to anticipate how the different categories of households will respond to variations in the exogenous variables. For instance, such an analysis could be used to compare the effect of the CAP reform on off-farm labour for agricultural households in which the farm wife is not currently employed off-farm to agricultural households where the farm wife is employed off-farm.

Annex. Data definitions