Philippe Grangier

Revisiting Quantum Mysteries

In this article we argue that in quantum mechanics, and in opposition to classical physics, it is impossible to say that an isolated quantum system "owns" a physical property. Some properties of the system, its mass for example, belong to it in a sense close to that of classical physics; but most often a property must be attributed to the system within a context. We give simple motivations for adopting this point of view, and show that it clarifies many issues in quantum physics. I. OUR QUANTUM WORLD.

In many broadcasts or publications intended for the general public one meets the assertions that in the "quantum world" an object can be in several places at the same time, or that two particles located at astronomical distances can influence each other instantaneously, or that particles pass through walls... However, the world thus described is nothing but ours. All those who are familiar with the popularization of quantum physics are used to hearing these sentences, which fascinate and challenge. They often lead to the ideas, quite frustrating for the uninitiated, that quantum mechanics is a discipline full of paradoxes, and incomprehensible otherwise than by its mathematical formalism; and that even the best physicists disagree on what it means [START_REF] Laloë | Do We Really Understand Quantum Mechanics?[END_REF][START_REF] Peres | Quantum Theory: Concepts and Methods[END_REF]. Quantum mechanics being at the basis of innumerable technologies (electronics, lasers, medical imaging...) which have changed our daily life, and having never been disproved, this difficulty in communicating its physical content to the layman is surprising. It may even appear embarrassing, in a context where many countries are investing billions of Euros to develop technologies related to the "second quantum revolution", even more difficult to explain than the first one.

A legitimate question is then : would it be possible to formulate different statements, referring specifically to this mysterious "quantum world", but not appearing as a series of contradictions or absurdities? This is what we propose here, avoiding also any abusive recourse to mathematics: they are indispensable, but will only be mentioned briefly in a second time.

II. EMPIRICAL EVIDENCE AND ITS

CONSEQUENCES.

Our starting point will be a series of empirical observations, that is to say, observations that relate to experiments that are quite feasible and that have been carried out, sometimes recently. We will start from concepts known at the beginning of the 20th century (light is a wave, matter is made of atoms with a nucleus and electrons...) and we will introduce new phenomena, which we will then have to explain.

The fundamental physical fact is that observations of microscopic objects show a discrete, or quantized, character, of which here are some examples:

• The possible wavelengths of light emitted by atoms take only certain particular values, each atom being characterized by the "spectrum" of wavelengths that it can emit.

• Light can extract electrons (particles) from matter, but to describe this phenomenon called the photoelectric effect, we are led to admit that light is also formed of particles (photons).

• We can perform experiments where objects classically considered as particles (electrons, atoms...) give rise to interference effects, which are classically associated with waves, and not with particles.

A rather inevitable conclusion of these observations is that microscopic objects (atoms, electrons, photons...) have behaviors that combine a discrete character (i.e. described by integers) with the possibility of interference (traditionally associated with continuous waves).

Quantum mechanics was built on the basis of these observations, by a few brilliant physicists who invented a somewhat bizarre mathematical formalism, able to account for all these observations, and also to make countless predictions of new phenomena. This formalism, or even these formalisms, because there are several equivalents, have never been put in default, but the physical nature of the objects and properties they describe has remained obscure. This questioning, however intrinsic to physics, has given rise to innumerable "interpretations" and paradoxical statements -the sentences quoted at the beginning of this text are a very partial example.

It is therefore clear that an "ingredient" is missing that would give a physical meaning to this very efficient formalism; this is what we propose in what follows.

III. UNDERESTIMATING EMPIRICAL EVIDENCE ?

The normal approach in physics since Newton has been to define objects, to attribute properties to them, and to measure these properties. One then asserts that the object "has" this property, for example that it has a position, a velocity, a momentum... Does this "natural" approach work in quantum mechanics? Although physicists are extremely reluctant to admit it, the answer is clearly no -and this "no", correctly interpreted, provides the empirical element that is missing in the construction. So let's look at a simple experiment that shows that it is impossible to say that a quantum particle "owns" a property. Of course some particle's properties, its mass for example, belong to it in a sense very close to classical physics; but most often the situation is not so simple.

One can for example measure the angular momentum of an atom along a certain direction u, or the polarization (equivalent to angular momentum) for a photon. Such a measurement gives a discrete result, as explained above, and we can therefore say for example that we obtain the value +1, in adequate units. Moreover, if we repeat this same measurement on the same particle, we find +1 again: the result can therefore be predicted with certainty, as one would expect in classical physics.

But let's complicate things a bit, and measure the angular momentum in another direction, let's say v. We find for example +2, then we come back to u, and we measure again. In a classical framework we would expect to find the previous +1 result, but in fact we don't necessarily get +1, but a random result! In general this new result can take any value in a certain set that is defined by the system studied, for example (-2, -1, 0, +1, +2) for a certain type of atom, each of these values having a certain probability of being obtained. One sometimes speaks of "measurement-induced perturbation", a phenomenon that also exists in classical physics; but the randomness described here is inescapable: it is not possible to eliminate it completely, nor to trace it back to a "true value" of the measured quantity.

Very often this simple but fundamental experiment is not considered as a fact, but is immediately buried in mathematical formalism, invoking operators that do not commute or other arguments that are not explanations, but mathematical descriptions. However, this experiment is essential since it shows that the value attributed to certain physical properties varies in an irremediably random way when the measurement process is modified. However, this is not an incomprehensible blur, since as long as the measurement does not change, the value obtained always remains the same: certainty and repeatability remain, but under more restrictive conditions than those observed in classical physics.

If one legitimately considers that certainty and reproducibility are minimum requirements for a realistic description of the physical world, the inescapable consequence of the above observations is that the object to which one must attribute physical properties is not "a system", but "a system on which a given measurement is made", since only in this case the result does not change. This is an essential difference with Newtonian physics, and it can be used as a basis to build up quantum physics as we will see now.

IV. CONTEXTS, SYSTEMS AND MODALITIES.

These observations being very quantum but not really mysterious, we can formalize them (still without mathematics!) by introducing some vocabulary [START_REF] Grangier | Contextual objectivity: a realistic inter-pretation of quantum mechanics[END_REF][START_REF] Auffèves | Contexts, Systems and Modalities: a new ontology for quantum mechanics[END_REF][START_REF] Auffèves | Deriving Born's rule from an Inference to the Best Explanation[END_REF][START_REF] Grangier | Completing the quantum formalism in a contextually objective framework[END_REF][START_REF] Grangier | Contextual inferences, (non)locality, and the (in)completeness of quantum mechanics[END_REF][START_REF] Farouki | The Einstein-Bohr debate: finding a common ground of understanding ?[END_REF] :

• let us designate the devices, arranged to carry out a particular measurement on a given quantum system, by the word "context";

• us refer to a particular measurement result, on a given system in a given context, by the word "modality". By definition a modality characterizes the system as well as it is physically possible, and, once obtained, it is ideally repeatable with certainty as long as the context is not changed;

• us admit the following fundamental principle, called "contextual quantization": whatever the context, a measurement on a given system gives one modality among N possible ones, where the value of N characterizes the system; these N modalities are mutually exclusive, i.e. only one can be realized at a time.

It follows from this principle that if there are other different modalities in other contexts, the link between the modalities of different contexts can only be probabilistic; otherwise one would have a "super context" with a number of mutually exclusive modalities greater than N , which would contradict the fundamental principle [START_REF] Auffèves | Deriving Born's rule from an Inference to the Best Explanation[END_REF].

In this approach the probabilities in quantum mechanics are thus physically necessary, since they are imposed by the contextual quantization. It is crucial to note that the conjunction of the two ingredients, quantization and contextuality, is required: for example, many probabilistic phenomena are discrete but classical (heads or tails, dice game...), and others have contextual aspects (for example in a poll it is known that the order of the questions influences the answers) but they remain classical because quantization does not play a constraining role.

V. A LITTLE BIT OF MATHEMATICS.

The last step of our approach is to mathematize the principle of contextual quantization, to obtain the quantum formalism. It is crucial to underline that, as usual in physics, this step cannot be "deductive" (we cannot show that the formalism is the only one possible), but must be "inductive" : we propose a formalism, the most general one possible, respecting the imposed physical conditions, and we verify deductively that it works by describing well the observed effects. The theoretical question is therefore : given a modality (among N ) in an initial context, resulting from a previous measurement, what is the probability of obtaining another modality (among N ) in a final context associated with a new measurement ? We are therefore looking for an N ×N matrix of probabilities between the initial context and the final context, which is generally called a stochastic matrix.

The fundamental (and actually unique) mathematical ingredient is then to associate a projector (N × N matrix) with the set of all modalities that are related to each other with certainty, either in the same context (it is then the same modality) or in different contexts (this link is observed empirically). This set constitutes an equivalence class of modalities that are linked "extracontextually", and it is therefore called an extravalence class.

By admitting this fundamentally quantum idea, and a few simple consistency arguments [START_REF] Auffèves | Deriving Born's rule from an Inference to the Best Explanation[END_REF], there is in fact no longer any choice: powerful mathematical theorems allow us to show that the only possible theory is quantum mechanics.

More precisely, Uhlhorn's theorem [START_REF] Uhlhorn | Representation of symmetry transformations in quantum mechanics[END_REF][START_REF] Chevalier | Wigner-Type Theorems for Projections[END_REF] shows that unitary transformations between projectors are necessary to preserve the mutually exclusive character of events in each context; and Gleason's theorem [START_REF] Gleason | Measures on the Closed Subspaces of a Hilbert Space[END_REF][START_REF] Cooke | An elementary proof of Gleason's theorem[END_REF] shows that Born's law is necessary to respect the general structure of a probability law (see details in the Appendix).

The probability matrix we are looking for has then a particular form: it is unistochastic, i.e. it is constituted by the squared moduli of the coefficients of a unitary matrix. It also follows that a context is associated to a family of N orthogonal projectors, and that a change of context is associated to a unitary transformation between these families, which reconstitutes the usual framework of quantum probabilities [START_REF] Auffèves | Deriving Born's rule from an Inference to the Best Explanation[END_REF]. We note that within this framework a usual quantum state |ψ is incomplete, because it lacks the specification of the context to define a true probability distribution [START_REF] Grangier | Completing the quantum formalism in a contextually objective framework[END_REF][START_REF] Grangier | Contextual inferences, (non)locality, and the (in)completeness of quantum mechanics[END_REF]; this is consistent with attributing a modality to a system within a context.

VI. NO PARADOX ANYMORE ?

We will not go further into the mathematical details here, but to conclude let us try to translate into this new language the paradoxical statements quoted at the beginning of this article, again avoiding any formalism:

• an object can be in several places at the same time:

we often hear also that a quantum bit is in several states at once, and that this corresponds to the notion of quantum superposition. But what does this "at the same time" mean? In fact, it is a question of detecting a modality, therefore a certainty, but a certainty in another context. To say that a particle is "at the same time" in the two arms of the interferometer means in fact that if these two arms interfere constructively, we can predict with certainty where the particle will go. On the other hand, if the measurement is made inside the interferometer, then the result is random, according to the principles explained above: the measurements either inside or outside the interferometer correspond to different contexts.

• two particles separated by an arbitrarily large distance can influence each other instantaneously: this is the phenomenon of entanglement, and there is in fact no instantaneous influence at a distance, but again the manifestation of a modality, which is a certain and reproducible property of the pair of particles, e.g. that the total angular momentum of the pair is zero. This can be verified in the appropriate context, but if one makes the measurement in a context where the particles are spatially separated one will obtain necessarily random (because of the change of context, see above), but strongly correlated results. This is again a consequence of Born's law, which still holds when the context changes, and illustrates nicely the predictive incompleteness of |ψ without a context [START_REF] Grangier | Contextual inferences, (non)locality, and the (in)completeness of quantum mechanics[END_REF].

• the quantum particles pass through the walls: this is the tunneling effect, an "obvious" effect for classical waves (it is due to so-called evanescent waves), but "impossible" for classical particles. But the classical notions of waves and particles are inadequate here, and one must consider probability amplitudes, which allow to physically visualize Born's law in this case. The change of context corresponds then to the passage from a representation where the momentum of the particle is defined, to another "incompatible" one where its position is defined. A usual quantum wave-packet is somewhere between these two extremes, but in any case speaking about particles or waves with a classical behavior is a fiction, sometimes useful but most often misleading.

The above examples do not necessarily require to fully determine the modalities, i.e. all the system's properties, but the general ideas remain valid. It may be noticed that contextual quantization is difficult to admit, even for physicists, since it asserts that properties/modalities belong to a system within a context, and not to a system considered as "alone in the universe", as classical physics would assume. But the notion in itself is not paradoxical, and if we admit that such is the behavior of the nature in which we live, then the statements related to the "quantum world" finally cease to appear as a long series of contradictions or absurdities.

VII. QUANTUM COMPUTING IN CONTEXT.

This point of view also underlines that statements made "out of context" (such as those concerning the famous Schrödinger's cat, both dead and alive) are meaningless. Any statement concerning a quantum system, even a large one such as a quantum computer, is only meaningful in combination with a relevant context in which the modalities under consideration can manifest themselves and be observed.

This allows us to come back to the technological promises mentioned at the beginning of this article. It is true that a quantum computer has, in principle, a gigantic computing power, based on the manipulation of entangled states involving a very large number of quantum bits. But this manipulation will only be possible if an adequate context is available, which will also be of a very high complexity -even if it is then a classical complexity.

Then is it certain that the ensemble formed by the quantum "heart" (the system) surrounded by classical instruments capable of interacting with it (the context) will be more efficient than a usual, classical supercomputer ? The answer to this question remains open, and the scientific and rational conclusion is that we have to go forward and see what happens -it may not be exactly what we expect, but this is very common in experimental physics, and it could be just as interesting.

Appendix : Uhlhorn's and Gleason's theorems.

As written above, the empirical facts that we want to describe mathematically are :

(i) in each context a measurement provides one modality among N possible ones, that are mutually exclusive. No measurement can provide more than N mutually exclusive modalities, and once obtained in a given context, a modality corresponds to a certain and repeatable result, as long as one remains in this same context.

(ii) the certainty and repeatability of a modality can be transferred between contexts, this fundamental property is called extracontextuality of modalities. All the modalities that are related together with certainty, either in the same or in different contexts, constitute an equivalence class that we call an extravalence class.

Our central mathematical ingredient is then to associate a rank-one projector P i (a N × N hermitian matrix such as P 2 = P ) to each extravalence class of modalities, in such a way that the N projectors associated with the N modalities within a given context are mutually orthogonal. In addition, we assume that the probability to get a given modality is a function of its associated projector only. The heuristic motivation for this choice is that it ensures that the events associated with modalities cannot be subdivided in more elementary events, as it would be the case with classical (partition-based) probabilities.

The choice of a specific orthogonal set of projectors associated with a context is not given a priori, but mutually exclusive modalities in the context should stay so, whatever choice is made for the projectors. It means that if two orthogonal projectors are associated with two mutually exclusive modalities, they should stay orthogonal whatever choice is made for the projectors associated with a "reference" (fiduciary) context. Then Uhlhorn's theorem [START_REF] Uhlhorn | Representation of symmetry transformations in quantum mechanics[END_REF][START_REF] Chevalier | Wigner-Type Theorems for Projections[END_REF] warrants that the transformations between the sets of projectors associated with different contexts must be unitary or anti-unitary; anti-unitary operators are associated with time reversal, so we will omit them here and consider only unitary ones.

We thus get a major result : once a set of mutually orthogonal projectors associated with a fiduciary context has been chosen, the sets of projectors associated to all other contexts are obtained by unitary transformations, so we are simply "moving" in a Hilbert space. There are also various arguments for using unitary (complex) rather than orthogonal (real) matrices; in our framework the simplest argument is to require that all permutations of modalities within a context are continuously connected to the identity, which is not possible with orthogonal matrices, but is possible with unitary ones.

The next step is to consider the probability f (P i ) to get a modality associated with projector P i . By construction a context is such that i=N i=1 P i = I, and i=N i=1 f (P i ) = 1 for any complete set {P i }. But these are just the hypothesis of Gleason's theorem [START_REF] Gleason | Measures on the Closed Subspaces of a Hilbert Space[END_REF][START_REF] Cooke | An elementary proof of Gleason's theorem[END_REF], so there is a density matrix ρ such that f (P i ) = Trace(ρP i ). If the value 1 is reached, then ρ is also a projector Q j and f (P i ) = Trace(Q j P i ) which is the usual Born's formula.

We have thus reconstructed the basic probabilistic framework of quantum mechanics. In order to use it, one must define explicitly the relevant physical properties and associated contexts, that may go from space-time symmetries (Galileo group, Lorentz group) [START_REF] Laloë | Symétries continues", to appear in "Collection: Une Introduction à[END_REF] to qubits registers. In any case, contextual quantization applies and sets the scene where the actual physics takes place.

An interesting question is whether this probabilistic formalism might be used outside physics, for its own virtues in a "quantum-like" analogy [START_REF] Khrennikov | Ubiquitous Quantum Structure : From Psychology to Finance[END_REF][START_REF] Khrennikov | Interpretations of probability[END_REF]. However, it can be observed that the requirements (i) and (ii) written above are strictly respected in quantum mechanics, hence the strong predictive power, whereas this is only loosely the case in other domains. Therefore whether or not a variant of this formalism might have a useful predictive (and not only descriptive) power outside physics remains an open -and quite stimulating -question.

Acknowledgements : The author thanks Alexia Auffèves, Roger Balian, Nayla Farouki and Franck Laloë for many interesting and stimulating discussions.


Dmitry A Testov

A P Severyukhin

B Roussière

N Arsenyev

F Ibrahim

M Lebois

I Matea

Yu Penionzhkevich

V Smirnov

E Sokol

I Stefan

D Susuki

D Verney

Jh Wilson

Studies of 123 Ag β-decay at ALTO

The neutron rich 123 Ag nucleus was populated via induced photofission of UC x at the ALTO ISOL facility. Its β-decay properties were studied by means of detecting β-delayed γ and neutron activities. The measured half-life T 1/2 = 0.350(20) s agreed with the earlier data; the β-delayed neutron emission branching P 1n = 1.01(24)% was reestablished using γ and neutron counting. Keywords β-decay • β-delayed neutron emission • beams of radioactive nuclei details in ref. [8]). In case of the neutron-rich nuclei in 51 the vicinity of A ≈ 130, the E1 strength could be sig-52 nificantly more important than in nuclei close to the 53 valley of β-stability. 54 In this context, the β-decay properties of heavy Ag 55 and Cd isotopes are ideal candidates for such studies. 56 Particularly, in very neutron-rich Ag nuclei the total 57 energy released through beta-decay, Q β , can go be-58 yond 10 MeV (in case of 130 Ag, for example, Q β = 59 15.42(30) MeV [9]) and subsequently even deeply bound 60 neutrons can decay into protons. When the Cd daugh-61 ter nucleus is produced in a high-energy configuration 62 above the neutron separation energy (S n ), it usually 63 de-excites through the neutron emission. On the other 64 hand, the rearrangement of nuclear matter could pro-65 ceed through collective modes of de-excitation in the 66

Introduction

The delayed neutron emission probability (P n ) and the half-life are the two key ingredients of β-decay that depend on the integrated quantities of the nuclear structure. The half-life provides information on an integrated total strength function, whereas P n value probes the part of the strength function above the neutron separation energy (S n ). Therefore, measurements of these values in experiments often provides the first insight into the nuclear structure. Although the mechanism of β-decay of nuclei is in principle understood, development of a satisfactory description coherent with new data sets for neutron-rich species is highly required. A significant experimental effort is currently underway to reveal the role of neutron excess in the shell evolution in these regions [START_REF]The Euroschool 323 Lectures on Physics with Exotic Beams[END_REF]. The current experimental study of β-delayed (multi) neutron emission in the regions of the chart of nuclides, which are important from the astrophysical point of view, is facing a new era: pure beams of neutron-rich nuclei of hard-to-reach mass regions of a e-mail: dumon@jinr.ru 78 Ni and 132 Sn, have become available at radioactive 31 ion beam facilities [START_REF] Blumenfeld | [END_REF].

46

The occurrence of non-negligible low-lying E1-strength 47 can influence the radiative neutron capture cross sec-48 tion by orders of magnitude and, consequently, also the 49 rate of the astrophysical r-process nucleosynthesis (see daughter isotope, involving also the most superficial nucleons. Namely, microscopic calculations by refs. [10,[START_REF] Severyukhin | Proceedings of EXON-2018[END_REF] showed that the Gamow-Teller (GT) transitions of deeply-bound neutrons can trigger coherent dipolar oscillations (like the PDR), which in turn engender a significant emission of E1 γ-radiation. It is worth mentioning that the inclusion of the coupling terms with more complex configurations than 1p -1h ones substantially enriches the low-energy spectrum of nuclei in the A ≈ 130 mass range, see, e.g., ref. [START_REF] Severyukhin | [END_REF].

The r-process abundance region around A ≈ 110, prior to the A = 130 peak, is one of the regions of intense interest, where the astrophysical models underestimate the abundances by an order of magnitude or more [13]. The newly measured P n values in this region will directly affect the final r-process abundance pattern between the abundance ,,trough" the region taneous fission studies [26] but modified to allow β delayed neutron measurements. TETRA response to the β-delayed neutron is fully understood by means of MCNP simulations [27] and by the reference measurements of β-decay of heavy 96,97 Rb isotopes [28]. The neutron efficiency n = 53(2) % was measured with a 252 Cf source. The Ge(HP) detector was calibrated using standard sources of 60 Co and 152 Eu. The absolute efficiency measured for 1.3 MeV γ-rays was γ ∼ 0.6 %.

The efficiency of the β detector β = 37(4)% was determined using ratios of area under peaks for identified transitions in β-decay daughters in the direct γ and β gated γ-spectra using the coincidence method described in ref. [22,28].

The measurement cycle was divided into three stages.

The data was collected for T meas = T beam + T decay : after the accumulation of a source during T beam , the beam was deviated to allow counting the decay of the source for T decay . At the third stage the data acquisition system was stopped and the tape was rewound for 2 m to transport the source, accumulated on the tape, outside the detection system. Adjusting the T beam and T decay parameters, see Table 1, it was possible to favour detection of decay radioactivity from 123 Ag. Since the half-life of 123 Ag is in a hundred ms range, in the offline analysing procedure for every cycle, only the data collected within τ meas = τ data +τ bg , defined as in Fig. 1 was sorted. The γ-ray spectra incremented within τ data and τ bg were denoted as S data and S bg correspondingly. ones quoted in ref. [18], see Table 2. The high production rates of Cd and In isotopes together with a small P 1n value for 123 Ag as well as large neutron coincidence window (100 µs due to the neutron moderation) resulted in a fact that the neutron tag did not lead to further purification of the recorded γ-ray spectra.

P 1n extraction procedure

The P 1n value of a neutron precursor (A, Z) is the ratio of the number of decays to the β-delayed neutron daughter (N β dn ) to the total number of β-decays of the sition probability, the p i 1n can be found as:

243 p i 1n = N β dn N total = N n n • γ i • β • I β i A i γ • (1 -T γ dead ), (1)
where A i γ is the number of counts under the correspond- 4.

264

After determining P 1n and T 1/2 the 123 Ag production yield (Φ) was estimated to be ≈ 7700 s -1 from the equation:

T beam 0 A beam (t) + τ decay T beam A decay (t) = N total (2)
where A beam (t) and A decay (t) are the activity of 265 123 Ag during the collection and decay measurements 266 respectively defined using the Bateman rules:

267 T beam 0 Φ • (1 -e -t•λ )δt+ + τ decay T beam Φ • e -(t-T beam )•λ (1 -e -T beam •λ )δt = = N total , (3)
where λ is the decay constant of 123 Ag.

268

The production yields of 123 Cd, 123 Cd m and 123 In,

32

  Reliable knowledge of the nuclear physics proper-33 ties for the extremely neutron-rich nuclei along the r-34 process path are needed to interpret the observational 35 data in the framework of proposed astrophysical mod-36 els [3-5]. As the neutron excess increases near the drip 37 line, we expect the properties of the giant resonances 38 and of the low-lying collective states to be dramatically 39 affected at extreme N/Z ratios [6]. The formation of 40 a neutron-skin may introduce a new type of collective 41 mode in neutron-rich nuclei corresponding to an out-of-42 phase vibration of the skin against the neutron-proton 43 inert core [7]. The concentration of the electric dipole 44 (E1) strength around the particle separation energy is 45 commonly called the pygmy dipole resonance (PDR).

3 2

 32 (A = 114-120) and the onset (A ≈ 123) of the A = 130 peak region. The β-delayed neutron emission of 123 Ag has been reported previously from the OSIRIS mass-separator in refs. [14, 15]. The mass separated heavy fission products have been studied using a neutron counter comprised of 30 He tubes. The weak neutron activities seen at the 122 and 123 masses were assigned to the silver isotopes. Consequently, due to 390(30) ms half-life measured for the 123 mass the neutron activity (still weak) was assigned to decay of 123 Ag. Later at the TRISTAN on-line isotope separator facility at Brookhaven National Laboratory, the half-lifes and P n values have been measured for 121-124 Ag [16]. In a single set of experiments, using mass-separated fission fragments, the neutron and β growth (decay) curves were measured simultaneously at masses 120-124 and at 127. The neutron decay curve resulted in a 300(10) ms half-life which was assigned to decay of 123 Ag. The P n value of 0.55(9)% was determined as a ratio of the neutron to β saturation activities, corrected by the corresponding efficiencies. Neutrons were straight detected by 40 3 He proportional tubes embedded in polyethylene. The more recent measurements of the β-delayed neutron emission branching for neutron-rich Ag isotopes, populated in a fragmentation reaction, have been done at the National Superconducting Cyclotron Laboratory at Michigan State University [17]. The yielded half-life of 272(24) ms was derived using time difference between the ion implantation and the correlated β-decay. The half-life was slightly shorter than reported previously [15, 16]. The P n value was determined as 1.0(5)% based on the number of β-neutron coincidence events divided by the number of β decays following the implantation of 123 Ag heavy ions. Neutrons were detected by the NERO array consisting of 60 3 He and BF 3 counters. Regardless the numerous studies the 123 Cd level scheme 120 populated in the decay of 123 Ag was only reported in 121 the study performed at the Buenos Aires Tandem ac-122 celerator facility [18]. The 123 Ag nuclei was produced as 123 a mass-separated fission product from the fast-neutron 124 fission of natural U. Then γ-spectrometry measurements 125 were performed using two Ge(HP) detectors. Spins and 126 parities for the most important states were suggested 127 upon log(ft) values obtained from measured γ-ray tran-128 sition intensities. 129 In the work presented here the neutron-rich 123 Ag 130 nucleus was revisited to study for the first time its β-131 decay by simultaneous detection of β-delayed neutron-132 and γ activities. The heavy Ag nuclei were populated 133 using induced photofission of uranium at the ALTO 134 ISOL facility [19, 20] and measured at the BEDO β-135 decay tape station [21] equipped with the TETRA neu-136 tron detector containing 80 proportional counters of 137 3 He [22]. Up to now at ALTO the studies were con-138 centrated in the region of heavy Ga nuclei crossing 139 the N = 50 neutron closed shell: competition of the 140 neutron emission process with the emission of high en-141 ergy γ-rays [10]; population in β-decay resonance low-142 energy structures [23]. In the present paper we report 143 the first measurement of β-delayed neutron precursor 144 in the 132 Sn region performed at ALTO. 145 Experimental details 146 Neutron-rich isotopes were produced via induced photofis-147 sion at ISOL facility ALTO [19, 20]. A 50 MeV, ∼ 7 µA 148 electron beam was applied to a UC x production target 149 heated up to 2000 o C to obtain low energetic beams 150 (∼30 keV) of radioactive isotopes. Photofission prod-151 ucts were extracted from the target with a hot plasma 152 universal ion source FEBIAD (Forced Electron Beam 153 Induced Arc Discharge) [24] with 1+ charge state and 154 delivered to the mass separator PARRNe. The reso-155 lution (m/δm ≈ 1500) was high enough to provide iso-156 baric selection of A = 123. The members of the isobaric 157 chain such as Pd and Rh were refractory metals whose 158 efficiency to release from the source was very low. 159 The modular detection system equipped with the 160 BEDO tape station [21] was installed behind the mass 161 separator PARRNe and was operated in the neutron 162 detection configuration. It consisted of highly efficient 163 neutron detector TETRA [22]; a ∼4π β scintillator and 164 a High-Purity coaxial Germanium γ-ray detector (EUROGAM-165 1 type [25]). Mass-separated beams were collected onto 166 an Al-coated Mylar movable tape at the geometrical 167 centre of the array. TETRA is one of the long-neutron 168 counters constructed at JINR Dubna originally for spon-169

Figure 2 (

 2 Figure 2 (top), displays the β gated γ-spectrum (S β data ) which reveals transitions characterising β-decay of the A = 123 isobaric multiplet (Ag, Cd, In). Whereas the background spectra S β bg , Fig. 2 (middle), shows transitions only in the longer lived In and Sn. The subtraction of both spectra resulted in S β sub spectrum plotted in Fig. 2 (bottom). In this spectrum transitions in 123 Cd were favoured to be on the positive part of the scale whereas the γ-ray transitions in the longer-lived daughters are ,,reversed" to the negative part. This method allowed us to clearly observe five transitions previously attributed to 123 Cd [18]. The relative intensities measured in both S β data and S β sub are in agreement with the

223Fig. 2 β

 2 Fig. 2 β gated γ-spectra integrated over 63 cycles recorded for the mass A = 123 settings. S β data (top), S β bg (middle), S sub (bottom) see text for details. Transitions are indicated in the daughter nuclei: 123 Ag→ 123 Cd (•); 123 Cd→ 123 In ( ); 123 Cd m → 123 In ( ); 123 In→ 123 Sn ( ); 123 In m → 123 Sn ( )

244T 1 / 2

 12 ing peak in the S β data (S β sub ) spectra, n , β and γ i are 245 efficiencies of neutron, β and γ (for i-th transition) de-246 tection respectively and T dead γ is the dead time of the 247 γ electronic channel. The dead time was estimated by 248 monitoring the occupation fraction of the channel. The 249 average dead time over τ meas was T dead γ = 34(6)%. The 250 adopted P 1n value then was calculated as the weighted 251 average of p i 1n extracted for different transitions. 252 This procedure was applied twice to derive the P 1n 253 value using S β data and S β sub spectra. In case of S β data the 254 N β dn was measured as the number of neutrons detected 255 within τ data and corrected for the neutron background. 256 In case of S β sub the N β dn was defined as the differ-257 ence between the number of neutrons detected within 258 τ data and τ bg . The 1.01(24) % and 1.03(36) % values 259 for P 1n , obtained using S β data and S β sub spectra respec-260 tively, are consistent within the error bars. The adopted 261 values P 1n = 1.01(24) % (as the most precise) and 262 = 0.350(20) ms are in good agreement with the 263 results of different experiments cited in Table

269

  delivered with the beam in the present experiment and 270 reported in

4 275

 4 ConclusionsThe paper reports on the first measurement of a de-276 layed neutron precursor in the neighbourhood of 132 Sn 277 produced at ALTO facility. β-decay of the neutron-rich 278 123 Ag was investigated using a A = 123 mass-separated 279 beam delivered to the neutron detector TETRA set up 280 at the BEDO tape station. The detector arrangement 281 allowed simultaneously recording of β-, neutron-and γ-282 radiation accompanying the decay of 123 Ag for the first 283 time. In addition to the 123 Ag ground state, the iso-284 meric and ground states in 123 Cd and 123 In were also 285 populated by induced uranium photofission. The iden-286 tification of the isotopes was reliably performed based 287 on the recorded β gated γ-spectra. Due to the fact that 288 neutron emission was not energetically possible in the 289 members of the A = 123 isobaric multiplet, other than 290 Ag, the observed neutron activity was attributed to the 291 β-n decay of 123 Ag.

314

  The authors wish to acknowledge support from the 315 bilateral agreement JINR-CNRS and the TETRA col-316 laboration. We thank the technical staff of the Tan-317 dem/ALTO facility for their assistance with the exper-318 iments and for providing excellent quality radioactive 319 beams. The use of one Ge detector from the French-320 UK IN2P3-STFC Gamma Loan Pool is acknowledged.321

Table 1

 1 Tape-cycle parameters in ms used for the A =123 settings: T beam and T decay are the duration of the beam collection and the beam-off source decay running acquisition time correspondingly; τ data , τ bg and τ decay are as defined in Fig.1and N cycles is the total number of tape cycles.

	T beam T decay τ data τ decay	τ bg	N cycles
	1000	4000	1500	500	1500	63
			T meas	Hardware cycle parameters during data taking
	T beam			T decay
	τ data	τ decay	τ bg = τ data	Software cycle parameters for	t
						data analysis
			τ meas
	Fig. 1 Time-cycling parameters for the data taking and anal-
	ysis

Table 2

 2 Transitions observed in 123 Ag

	β---→ 123 Cd decay.

precursor (N total ). The experimental methods to derive 222 P 1n differ in a way how N β dn and N total are inferred.

Table 3 ,

 3 were estimated by using sets of

	271
	Bateman equations similar to Eq. 2 and 3, known half-

272

lives and the absolute intensities of β-decay to excited 273 states in daughter nuclei cited in the literature

[32]

.

274

Table 3

 3 Measured production yields (Φ) for A =123 mass isotopes implanted into the tape; P 1n and T 1/2 for 123 Ag as measured in this work, for others as recommended by the evaluator[32].

				T 1/2			P 1n , %	Φ
				s			%		s -1
		123 Ag		0.350(50)		1.01(24) 7.7(40)•10 3
		123 Cd		2.10(2)		-		∼ 4.7•10 4
		123 Cd m	1.82(3)		-		∼ 1.4•10 5
		123 In		6.17(5)		-		∼ 1.4•10 6
		350
		300
	Count / 100 ms	200 250
		150
		background
		100
		0	0.5	1	1.5	2	2.5	3	3.5	4
						Time, [s]
	Fig. 3 Neutron activity curve from photo-fission fragments
	recorded for A = 123 mass separator settings. The curve de-
	noted by black points was accumulated within 93 cycles, see
	Table. 1; the red curve is the result of the obtained fit with
	T 1/2 = 0.350 (20) ms.

Table 4 T

 4 1/2 and P 1n of 123 Ag neutron precursors. The type of β-delayed activity for half-life (γ, β, neutron or i-β -ion-β correlated decay curve) is shown by the superscript; 123 Ag production method is indicated as mass. sep. or laser for ISOL mass separated and selectively laser ionized fission fragments correspondingly; or frag. -reaction of fragmentation. Lund, and G. Rudstam, Phys. Rev.C 13, 321 (1976) 15. E. Lund, and G. Rudstam, Phys. Rev.C 13, 1544 (1976) 16. P. L. Reeder, R. A. Warner, and R. L. Gill, Phys. Rev.

	T 1/2	P n	Measured	Neutron	Production	Facility
	ms	%	activities	detector	method
	350(20) n 1.01(24)	γ, β, n	TETRA	mass. sep.	ALTO, present
	272(24) β	1.0(5)	β, n, ion	NERO	frag.	MSU, [17]
	293(7) n	-	β, n	not reported	laser	ISOLDE, [29]
	300(10) n	0.55(9)	β, n	40 x 3 He	mass. sep.	TRISTAN, [16]
	350(40) γ	-	γ	-	mass. sep.	TANDAR [18]
	300(20) γ	-	γ, β	-	mass. sep.	TRISTAN, [30]
	390(30) n	-	β, n	30 x 3 He	mass. sep.	OSIRIS [14, 15]
	0.296(7)	0.56(9)			recommended by evaluator [31]
	14. E.

C 27, 3002(R)

(1983)


Saeid Homayouni

Christian Germain

Olivier Lavialle

Gilbert Grenier

Jean-Pascal P Goutouly

Cornelis Van Leeuwen

Jean-Pierre P Da Costa
email: jean-pierre.dacosta@ims-bordeaux.fr.tél.33540002634

Abundance weighting for improved vegetation mapping in row crops. Application to vineyard vigour monitoring

Keywords: Remote sensing, Multispectral data, Precision viticulture, Row crop, Vine, Vegetation index, Spectral unmixing, ICA, Deformable models. d'améliorer Télédétection, Données multi spectrales, Viticulture de précision, cultures en rangs, Vigne, Indice de végétation, Séparation spectrale, ACI, Modèles déformables

come   L'archive ouverte pluridisciplinaire

Résumé

Nous présentons une approche globale pour la cartographie de la vigueur sur des cultures en rangs, basée sur l'étude d'images de télédétection multi spectrales en haute résolution. Notre principale contribution consiste à prendre en compte l'abondance de végétation pour le calcul d'indices de vigueur. Développée dans un contexte viticole, cette approche est transposable à toute culture en rang, notamment en arboriculture. Elle se fonde à la fois sur les propriétés spatiales et spectrales des images. L'information spectrale est exploitée à l'échelle du pixel par un algorithme d'ACI pour produire des cartes d'abondance de végétation. Concernant l'information spatiale, un algorithme de modèles déformables permet d'ajuster un réseau de rectangles aux différentes plantes. Les deux types d'information sont alors combinés pour calculer des indices de vigueur, pondérés par les mesures d'abondance, et d'affecter ces indices à des plants spécifiques de plantes. Les mesures ainsi établies sont finalement utilisées pour la cartographie intra-parcellaire de la vigueur. Une procédure de validation a été mise en place sur une parcelle viticole. Il est montré que la prise en compte de l'abondancedirectement ou comme pondération dans le calcul d'indices de végétations -permet

Introduction

Estimation, evaluation and monitoring of biomass are key factors for the enhancement of productivity and profitability in agriculture. Vigour, defined as the biomass annually stored per area unit [START_REF] Champagnol | Eléments de physiologie de la vigne et de viticulture générale[END_REF], is an objective measure of biomass. Biomass production concerns various parts of the plant including roots, woods (trunk and shoots), leaves and fruits. Leaves are of particular interest since they are the most visible part of the whole biomass production system, their characteristics being directly affected by the health, stress and diseases of the plants. Quantifying the amount of leaves can thus help in the monitoring of crops during the production season not only in quantity but also in quality.

Indeed, in some cases such as viticulture, the highest quality potential is obtained when vine vigour is moderate. Vine vigour is highly variable among vineyard blocks as well as inside the blocks. The reasons for this variability are mainly soil type, soil depth and topography [START_REF] Bramley | Managing variability in viticultural production[END_REF]. Excessive vigour induces high yield and shaded canopy, resulting in quality losses, whereas very weak vigour leads to yield decrease and financial losses. Reliable vigour monitoring is thus essential to the balanced management of vine production.

Thanks to recent advances in remote sensing, there has been a growing interest in its application to precision agriculture (e.g. [START_REF] Brisco | Precision agriculture and the role of remote sensing: a review[END_REF][START_REF] Tisseyre | Precision viticulture: precise location and vigour mapping aspects[END_REF][START_REF] Bannari | Synergy among GIS, GPS and Hyperspectral Remote Sensing in Precision Farming[END_REF][START_REF] Hall | Characterising and mapping vineyard canopy using high spatial-resolution aerial multispectral images[END_REF]. In viticulture, multi/hyper spectral airborne/satellite imagery has made possible a flexible mapping of yield, soil and diseases (e.g. [START_REF] Bramley | Managing variability in viticultural production[END_REF][START_REF] Lamb | Monitoring vineyard variability from the air[END_REF][START_REF] Carothers | Imagery technology meets vineyard management[END_REF][START_REF] Dobrowski | Grapewine dormant pruning weight prediction using remotely sensed data[END_REF][START_REF] Johnson | Mapping vineyard leaf area with multispectral satellite imagery[END_REF][START_REF] Hall | Characterising and mapping vineyard canopy using high spatial-resolution aerial multispectral images[END_REF][START_REF] Zarco-Tejada | Assessing vineyard condition with hyperspectral indices: Leaf and canopy reflectance simulation in a row structured discontinuous canopy[END_REF]. In particular, various indicators of vigour based on leaf reflectance are accessible by remote sensing imagery, e.g. [START_REF] Rouse | Monitoring the vernal advancement and retrogradation (green wave effect) of natural vegetation[END_REF], [START_REF] Richardson | Distinguishing vegetation from soil background information[END_REF], [START_REF] Huete | A soil-Adjusted Vegetation Index (SAVI)[END_REF].

Such indicators, called vegetation indexes (i.e. VI) are computed at the pixel level by combining various bands of the spectral reflectance. They make use of spectral information only without taking into account any spatial information. Simple and fast to compute, they allow the evaluation of vigour while avoiding the direct labour-intensive measurement of biologic features such as the total leaf area. VI maps have been widely used especially for soil covering crops like wheat and corn. In contrast, for perennial woody species conducted in rows such as vines, the presence of a significant amount of bare soil (or cover crop) on remotely sensed images makes biomass quantification by means of VIs less obvious.

Applied to crops such as vines, the precision agriculture concept implies that the plot can not be processed as a whole. Each row, or even each plant, has to be considered separately so that vegetation variability can be described at the plant level. Inter-rows, which often consist of bare soil or grass, should not interfere in the description of the canopy. For such a purpose, very high-resolution images, acquired with either multispectral or hyperspectral sensors, are necessary to attain accurate crop versus non-crop discrimination. Image classification algorithms, e.g. applied to vegetation indexes, can then make use of spatial or/and spectral information to discriminate between crop and non-crop pixels. For instance, in the case of vines, [START_REF] Hall | Characterising and mapping vineyard canopy using high spatial-resolution aerial multispectral images[END_REF] propose an algorithm ("Vinecrawler") that labels image pixels as vine, if their NDVI values [START_REF] Rouse | Monitoring the vernal advancement and retrogradation (green wave effect) of natural vegetation[END_REF] exceed a predefined threshold. They are aggregated into rows which are finally processed as a whole. Semi-automated frameworks have also been applied [START_REF] Marguerit | La télédetection à haute résolution pour spatialiser les variations intra-parcellaires de la vigueur de la vigne[END_REF][START_REF] Ferreira | Transformation of high resolution aerial images in vine vigour maps at intra-block scale by semi automatic image processing[END_REF]) to extract vigour information and map it spatially in a row-column system. In these works, the pixels are considered as pure pixels. However, remotely sensed optical images are concerned by spectral mixing i.e. the mix of several contributions in a single pixel. In the particular case of row crops, the pixels can not be classified by basic algorithms into crop or non-crop pixels. The reason is that most pixels result from a mix of crop, inter-row soil, grass and even shadows.

Therefore, soft classification techniques or unmixing approaches have to be considered to discriminate between crop and non-crop, or at least to estimate, for each pixel, the abundance of vegetation. Most unmixing approaches are based on supervised or semi-supervised techniques where the spectra of the species or materials are known or are easily available from images [START_REF] Pacheco | Contribution of Hyperspectral Remote Sensing for LAI Estimation in Precision Farming[END_REF][START_REF] Bannari | Estimating and mapping crop residues cover on agricultural lands using hyperspectral and IKONOS data[END_REF]. Generally such methods deal with non-structured crops at the farm level. Previously, we proposed an unsupervised method for partial unmixing which aimed at evaluating the amount of canopy in each pixel [START_REF] Homayouni | Partial unmixing of multi or hyperspectral images using ICA and Fuzzy Clustering techniques. Application to vegetation mapping on vineyards[END_REF]. It was applied to viticulture, at the plot level.

In this paper, a non supervised framework is proposed to transform high resolution multispectral images of row crops into vegetation vigour maps. Its main point is to take advantage of both spectral and spatial features to produce improved vigour indexes that take the row layout into consideration. Spectral information is used at pixel level to compute abundance coefficients which are used as weights in the computation of vegetation and leaf area indexes. Spatial information is used to bring back spectral measurements to specific row and plant positions.

Section 2 is dedicated to the description of our framework. In section 2.1, we recall the main principles of the row detection algorithm presented in [START_REF] Bobillet | Row detection in high resolution remote sensing images of vine fields[END_REF]. It aims at an accurate positioning of rows and individual vines for further spatialization of spectral measurements. In section 2.2, we address the problem of spectral mixing. A method is proposed for the semi-automated production of abundance maps based on independent component analysis. Then, in section 2.3 a geometric procedure is proposed to combine spatial information about rows and spectral information from abundance maps and vegetation indexes. It results in plant by plant vigour estimations. In section 3, the whole procedure is applied and validated on multispectral images of vineyards. Plant scale measurements are used to produce vigour maps by interpolation. They are finally correlated to ground measurements for global validation of the framework.

A new framework for vigour analysis

The row detection algorithm

Most computer vision methods for detecting crop rows are implemented in real-time systems for automatic guidance of agricultural implements (e.g. [START_REF] Keicher | Automatic guidance for agricultural vehicles in Europe[END_REF]Seufert 2000, Hague and[START_REF] Hague | A bandpass filter-based approach to crop row location and tracking[END_REF]. Images are taken by on-board cameras and show only a small number of rows.

The Hough transform is often used to recover the linear structures [START_REF] Marchant | Tracking of row structure in three crops using image analysis[END_REF][START_REF] Keicher | Automatic guidance for agricultural vehicles in Europe[END_REF]. Usually, the very high resolution of the considered images and the small number of rows lead to sufficiently accurate results. When lower computational time is required, tracking algorithms based on Kalman filtering [START_REF] Hague | A bandpass filter-based approach to crop row location and tracking[END_REF] or linear regression approaches [START_REF] Søgaard | Determination of crop rows by image analysis without segmentation[END_REF] are also implemented.

Unlike the real-time context, computational time is not a critical issue for accurate location of rows in crop monitoring by remote sensing. On the other hand, row position and row angle estimations have to be very precise, due to the image size and to the length of the rows. The specific arrangement of individual crops suggests the use of the Hough transform. However, previous attempts, not reported in this paper, have shown that the Hough transform is not adapted to locate vine rows. The large number of linear structures and the presence of undesirable alignments make the row detection difficult.

In order to retrieve the rows with high accuracy, we have chosen an algorithm based on active contour models, i.e. snakes [START_REF] Kass | Snakes: Active Contour Models[END_REF], previously described in [START_REF] Bobillet | Row detection in high resolution remote sensing images of vine fields[END_REF].

This algorithm implements an active contour network which aims at fitting a line to each row through a global convergence process. Prior segmentation of the parcel is needed so that peripheral vegetation, buildings, roads etc. do not interfere in the row detection process. Figure 1 shows an example of segmentation obtained on a vine parcel using the algorithm described in [START_REF] Da Costa | Delineation of Vine Parcels by Segmentation of High Resolution Remote Sensed Images[END_REF]. This prior segmentation step can be manual or automatic. Both the segmentation and the row detection algorithms operate on greyscale images. The multispectral image under consideration has thus to be transformed into a grey scale image which discriminates the crop rows from their background. This is done easily by computing any vegetation index, e.g. NDVI [START_REF] Rouse | Monitoring the vernal advancement and retrogradation (green wave effect) of natural vegetation[END_REF] using the red and infra-red spectral bands.

In spite of their apparent regularity, row crop images may show some particularities that make the detection process difficult. Rough measurements of average row spacing and orientation are not sufficient for a fine detection. Indeed, row spacing and orientation are only approximately constant over the field. For instance, perspective distortion effects can explain a gradual variation of row orientation across the image.

The active contour model is designed according both to ideal row properties and to the particularities that may occur. Rows are considered as roughly parallel segments. The model chosen in [START_REF] Bobillet | Row detection in high resolution remote sensing images of vine fields[END_REF]) is a network of quasi-parallel segments.

The geometry of the segment network is controlled by the minimization of a pre-defined energy E:

int (1 ) ext E E E µ µ = + - , (1)
where E int and E ext are respectively the internal and the external energies. E int reflects the internal constraints that the snake undergoes. The external energy E ext takes into account the effect of the image. It enables the attraction of the snake towards the desired state. µ is a weighting parameter. Formulations of E int and E ext and details of implementation can be found in [START_REF] Bobillet | Row detection in high resolution remote sensing images of vine fields[END_REF]. Figure 2 shows an example of row detection results obtained on the parcel of figure 1 using this method. The lines appear to fit almost perfectly the central axis of each vine row overcoming the irregularity of foliage and the absence of some vine plants. The network of lines obtained by the active contour algorithm allows to go beyond the simple detection of crop rows. Indeed, as many perennial crops, vine is usually planted regularly i.e.

with constant distance between rows and between vine stocks. Such information can be used to place a regular network of rectangles along the detected rows (see Figure 3). The length of the rectangles corresponds to the distance between two consecutive stocks. Their width corresponds to the trimming width. These features depend on the agricultural practices used in the region or even in the agricultural estate. They can be provided to the algorithm by the end-user. Once the exact positions and occupation areas of individual plants are determined, the quantification of crop vigour can be carried out at the scale of the plant by taking spectral information into account. This process is described below.

Estimation of vegetation abundance maps by partial spectral unmixing

Modelling the remotely sensed data

A VNIR (Visible and Near Infrared) remote sensing system is an electro-optical system designed to observe the radiations reflected by ground objects at wavelengths generally comprised between 0.4 and 2.5 µm. In such a system, the outputs, i.e. the image pixel data or digital numbers (DNs), are assumed to be linear combinations of the inputs, i.e. the spectral reflectances or radiances [START_REF] Schowengerdt | Remote Sensing: Models and Methods for Image Processing[END_REF]. This assumption, usually valid at a macroscopic level, implies that various effects (e.g. topographic variations, atmospheric effects, specular reflectance, etc.) are negligible. Several physical models have been proposed

to represent such observation systems and to analyze their effects on observed spectra [START_REF] Schowengerdt | Remote Sensing: Models and Methods for Image Processing[END_REF]. Let ( , , )

DN x y λ be the digital number recorded by the sensor at pixel ( , )

x y and wavelength λ . It can be expressed as follows:

( , , ) ( ( , , ), )

i DN x y f R x y P λ λ = , ( 2
)
where R is the spectral reflectance of the ground sampling unit corresponding to (x,y) at wavelength λ . The i P are the physical parameters involved in the whole remote sensing system: the atmosphere, the scene geometry and the sensor features. If a linear model is considered, the DNs are linearly related to the reflectance or at-sensor radiance. A multispectral image can thus be expressed in matrix form by:

n m n p p m n m X R A N × × × × = × + . ( 3
)
n p R × contains the reflectance (signature or end-member) of p materials in n spectral bands. In the contexts of detection and classification of spectral classes, p m A × has to be estimated.

This estimation can be viewed as an inverse problem for which Blind Sources Separation techniques (BSS), including Independent Component Analysis (ICA), are appropriate solutions.

Independent Component Analysis

ICA brings a solution for BSS under the assumption of linear mixing i.e. when the multivariate observations are assumed to be unknown linear mixtures of some unknown latent variables, the independent components (i.e. sources or factors):

X M S N = × + . ( 4
)
X is the observation matrix and S the sources. M is called the mixing matrix and N represents additive noise. The aim of ICA is to estimate S using an inverse model:

Ŝ W X = × , (5)
where W, the separating matrix is an estimate of 1 M -. In this regard, ICA is similar to Principal Component Analysis (PCA). However, while PCA minimizes the correlation between estimated components, ICA aims at the independence of the components. The estimation of independent components is possible under the following assumptions [START_REF] Hyvarinen | Independent Component Analysis: algorithms and applications[END_REF]:

-the mixture is actually linear, -the sources are mutually independent, -the number of Gaussian sources is at most one, -the number of sources is less than the number of observations.

ICA algorithms are based on the optimization of a criterion quantifying the independence of components. This criterion is usually based on higher order statistical moments. The optimization of this function will lead to maximize the independence of components, i.e. to minimize their mutual information. In this regard, several algorithms for ICA are available in literature including Jade [START_REF] Cardoso | Blind Beamforming for non-Gaussian Signals[END_REF] which is based on the joint probability function, FastICA [START_REF] Hyvarinen | Independent Component Analysis: algorithms and applications[END_REF] relying on the minimization of contrast functions (typically kurtosis), or the method proposed by [START_REF] Robila | A Fast Source Separation Algorithm for Hyperspectral Imagery[END_REF] which minimizes mutual information.

Regarding multispectral images, ICA is generally used according to two strategies. The first one, (Bayliss et al. 1998, Kosaka and[START_REF] Kosaka | ICA aided linear spectral mixture analysis of agricultural remote sensing images[END_REF] considers as sources the spectral signatures of the various classes (the end-members). In contrast, in the second one, the sources are class abundances (e.g. [START_REF] Robila | A Fast Source Separation Algorithm for Hyperspectral Imagery[END_REF]Varshney 2002, Shah 2004). Abundance values range continuously from low values to high values corresponding respectively to the absence and to the presence of an individual class within a specific pixel.

In the present work, we chose the second strategy. For vigour analysis in row crops, the source of interest is the one related to crop abundance, e.g. vine abundance in the case of viticulture. Other sources are related to soil, other types of vegetation, etc. The assumption of mutual independence of the sources entails that the vine class abundance does not depend on other possible sources as for instance bare soil or grass. Although this assumption does not hold -abundances sum to 1 -ICA usually provides relevant sources from which meaningful information about vegetation abundance can be extracted. Concerning non-Gaussianity, the useful source, i.e. the one related to vegetation, is usually non-Gaussian. Indeed, in practice, the crop abundance source is coarsely bi-modal as it is composed of rather crop-like pixels with high crop abundance values and rather non crop-like pixels with low crop abundance values. Other possible sources depend on the image content. They may be related to non-crop vegetation species (e.g. grass, trees), to ground targets or to soil variations. Gaussian-like sources usually correspond to residual noise and bear no information about vegetation abundance.

Component selection

In general, ICA implementations are unsupervised and do not allow any component ranking.

Neither the rank nor the energy of the components makes any sense in ICA. Consequently, even if the ICA algorithm provides a "crop-like" source, the latter must be retrieved among all the sources. Although this task seems straightforward if done manually, making it automatic is not so easy. The selection of meaningful components is often carried out using statistical criteria (e.g. higher order moments, entropy). However, it seems difficult in our case to select the crop component by using the statistical characteristics of its distribution. For this reason, we tackled the problem of component selection from a biological point of view. Indeed, in the agricultural context of vigour mapping, we can benefit from additional information provided by vegetation indices. If the available spectral data comprise sufficient information to compute a vegetation index I, then the crop component V can easily be selected by choosing, between all extracted sources i S , the one which is the most correlated to I:

1 , , arg max ( , )

p i S S V Corr S I = … . ( 6
)
Though efficient in most of cases, this criterion may not work in the presence of grass between the rows as it could result in an independent component by itself. However, such a case has not occurred in practice since grass, when present, is usually associated with vine, composing together a vegetation component. Indeed, spectral information extracted from multi-spectral data, is not sufficient to separate vegetation species such as grass and vine.

Such discrimination requires enhanced spectral information. The presence of grass will not be addressed here.

Component normalization

As mentioned previously, there are constraints imposed by the spectral mixing model that ICA can not account for. The signs and the magnitudes of the components can not be directly determined and the additivity property mentioned earlier is not verified. The components can not be directly considered as abundances. In particular, the vine component identified by maximizing its correlation to the vegetation index is not necessarily positive nor bounded to

[ ] 0,1 .
Some authors propose to take the constraints of the mixing model into account in the source separation algorithm itself (Chang 2003, Nascimento and[START_REF] Nascimento | Does independent component analysis play a role in unmixing hyperspectral data? In Pattern Recognition and Image Analysis[END_REF], the outputs of which can be directly considered as abundances. In our case, the crop abundance is obtained a posteriori using a soft classification scheme. The classification is performed by a Fuzzy C-Means technique (FCM) [START_REF] Bezdek | FCM: The Fuzzy CMeans Clustering Algorithm[END_REF]. Compared to classical clustering techniques (e.g. K-means), FCM is more robust to outliers and has the advantage of providing for each pixel a membership degree to each class. The only requirement is to set the number of clusters a priori. Regarding the vine component, the values of the pixels relate to the presence or the absence of vine. In the best case, the histogram of this component may be bi-modal (vine and non-vine pixels). Applied to this vine component, the FCM technique is thus expected to provide a measure of the abundance of vine i.e. an estimation of the percentage of the pixel area covered by vine. From now on, this vine abundance will be denoted A. Figure 4 gives a schematic illustration of vine abundance estimation.

Precise vigour estimation along rows

The use of the active contour model described above makes it possible to take measurements along crop rows and to assign the measured feature to a precise locus on the row. The feature to be measured will depend on the type of crop, on the agricultural practices, and on the type of information needed by the end-user. The width of the canopy, though an intuitive feature,

does not always bear meaningful information about crop vigour since, in horticulture, the canopy is sometimes controlled by lateral trimming. Such an agricultural practice is quite common in viticulture, for instance in the region of Bordeaux. In such cases, vegetation vigour must be assessed by other means, for example by measuring a vegetation index (VI) related to the biological status of the plant or by estimating vegetation thickness through a leaf area index (LAI). Below, we describe the method we chose to compute these indexes and to average them at the scale of the plant, combining both spatial information about plant layout and spectral information about crop abundance.

About vegetation indexes

A Vegetation Index (VI) is a measure without dimension which is usually a non linear combination of spectral bands of multispectral data. The formulations of VIs are based on the reflection properties of leaves [START_REF] Gates | Spectral properties of plants[END_REF]. Advantages are their simplicity of implementation and their low computational cost. However, their use is limited by some important drawbacks: they are non linear and saturated for highly vigorous vegetation, and are hindered by atmospheric and soil effects. There are two main categories of VIs: ratio based and orthogonal based (see Table 1). Ratio based VIs have long been studied in agricultural and environmental applications. The various formulations attempt to reduce the sensitivity to exterior conditions [START_REF] Zarco-Tejada | Assessing vineyard condition with hyperspectral indices: Leaf and canopy reflectance simulation in a row structured discontinuous canopy[END_REF]. They can also be used to discriminate vegetation from its environment or as measures of plant vigour [START_REF] Rouse | Monitoring the vernal advancement and retrogradation (green wave effect) of natural vegetation[END_REF][START_REF] Hall | Characterising and mapping vineyard canopy using high spatial-resolution aerial multispectral images[END_REF]. Orthogonal indexes, such as PVI [START_REF] Richardson | Distinguishing vegetation from soil background information[END_REF] or SAVI [START_REF] Huete | A soil-Adjusted Vegetation Index (SAVI)[END_REF], take the spectral reflection of soil into account but need prior calibration. They are particularly relevant in the case of leaf area measurements.

Table 1. Some ratio-based and orthogonal-based vegetation indexes. R and IR respectively denote the reflectance magnitudes in the Red and Infra-Red channels. L, a and b are constants used to compute orthogonal indexes.

Formula References

Difference Vegetation Index DVI = IR -R [START_REF] Tucker | Red and Photographic Infrared Linear Combinations for Monitoring Vegetation[END_REF] Simple Ratio Index SRI = IR / R [START_REF] Pearson | Remote mapping of standing crop biomass for estimation of the productivity of the short-grass prairie, Pawnee National Grasslands, Colorado[END_REF] Normalized Difference Vegetation Index NDVI = (IR -R) / (IR + R) [START_REF] Rouse | Monitoring the vernal advancement and retrogradation (green wave effect) of natural vegetation[END_REF] Infrared Percentage Vegetation Index IPVI = IR / (IR+R) [START_REF] Crippen | Calculating the vegetation index faster[END_REF] Perpendicular Vegetation Index PVI = (IR-a.R-b)/√(1+a²) [START_REF] Richardson | Distinguishing vegetation from soil background information[END_REF] Soil Adjusted ratio Vegetation Index 1 SAVI1 = (1+ L).(IR-R)/(IR+R+L) [START_REF] Huete | A soil-Adjusted Vegetation Index (SAVI)[END_REF] Soil Adjusted ratio Vegetation Index 2 SAVI2 = IR /(R + b/a) [START_REF] Major | A ratio vegetation index adjusted for soil brightness[END_REF] Transformed Soil-Adjusted Vegetation Index TSAVI=a*(IR-a*R-b)/(a*IR+R-a*b) [START_REF] Baret | TSAVI: a vegetation index which minimizes soil brightness effects on LAI and APAR estimation[END_REF] 2.3.2 An estimated leaf area index.

The Leaf Area Index, i.e. LAI, defined as half the green leaf area per area unit [START_REF] Welles | Instrument for indirect measurement of canopy architecture[END_REF], is known to be a realistic indicator of vigour. As its direct measurement potentially affects the health of the leaves, it can not be carried out over the whole parcel. Pruning weight or PW, i.e. the weight of the shoots pruned after harvest, is another vigour related feature. Its measurement is delayed in winter and can thus be exhaustive. However, both LAI and PW are labour intensive and time consuming and cannot be used as a routine for crop monitoring.

Instead, indirect estimation of LAI is possible, as described in [START_REF] Ross | The Radiation Regime and Architecture of Plant Stands[END_REF]. The estimation eLAI is obtained as follows: ln( ).cos( ) / ( )

eLAI P G α α = - , ( 7
)
where P is the probability that a beam of radiation with incidence angle α passes through the canopy, supposed uniform and composed of leaves with random angular and spatial distributions. ( ) G α is the mean projection coefficient of unit foliage area on a plane perpendicular to α.

To estimate eLAI from spectral data, cos( ) / ( ) G α α can be determined at 0.5 for plants which have randomly distributed leaf angles such as agricultural crops [START_REF] Norman | Modelling the Complete Crop Canopy[END_REF]). The incidence angle α corresponds to the sensor viewing zenith angle. Airborne optical remote sensors are usually pointed at a view angle of 0 (nadir looking). P represents the gap (nonvegetation) fraction, which is determined by spectral unmixing as follows:

1 P A = -, where A is the vine abundance. eLAI is then derived according to the following formula:

2 ln(1 ) eLAI A = - -. (8)
This leaf area index, together with the abundance-weighted vegetation indexes, will now be evaluated and compared within the experimental framework described hereafter.

Enhancing vegetation and leaf area indexes by abundance weighting

Any vigour indicator, i.e. vegetation or leaf area index, is all the more interesting as it can be computed at the scale of the plant and assigned to a specific plant position. A straightforward way to do this is to make use of the rectangle networks extracted through the active contour algorithm (see section 2.1). The vigour indicator is averaged over the pixels comprised in the rectangle and assigned to the centre of the rectangle.

In practice, for vegetation indexes, the presence of other classes inside the rectangles (i.e. soil, shadows, etc.), together with the mixing phenomenon, hinder the correct assessment of vigour. However, the effect of these non-vegetation pixels is easily reduced by weighting the vegetation index by the vine abundance:

rectangle rectangle / th th k i i i i k i k VI a VI a ∈ ∈ = ∑ ∑ (9)
where k VI is the vigour associated with the th k stock, i VI is the vegetation index value on pixel i and i a is the abundance of vegetation on pixel i. Finally, the vine vigour values k VI are assigned to the rectangle centroids so as to constitute a regular grid of pointwise values for mapping purposes.

Concerning eLAI, a simple average computed over the pixels comprised in the rectangles is sufficient, making the assessment of vigour even more straightforward.

Experimental evaluation

The measurements carried out using the methods described above are point scale information.

Herein, they can be used directly when information about individual plants is needed. On the contrary, they are not adapted for a global representation of the parcel vigour variability. In order to make such data easy to interpret and to use by experts and even by farmers, maps are necessary. This is done by interpolation of the plant-scale measurements.

After a brief presentation of the vineyard parcel and of the data we used for experimental validation, we will show some vigour maps obtained from plant-scale vigour measurements.

These maps will help in the qualitative comparison between the various vigour indexes produced by our global framework and the ground truth data collected on the parcel. Finally, statistical correlation results between ground truth and image-based measurements will provide a quantitative evaluation of the global framework.

Images and ground truth data

The evaluation is carried out on vineyards of the Bordeaux area. The image analysis framework developed in this work, is applied to multi spectral aerial silver photographs acquired in July 2006 and digitized at a resolution of 4800 dpi. These images are composed of green, red and infrared spectral bands. The camera was taken on board a light airplane. Flying altitude was low to allow very high spatial resolution i.e. 7-10 cm/pixel. An example of such data is shown in figure 5, in the form of a false colour image. Radiometric and geometric targets, which appear as white spots in the image, are ground control points meant for the geo-referencing of images. The precision of the geo-referencing is related to the accuracy of the GPS, typically 5m or 1m at best with post-treatment, which is sufficient for standard vigour cartography. In contrast, the geo-positioning of ground measurements and image data requires a much better accuracy since both image resolution (less than 10cm/pixel) and plant size (0.5m × 1m) are smaller than the accuracy of standard GPS. In our case, we chose not to use GPS-based positioning but rather to take advantage of the strict regularity of plant layout. Indeed, the exact geometry of the parcel is known, including the number, positions and lengths of the rows and the distances of the plants from the beginning of the row. It is thus possible to perform geo-referencing of ground measurements and image data, within a local coordinate system with no reference to terrestrial coordinates. As the beginning of the rows can be determined on images with a resolution of about 2 to 3 pixels because of foliage irregularity, geo-positioning accuracy falls down to 20 or 30 cm, out-performing standard GPS resolution.

Vegetation vigour maps

Various vigour-related indexes were extracted. Both ratio-based and orthogonal vegetation indexes (section 1.2) were computed. The eLAI, estimated Leaf Area Index, was deduced from abundance maps. Three indexes, NDVI, PVI and eLAI are illustrated in figure 6.

These indexes were combined with geometric information to produce plant-scale vigour measurements according to the geometric framework described in section 2.3. These measurements were inserted into a GIS (ArcGIS) and manipulated as vector data for mapping.

Within block vigour maps were created by inverse distance interpolation. The three corresponding vigour maps are shown in figure 6. Though we will not comment further the interest of such maps for agricultural monitoring, it is worthwhile mentioning some properties of these maps. First, they allow enhancing the variability of vigour, difficult to observe in the original index images. Secondly, the global structures of the three maps are similar with particularly strong resemblance between PVI and eLAI. Figure 7 shows the scatter plots between the two vegetation indexes, NDVI and PVI, and the leaf area index eLAI. The data are the measurements carried out on the experimental plots described above. Less than 10% outliers were excluded which corresponded to missing canopy (missing plants or deviated canopy). While the relation between NDVI and eLAI is rather loose, the linearity of the correlation between PVI and eLAI is clear, with a high correlation coefficient.

Correlation between vigour maps and ground truth data

An experimental validation of the framework was carried out by statistical analysis. eLAI measurements and ground truth pruning weights were compared, at the scale of the experimental plots described in section 3.1. The scatter plot between the two statistical series is shown in Figure 8. 2 groups the correlation coefficients obtained in two cases, depending on whether the vegetation indexes are uniformly weighted or weighted by the vine abundance-values. The results show that the use of abundance values as weights allows enhancing the correlation between image-based and ground measurements. In other words, the abundance values seem to reduce the effect of non-vegetation pixels and thus to refine vegetation indexes. Besides, it can be noted that DVI, a very simple vegetation index, appears to outperform other indexes after abundance weighting, even orthogonal ones, in spite of the relative complexity of their computation. Finally, it appears that eLAI, directly computed from the abundance values, is as correlated to pruning weights as the best vegetation indexes. A validation procedure was carried out on experimental vine plots where ground pruning weight measurements were available. Various image-based vigour indicators were computed and showed comparable performances. For all indicators, enhanced results were obtained using vegetation abundance as a weight for vigour computation. Besides, the estimated Leaf Area Index, eLAI, directly computed from the abundance values, proved to be as correlated to pruning weights as the best vegetation indexes, including orthogonal vegetation indexes. This result is noteworthy, considering that such indexes are quite difficult to compute, for they require to calibrate the image according to local soil characteristics. This makes of eLAI, computed using the unsupervised algorithm proposed in this paper, an interesting alternative to standard indexes, especially for large commercial applications where calibration is not systematic.

Finally, this work showed very promising results. The degree of correlation between ground data and image-based measurements confirms, in the case of row crops, the potential of multispectral imagery for canopy characterization. Furthermore, the significant improvement obtained using abundance information suggests to take into account abundance based weighting to improve the accuracy of vigour assessment.

Figure 1 .

 1 Figure 1. Left: typical high-resolution remote sensing image of a vineyard in the visible spectrum. Right: segmented version.

Figure 2 .

 2 Figure 2. Left: Row detection result. Middle: detail #1. Right: detail #2.

Figure 3 .

 3 Figure 3. Left: Row detection result. Middle: detail #1. Right: detail #2.

A

  × is a combination matrix: in the case of remote sensing applications, it contains the abundances ij a of the p materials on each of the m pixels. n m N × is a residual matrix. Finally, n m X × contains the recorded pixel data (DNs). If the elements of p m A × satisfy the constraints this model is similar to the Linear Spectral Random Mixture model proposed in[START_REF] Chang | Hyperspectral Imaging: Techniques for spectral detection and classification[END_REF]).

Figure 4 .

 4 Figure 4. From multispectral images to the vine abundance map

Fig. 4 .

 4 Fig. 4. Left: segment network obtained through the row detection algorithm. Middle: rectangle network corresponding to the vine stocks. Right: the rectangle centroids.

Figure 5 .

 5 Figure 5. False colour image of an experimental plot with radiometric/geometric targets.

Figure 6 .

 6 Figure 6. The three vigour-related indexes and corresponding vigour maps obtained by inverse distance interpolation. From top to bottom: NDVI, PVI and eLAI. Low vigour is represented in white, high vigour in dark green.

Figure 7 .

 7 Figure 7. Scatter plots. Left: between NDVI and eLAI. Right: between PVI and eLAI.

Figure 8 .

 8 Figure 8. Scatter plot between eLAI and Pruning Weight, with the linear regression trend.

Future

  works will concern the refinement of the approach. Particularly, we intend to test and improve the whole framework on enhanced image data. Actually, the poor spectral resolution of the multispectral images used in this work certainly hinders the quality of vegetation index estimation, thus penalizing vigour estimation. With such spectral resolution, discrimination between different kinds of vegetation, e.g. grass and vine, is not possible either, which makes the present approach of limited use when inter-rows are covered by grass. The use of hyperspectral data, with both numerous and narrow spectral channels, should surely open up new horizons in future. Modern hyperspectral sensors, beyond being both technically and economically competitive, are also much less bulky than previous heavy acquisition systems thus allowing easy and cheap airborne acquisitions. Not only enhanced vigour characterization and mapping but also species discrimination should then be attainable, provided that sufficient spatial resolution is available.

Figure 1 .

 1 Figure 1. Left: Typical high-resolution remote sensing image of a vineyard in the visible spectrum. Right: segmented version.

Figure 2 .

 2 Figure 2. Left: Row detection result. Middle: detail #1. Right: detail #2.

Figure 3 .

 3 Figure 3. Positioning individual plants along crop rows. Left: segment network obtained through the row detection algorithm on an experimental plot. Middle: rectangle network corresponding to the vine stocks. Right: the rectangle centroids.

Figure 4 .

 4 Figure 4. From multispectral images to the vine abundance map

Figure 7 .

 7 Figure 7. Scatter plots. Left: between NDVI and eLAI. Right: between PVI and eLAI.

Figure 8 .

 8 Figure 8. Scatter plot between eLAI and Pruning Weight, with the linear regression trend.

Table 2 .

 2 Correlations between Vegetation Indexes and Pruning Weight.

	Uniform weighting Abundance-based weighting

A semi-automated framework is proposed to transform high resolution multispectral images of row crops into vegetation vigour maps. Though developed in a viticulture context, this method is generic enough to be adapted to any row crop, especially in arboriculture. It takes advantage of both spectral and spatial features extracted from image data. Spectral information is used at pixel level to process partial unmixing and to compute enhanced vegetation and leaf area indexes. Spatial information is used to bring back spectral measurement to specific row and plant positions. Resulting measurements are finally used for within block vigour mapping or plant scale characterization of crops.

allowing image acquisitions.

Figure 5. False colour image of an experimental plot with radiometric/geometric targets.

Acknowledgments

This work was carried out with the financial support of the FEDER Interreg IIIB program, within the PIMHAI project. We are grateful to the experimental domain of Couhins, for