Chen Beijing

Huazhong Shu

Hui Zhang

Gouenou Coatrieux

Limin Luo

Jean-Louis Coatrieux
email: coatrieux<jean-louis.coatrieux@univ-rennes1.fr

Combined invariants to similarity transformation and to blur using orthogonal Zernike moments

Keywords: Zernike moments, circularly symmetric blur, combined invariants, pattern recognition, template matching

The derivation of moment invariants has been extensively investigated in the past decades. In this paper, we construct a set of invariants derived from Zernike moments which is simultaneously invariant to similarity transformation and to convolution with circularly symmetric point spread function (PSF). Two main contributions are provided: the theoretical framework for deriving the Zernike moments of a blurred image and the way to construct the combined geometric-blur invariants. The performance of the proposed descriptors is evaluated with various PSFs and similarity transformations. The comparison of the proposed method with the existing ones is also provided in terms of pattern recognition accuracy, template matching and robustness to noise. Experimental results show that the proposed descriptors perform on the overall better.

INTRODUCTION

Recognition of objects whatever their position, size and orientation is an important concern. In the past decades, many techniques including moment invariants , Fourier descriptors and point set invariants have been reported in the literature. Among

[1 ]- [START_REF] Abu-Mostafa | Recognitive aspects of moment invariants[END_REF] [6 ]

[7 ]- [START_REF] Lowe | Distinctive image features from scale-invariant key points[END_REF] them, moment invariants have been extensively used for image description in object recognition , , image classification and [START_REF] Khotanzad | Invariant image recognition by Zernike moments[END_REF] [12 ] [START_REF] Yang | Classification of rotifers with machine vision by shape moment invariants[END_REF] scene matching . However, much less attention has been paid to invariants with respect to changes of the image intensity function [START_REF] Lin | Template matching using the parametric template vector with translation, rotation and scale invariance[END_REF] (known as radiometric invariants) as to joint radiometric-geometric invariants.

Since real sensing systems are usually imperfect and environmental conditions are changing during the acquisition, the observed images often provide a degraded version of the true scene. Image blurring is an important class of degradations we have to face in practice due to the camera defocus, atmospheric turbulence, vibrations, and by sensor or scene motion . Blurring can be usually described by a [START_REF] Flusser | Recognition of blurred images by the method of moments[END_REF] convolution of an unknown original image with a space invariant point spread function (PSF). A conventional way to carry out blur object recognition is first to deblur the image, and then to apply the recognition methods . Unfortunately, the blind image deconvolution [START_REF] Savakis | Blur identification by residual spectral matching[END_REF]- [START_REF] Jung | A novel multiple image deblurring technique using fuzzy projection onto convex sets[END_REF] is an ill-posed problem. Moreover, the deconvolution process may introduce new artifacts to the image. To avoid these disadvantages, it is of great importance to find a set of invariants that is not affected by blurring. This is the objective of this paper.

The first paper on this subject was reported by Flusser and Suk who derived invariants to convolution with an arbitrary [START_REF] Flusser | Recognition of blurred images by the method of moments[END_REF] centrosymmetric PSF. These invariants have been successfully used in pattern recognition , in blurred digit and character [START_REF] Flusser | Degraded image analysis: an invariant approach[END_REF]- [START_REF] Stern | Recognition of motion-blurred images by use of the method of moments[END_REF] recognition , , in medical image registration , and in focus/defocus quantitative measurement . Other sets of blur [START_REF] Lu | Blurred image recognition based on phase invariants[END_REF] [27 ]

[28 ] [START_REF] Zhang | A new focus measure method using moments[END_REF] invariants have been derived for some particular kinds of PSF like the axisymmetric blur invariants and motion blur invariants , [START_REF] Flusser | Image feature invariant with respect to blur[END_REF] [31 ] [

. More recently, Flusser and Zitova introduced the combined blur-rotation invariants and applied them to satellite image 32 ] [START_REF] Flusser | Combined invariants to linear filtering and rotation[END_REF] registration and camera motion estimation . Zhang et al. proposed a method to get a set of affine-blur invariants. However, [START_REF] Flusser | Invariant-based registration of rotated and blurred images[END_REF] [35 ] [START_REF] Zhang | Determination of blur and affine combined invariants by normalization[END_REF] in their approach, the affine invariance is achieved through a normalization process. Suk and Flusser derived explicitly a set of combined invariants with respect to affine transform and to blur . These blur invariants have been further extended to -dimensions in [START_REF] Suk | Combined blur and affine moment invariants and their use in pattern recognition[END_REF] N continuous case as well as in discrete form . The above mentioned methods are mainly based on geometric or complex moments.

[38 ] [START_REF] Candocia | Moment relations and blur invariant conditions for finite-extent signals in one, two and N-dimensions[END_REF] The resulting invariants have information redundancy and are more sensitive to noise.

We have recently proposed an approach based on the orthogonal Legendre moments to derive a set of blur invariants, and we [START_REF] Zhang | Blurred image recognition by Legendre moment invariants[END_REF] have shown that they are more robust to noise and have better discriminative power than the existing methods. However, as pointed out in , one weak point of Legendre moment descriptors is that they are only invariant to translation, but not invariant under image rotation [START_REF] Zhang | Blurred image recognition by Legendre moment invariants[END_REF] and scaling. [43 ]

In this paper, we propose a new method to derive a set of combined geometric-blur invariants based on orthogonal Zernike moments.

We further assume that the applied PSF is circularly symmetric. The reasons for such a choice of PSF are as follows :

(1) The majority [START_REF] Flusser | Invariants to convolution with circularly symmetric PSF[END_REF] of the PSFs occurring in real situations exhibit a circular symmetry; (2) Since the PSFs having circular symmetry are a subset of centrosymmetric functions, it could be expected that we can derive some new invariants. In fact, the previously reported convolution invariants with centrosymmetric PSF include only the odd order moments. Flusser and Zitova have shown that there exist even order [START_REF] Flusser | Invariants to convolution with circularly symmetric PSF[END_REF] moment invariants with circularly symmetric PSF.

The organization of this paper is as follows. In Section II, some preliminaries about the radial moments, Zernike moments and image blurring are given. In Section III, we first establish a relationship between the Zernike moments of the blurred image and those of the original image and the PSF, and we then explain how to construct a set of blur invariants and joint radiometric-geometric invariants.

Experimental results on the proposed descriptors performance are provided in Section IV. Section V concludes the paper.

'

Preliminaries

This section presents the definition of radial and Zernike moments, and reviews the concept of image blurring. The radial moment of order with repetition of image intensity function ( , ) is defined as p q f r θ [START_REF] Mukundan | Moment Functions in Image Analysis Theory and Applications[END_REF] The Zernike moment of order with repetition off ( , ) is defined as p q r θ [START_REF] Mukundan | Moment Functions in Image Analysis Theory and Applications[END_REF] where ( ) is the real-valued radial polynomial given by R p,q r points out that the radial polynomial ( ) is symmetric with , that is, ( ) ( ), for 0. Thus, we can Equation (3) R p,q r q R p, q -r = R p,q r q ≥ consider the case where 0. Letting 2 in ( ) with 0, and substituting it into ( ) yields q ≥ p = q + l 3 l ≥ 2

where Let be a rotated version of , i.e. ( ) ( ), where is the angle of rotation, and let be the Zernike moments of . It

f ′ f f ′ r, θ = f r, θ -β β f ′
can be easily seen from ( ) that 2

Let ( ) be a blurred version of the original image ( ), under the condition that of imaging system being linear shift-invariant, g x, y f x, y the blurring can be usually described by the convolution where ( , ) is the PSF of the imaging system, and denotes the linear convolution. h x y

*

In this paper, we assume that the PSF, ( , ), is a circularly symmetric image function, and that the imaging system is h x y energy-preserving, which leads to, Under the assumption of ( ), the Zernike moments of ( , equal those of any rotated image . Combining this fact with ( ), we 8 h r )

θ h ′ 6
get is verified if and only if either or 0. Thus, an important property of circularly symmetric functions can be Equation [START_REF] Lowe | Distinctive image features from scale-invariant key points[END_REF] q = stated as follows.

Proposition 1

If 0 and ( , ) is a circularly symmetric image function, then for any non-negative integer . q ≠ h r θ l

Method Zernike Moments of the Blurred Image

In this subsection, we establish the relationship between the Zernike moments of the blurred image and those of the original image and the PSF. To that end, we first consider the radial moments. Applying ( ) to blurred image ( , ), we have 1 g x y

Applying ( ) to blurred image ( , ) ( , ) and using ( ), we obtain 4 g x y = g r θ 11

From ( ), the radial moments can also be expressed as a series of Zernike moments where , 0 , is the inverse matrix of . Both and are lower triangular matrices of size ( 1) ( 1), the ≤ j ≤ i ≤ l l + × l + elements of are given by ( ). The elements of are given by 5 [START_REF] Shu | A general method to derive the relationship between two sets of Zernike coefficients corresponding to different aperture sizes[END_REF] From ( ), we have 13

By introducing ( ) and ( ) into ( ), we obtain 15 [START_REF] Savakis | Blur identification by residual spectral matching[END_REF] 12

Based on ( ), we have the following theorem. 17

Theorem 1

Let ( , ) be the original image function and the PSF ( , ) be circularly symmetric, ( , ) be a blurred version of ( , ), then f r θ h r θ g r θ f r θ the following relation stands for any 0 and 0, where the coefficients ( , , , ) are given by q ≥ I ≥ A q l i j

Proof

For circularly symmetric function ( , ), using Proposition 1, we have if 0, thus, ( ) can be simplified as h r θ q +n m -≠

17

The proof has been completed.

Blur Invariants of Zernike Moments

Based on Theorem 1, it becomes possible to construct a set of blur invariants of Zernike moments which is described in the following theorem.

Theorem 2

Let ( , ) be an image function. Let us define the following function :

f r θ I ( f ) N N R × →
Then, ( , ) is invariant to circularly symmetric blur for any 0 and 0. The number is called the order of the I q +2l q ( f ) q ≥ l ≥ p = q +2l invariant.

The proof of Theorem 2 is given in . Some remarks deserve to be made. Appendix A

Remark 1

By using the symmetric property of , ( ) with it can be easily proven that (| | 2 , for < 0 is also invariant to R p q r q, I q + l q) ( f ) q convolution.

Remark 2

It can be deduced from ( ) that . Thus, only will be used as invariant for the case 0. 20 q = Remark 3

If we use the Zernike central moments instead of Zernike moments in ( ), then we can obtain a set of invariants ( 2 , ) 20

I ̄q + l q ( f )
that is invariant to both translation and to blur.

Based on Theorem 2, we can construct a set of blur invariants of Zernike moments with arbitrary order and express them in explicit form. The invariants up to sixth order are listed in . Appendix B

Lemma 1

Let be a rotated version of , i.e., ( ) (

), where denotes the rotation angle, then the following relation holds for any

f ′ f f ′ r, θ =f r, θ-β β q ≥ 0 and 0 I ≥

Proof

We demonstrate this lemma by induction about The proof is trivial for 0. Assume that the assertion is true for 1, 2, , 1, then l. l = … lusing ( ), we get 6

The proof is thus completed

Lemma 2

Let ( , ) be an image function. It holds for any 0 and 0 that f r θ q ≥ l ≥ where the superscript denotes the complex conjugate.

*

The proof of Lemma 2 is very similar to that of Lemma 1 and it is thus omitted.

Combined Invariants of Zernike Moments

In this subsection, we construct a set of combined geometric-blur invariants. As we have already stated, the translation invariance can be achieved by using the central Zernike moments.

shows that the magnitude of ( 2 , ) is invariant to rotation. Equation [START_REF] Jung | A novel multiple image deblurring technique using fuzzy projection onto convex sets[END_REF] I q + l q ( f ) However, as indicated by Flusser , the magnitudes do not yield a complete set of the invariants. Herein, we provide a way to build up [START_REF] Flusser | Combined invariants to linear filtering and rotation[END_REF] such a set. Let and be two images having the same content but distinct orientation ( ) and scale ( ), that is, ( , ) ( ,

), the f ″ f π λ f ″ r θ = f r/ λ θ-β
Zernike moment of the transformed image is given by Using ( ) and ( ), we have 4 13

Therefore, we have the following theorem:

Theorem 3
Let with and . Then is invariant to both image rotation and scaling for any non-negative integers and . q l

The proof of Theorem 3 is given in . Appendix A

Remark 4

Many other choices of and are possible. In fact, they can be chosen in such a way that , where ( , ) (

θ f Γ f Γ f ″ = λ Γ f θ f ″ = θ f -β f ″ r θ = f r/ ,
) is the transformed image of . However, it is preferable to use the lower order moments because they are less sensitive to noise λ θ -β f than the higher order ones . If the central moments are used, can be chosen as .

[46 ] θ f

Theorem 4

For any 0 and 0, let q ≥ l ≥ where ( 2 , ) is defined in ( ). Then, ( 2 , ) is both invariant to convolution and to image scaling and rotation. I q + m q ( f ) 20 SI q + l q ( f )

The proof is given in . For simplicity, the invariants defined in ( ) are hereafter denoted by ZMIs. The combined invariants appendix A 26 up to sixth order are listed in . Appendix C

Experimental Results

The following experiments illustrate the invariance of our ZMIs to various PSFs and similarity transformation, as its robustness to different kinds of noise. Comparison with existing methods in terms of recognition accuracy and template matching is also provided.

Let

for 0, where ( , ) is defined in ( ) and denotes the nearest integers not P ≥ SI p q 26 ⌊ x ⌋ greater than , and let ( ) ( , , ,

). The relative error between the two moment invariant vectors corresponding to an image x I ̃p = SI 0 SI

Test of Invariance

For this experiment, a set of eighteen butterfly images shown in , whose size is 128 128, has been chosen from the public Fig. 1 × Butterfly database as the original images. In order to evaluate the invariance with respect to rotation and blurring, these images were rotated by different angles from 0 to 90 every 5 , and a bilinear interpolation was used when required. Then, the normalized uniform disk °°°b lur with different sizes from 1 1 (no blurring) to 31 31 with interval 2 (sixteen masks in total) was applied to every rotated image. So,

× ×
the actual size of the blurred images in the experiments of this subsection is 212 212. Note that the original images have been zero

× padded to meet the actual size in order to avoid the boundary effect. Both the original images and the blurred images are then mapped inside the unit circle, and the Zernike moments are computed with the method reported in . The combined rotation-blur invariants [START_REF] Xin | Accurate computation of Zernike moments in polar coordinates[END_REF] ZMIs defined in ( ) of order up to 5 were computed for all 5472 images. The relative errors of ZMIs between each transformed 26 p = image and its original image were computed using ( ).

shows the mean values of ( ) where ( 1, 2, , 18) denotes the 27 Fig. 2 E p f ,g i i

f i i = …
images shown in . It can be seen that they are very low (less than 0.0035). The mean relative error reaches its maximal value Fig. 1 (a) (r) at the angle 45 due to the interpolation effect. The maximal value of the standard deviation (STD) is equal to 0.0035. We then tested β = °the combined scale-blur invariance. Eighteen images were blurred with the same masks, and then were scaled by a factor varying from 0.5 to 2 with interval 0.1, forming a set of 4608 images.

shows the mean values of the relative errors ( ) ( 1,2, , 18) (the Fig. 3 E p f ,g

i i i = …
maximal value of the standard deviation is 0.0043). It can be observed from and that the errors caused by blurring and similarity Figs. 2 3 transformation are very small (less than 0.006).

In the second experiment, the image shown in was convoluted with various PSFs and undergone similarity transformation Fig. 1 (a) including translation, scaling and rotation (

). The proposed ZMIs defined in ( ) of order from 2 to 5 were calculated for each Fig. 4 26 p image (the central Zernike moments were used in this experiment, and the moment invariants of order 0 and 1 were respectively used to achieve the scale invariance and translation invariance). depicts the invariant values. From this table, it can be seen that excellent Table 1 results have been obtained whatever the similarity transformation and image degradation.

We also compared our ZMIs with the complex moment invariants (CMIs) reported in and the Legendre moment invariants [START_REF] Flusser | Invariants to convolution with circularly symmetric PSF[END_REF] (LMIs) presented in in terms of blur invariance. This is because these methods have different behavior regarding to similarity [START_REF] Zhang | Blurred image recognition by Legendre moment invariants[END_REF] transformation: LMIs are invariant to translation only and CMIs are invariant to rotation and scaling. We do not include Zhu s method in ' this comparison due to the fact that the set of invariants derived in , is a subset of both CMIs and ZMIs. Eighteen images were ZMIs are depicted in . It is clear that the errors increase significantly with the size of the mask going up and our ZMIs behave better Fig. 5 than the two other types blur invariants.

Classification Results

In this evaluation, we also use the images shown in . The testing set was generated by adding disk blur, averaging blur and Fig. 1 Gaussian blur with zero-mean and STD 1,2. The mask with sizes 3 3, 5 5, 7 7, , 17 17, 19 19 pixels has been used, forming a

= × × × … × ×
set of 648 images. This was followed by adding a white Gaussian noise with different standard deviations and salt-and-pepper noise with different noise densities. Because the actual size of the PSF is usually unknown in practical application, in order to evaluate the performance of the different methods under such a situation, we take the size of the blurred image equal to the original image one (128 × 128) instead of its actual size (146 146) in this experiment even if the size of the mask is given. In that case, the boundary effect is then

× present.
shows some examples of the blurred and corrupted images. The Lance-Williams distance is used here as the classification Fig. 6 measure. This distance between the two images and is defined by their moment invariant vectors then decreases for all three methods. In other words, there should exist an optimal order for each type of moment invariants. This behaviour has also been observed and pointed out by Liao and Pawlak in image reconstruction due to the noise influence . In this [START_REF] Liao | On image analysis by moments[END_REF] experiment, the optimal order for CMIs is 7 (the feature vector includes 17 invariants), 7 for LMIs (the feature vector has 21 M = M = invariants), and 9 for ZMIs (with 26 invariants).

shows the detailed classification rates using the different moment invariants M = Table 2 with the optimal order. One can see from this table that in the noise-free scenario the recognition results are quite good whatever the method. The classification rates remain high for low and moderate noise levels but decrease significantly when the noise level goes up.

However, the proposed descriptors ZMIs perform better than other methods (all the rates are higher than 75 ) whatever the noise and the % noise level. Although this conclusion is drawn from only one experiment, it is in accordance with that reported in where the authors [START_REF] Teh | On image analysis by the method of moments[END_REF] pointed out that the Zernike moments have on overall better performance than other moments.

Template Matching

The objective of this additional test is to evaluate the performance of our descriptors in the case of localizing templates within a real outdoor scene image that has undergone similarity transformation and out-of-focus blur. For that purpose, two images were taken by digital camera (Panasonic DMC-FZ50) with different focus and different positions by rotating the camera. Then, nine circular areas with radius 10 pixels were extracted from to serve as templates (numbered from 1 to 9). The scale factor between two images is r = Fig. 8 (a)

obtained with the automatic scale selection . After that, the scaled template was shifted across the transformed and blurred image (

Fig.

)

. At each position, the invariants were calculated and compared with the invariants of the original template. For more detail about 8 (b) the matching procedure, we refer to Ref.

. [START_REF] Flusser | Degraded image analysis: an invariant approach[END_REF] Each moment descriptors with the optimal order obtained in the previous experiment ( 9 for ZMIs and 7 for CMIs) have been M = M = used here. Note that we did not test LMIs because they are not invariant with respect to image rotation and scaling. The matching position

' '
we consider corresponds to the location where the Lance-Williams distance ( ) reaches the minimum value, with representing the d f, g f template of the original image and the template of the transformed and blurred image. The matching results that have been obtained g based on these different moment invariants are summarized in . It can be seen that the proposed descriptors ZMIs match Fig. 8 (c) and (d) correctly for all nine templates and that it is not the case for CMIs.

Conclusions

In this paper, we have proposed a method to construct a set of combined geometric-blur invariants using the orthogonal Zernike moments. The relationship between the Zernike moments of the blurred image and those of the original image and the PSF has been established. Based on this relationship, a set of invariants to convolution with circularly symmetric PSF has been derived. The advantages of the proposed method over the existing ones are the following: [START_REF] Hu | Visual pattern recognition by moment invariants[END_REF] The proposed descriptors are simultaneously invariant to similarity transformation and to convolution. Using these invariants, the image deblurring and geometric normalization process can be well avoided;

(2) Like the method reported in , our method can also derive the even order invariants. The experiments conducted so far in very [START_REF] Flusser | Invariants to convolution with circularly symmetric PSF[END_REF] distinct situations demonstrated that the proposed descriptors are more robust to noise and have better discriminative power than the existing methods. ( 2 ) 0. T q l i j k = I q + l, q ( g ) -I q + l, q ( f ) =

The proof has been completed.

Proof of Theorem 3

can be written in a matrix form as Equation [START_REF] Flusser | Recognition of images degraded by linear motion blur without restoration[END_REF] Applying ( ) to the transformed image , it can also be expressed in a matrix form as 25 f ″

Based on the definition of and , it can be easily verified that

Γ f θ f
Substitution of ( ) and ( ) into ( ), and using the identity , we obtain A15 A17 A16 12 21 The proof has been completed.

To prove Theorem 4, we need first the following Lemma.

Lemma 3

Let Then, 2 , ) is invariant to image scaling and rotation for any 0 and 0. CL(q + l q ( f ) q ≥ l ≥ Lemma 3 can be easily proven by mathematical induction and it is thus omitted.

Proof of Theorem 4

By Theorem 1, ( 2) is invariant to convolution for any 0 and 0. Since is also invariant to convolution, it is obvious I q + k, q ( f ) q ≥ k ≥ that ( 2 ) defined in ( ) is invariant to convolution. To demonstrate its invariance with respect to image scaling and rotation, we SI q + l, q ( f ) 26

will prove the following result where ( 2 , ) is defined in ( ). CL q + l q ( f ) A19

We demonstrate ( ) by mathematical induction about . A20 l For 0, using ( ), ( ) and ( ), we have l = A19 20 25

Assume that the relationship ( ) is valid for 1, 2, , 1. Then, using ( ) and ( ), we have A20

… l - 20 25
For ( 2 ) , using the assumption ( 2 )

( 2 ) for 0, 1, , 1, and ( ) and ( ), we deduce from ( ) CL q + l, q ( f ) CL q + i, q ( f ) = SI q + i, q ( f ) i = … l - By combining ( ), ( ) and ( ), we deduce ( ) ( ) . By Lemma 3, is invariant to image A25 A29 A31 SI q 2l,q

+ (f) = CL q 2l,q + (f) CL(q 21, q) + (f)
scaling and rotation, thus, is also invariant to image scaling and rotation. SI(q 2l,q)

+ (f)
The proof has been completed. Eighteen objects selected from a butterfly database Fig. 2 The mean relative errors of our combined invariants ZMIs for different rotated and blurred versions of the images shown in . Fig. 1 Fig. 3 The mean relative errors of our combined invariants ZMIs for different scaled and blurred versions of the images shown in . Fig. 1 Fig. 4 The similarity transformed and blurred images of ( is the translation (in pixel) along the x-axis, the translation along the y-axis, Fig. 1 The invariants of images showed in Fig. 4 Fig. 4(a

[ 41 ] [ 42 ]

 4142 degraded by the normalized uniform disk blur with sixteen different sizes from 1 1 to 31 31. Three types of blur invariants of order up × × to 5 are calculated for the set of 288 images. The mean values of the relative errors (

  , , , denote the blur invariants, is the total number of invariants used in the experiment and | | the magnitude of j = … n n x the complex number . x We have computed the ZMIs, CMIs and LMIs up to order 1, 3, , 15, 17. The mean classification rates under different noise M = … conditions for different values of are shown in . It can be observed that the rate first increases, reaches the maximum value and M Fig.

7

 7

  can be written as 20 Substitution of ( ) into ( ), we get A5 A4 By shifting the indices in the above equation, we have where By using ( ), we have 19 ( ) can thus be rewritten as A8 By shifting the indices in the last part of the above equation, we get where By changing the order of summation in ( ), we have A12 Since the lower triangular matrixis the inverse of the matrix , that is, , where is the identity matrix, we have ,

  have shifted the indices in the last step of the above equation. Subtracting (

Appendix B (

 ( Fig. 1Eighteen objects selected from a butterfly database

Fig. 5

 5 Fig. 4The similarity transformed and blurred images of ( is the translation (in pixel) along the x-axis, the translation along the y-axis, Fig.1(a) Δx Δy

Fig. 6 Fig. 7 Fig. 8

 678 Fig. 6 Examples of image with different blur, different mask sizes and different noises ( is the STD of noise, is the density ofδ

  Fig. 4(a) Fig. 4(b) Fig. 4(c) Fig. 4(d) Fig. 4(e) Fig. 4(f) Fig. 4(g) Fig. 4(h)

  Zhu et al. and Ji and Zhu proposed the use of the Zernike moments to construct a set of combined blur-rotation

	[41 ]		[42 ]
					p	with repetition	p, Z p, p ,	has been used in
	the derivation of invariants. Since	Z p, p	corresponds to the radial moment	D p, p	or the complex moment	C , p	if neglecting the normalization

invariants. Unfortunately, there are two limitations to their methods: (1) Only the Gaussian blur has been taken into account, which is a special case of PSF having circularly symmetry; (2) Only a subset of Zernike moments of order 0 factor, the set of invariants constructed by Zhu et al. is a subset of that proposed by Flusser .

Table 2

 2 The classification rates ( ) of the LMIs, CMIs and the proposed ZMIs in object recognition with the optimal value of

	4(h)

[START_REF] Jung | A novel multiple image deblurring technique using fuzzy projection onto convex sets[END_REF]

[START_REF] Jung | A novel multiple image deblurring technique using fuzzy projection onto convex sets[END_REF]

[START_REF] Jung | A novel multiple image deblurring technique using fuzzy projection onto convex sets[END_REF]

[START_REF] Jung | A novel multiple image deblurring technique using fuzzy projection onto convex sets[END_REF]

[START_REF] Jung | A novel multiple image deblurring technique using fuzzy projection onto convex sets[END_REF]

[START_REF] Jung | A novel multiple image deblurring technique using fuzzy projection onto convex sets[END_REF]

[START_REF] Jung | A novel multiple image deblurring technique using fuzzy projection onto convex sets[END_REF]

[START_REF] Jung | A novel multiple image deblurring technique using fuzzy projection onto convex sets[END_REF]

[START_REF] Jung | A novel multiple image deblurring technique using fuzzy projection onto convex sets[END_REF]

[START_REF] Jung | A novel multiple image deblurring technique using fuzzy projection onto convex sets[END_REF]

[START_REF] Jung | A novel multiple image deblurring technique using fuzzy projection onto convex sets[END_REF] Sixth order

[START_REF] Jung | A novel multiple image deblurring technique using fuzzy projection onto convex sets[END_REF]

[START_REF] Jung | A novel multiple image deblurring technique using fuzzy projection onto convex sets[END_REF]

[START_REF] Jung | A novel multiple image deblurring technique using fuzzy projection onto convex sets[END_REF]

[START_REF] Jung | A novel multiple image deblurring technique using fuzzy projection onto convex sets[END_REF]

Appendix A Proof of Theorem 2

We prove this theorem by mathematical induction about . l For 0, using ( ), ( ) and , it can be easily deduced from ( ) that l = 18 19 20 Assume that Theorem 1 is valid for 1, 2, , 1, then we get … l -By using ( ), we have


Guy Henry
email: guy.henry@cirad.fr

Eduardo J Trigo
email: ejtrigo@gmail.com

Grupo Ceo

S A Hipólito Yrigoyen

The Knowledge Based Bio-Economy at work: from large scale experiences to instruments for rural and local development

Keywords: economy, large vs. small scale, rural development, policy bio-économie, grande échelle versus petite échelle, développement rural, politique

The new bio-economy is increasingly seen as a workable alternative to move today's economies in the direction of more sustainable natural resource use and economic growth. Most of existing experiences, both from science as well as from implementation of the Knowledge Based Bio-Economy (KBBE) concepts, refer to relatively strong science basis and relatively large scales of application. This paper proposes to review existing advances in the implementation of the KBBE in small scale situations and identify what the key research, institutional and policy issues that need to be taken into account to further promote KBBE approaches to rural and local development. As such, the analytical approach used includes an extensive literature review in the form of a desk study, followed by a comparative analysis and the formulation of recommendations regarding S&T and policy agendas.

Key words : bio-

INTRODUCTION

The new bio-economy, a concept synthesized by the OECD as "the aggregate set of economic operations in a society that uses the latent value incumbent in biological products and processes to capture new growth and welfare benefits for citizens and nations" (OECD, 2006), is increasingly seen as a workable alternative to move today's economies in the direction of more sustainable natural resource use and economic growth. The transition to economies and societies based on this concept is already underway and there is clear evidence of its potential and impacts in a wide arch of application sectors, going from food and health to transportation, construction and, even, the recreational industry. Most of existing experiences, both from science as well as from implementation of the Knowledge Based Bio-Economy (KBBE) concepts, refer to relatively strong science basis and relatively large scales of application. The questions and challenge is whether the new concepts are of relevance for the small scale and what are the needed policies to promote and guide investments and the new institutions to ensure the safety of the new technologies and the equitable distribution of the benefits of the ensuing economic growth. Existing evidence, however, points in the direction that KBBE opportunities can be linked to nearly any kind of plant material and as such bio-based industries are well-suited for local production, at small and medium scales, and as engines for rural development and income generation [START_REF] Trigo | Towards a Latin American Knowledge based Bio-Economy -A Concept Note. International ALCUE-Food Forum "Towards a Latin American KBBE: Prospects and Limitations[END_REF]. This paper proposes to review existing advances in the implementation of the KBBE in small scale situations and to identify what the key research, investment, institutional and policy issues are that need to be taken into account to further promote KBBE approaches to rural and local development. As such, it focuses on a desk study of the existing literature, as basis for a comparative analysis of on-going alternatives and the formulation of recommendations regarding S&T and policy agendas. It is expected that the discussion and findings in the paper will contribute to the growing literature on bio-economy and bio-energy aspects etc. but especially it is an attempt to fill in the gap regarding relevance/benefits/policies for the integration of small-scale farmers and processors with advanced knowledge/biotechnologies in LDC and emerging economies.

The paper is structured in three sections additional to this introduction. The second section focuses on the review of the KBBE concepts and general applications, emphasizing, in particular, those dimensions that are of greater relevance for improved natural resources use, rural and local development, and poverty alleviation. The third section reports on a number of actual experiences of small-scale applications with emphasis on Latin America. The fourth and final section, builds up on the lessons learnt both in Europe as well as in the experiences reported for LDCs. It also proposes and discusses the main components of a research and policy agenda for the better use of KBBE concepts and opportunities in small scale situations in general, and those linked to rural and local development in particular.

REVIEW OF KBBE CONCEPTS AND APPLICATIONS

The bioeconomy, as indicated above, is defined as the utilization of biology -and biological processes -in economic activities. There is nothing new about this, since human societies have been not only utilizing, but have been dependent on biological processes going back to the beginning of time. Actually, if we associate food with mankind survival, the bioeconomy is at the centre of historical processes and societies, as we know them today. Even the idea of added knowledge as a differentiating issue, is not adding anything new, as there are countless examples of progress been associated to processes based on different forms of

The Knowledge Based Bio-Economy at work: from large scale experiences to instruments for rural and local development -G. Henry and E.J. Trigo life getting to be more complex and effective. But somehow, until now, we have been essentially working within species or "sector" issues, with advances constrained to given species or process improvement, essentially related to food and medicine, while engineering and chemistry provided the materials for economic evolution and wellbeing. The essence of the new bioeconomy, is the increasing scope of applications and the rapid disappearance of those relatively "clean" frontiers.

In the last fifty years since the discovery of the DNA structure, the rapid emergence of biotechnology and the new biology and, in more recent times, their interface with advances in information and communications technologies and the nano-sciences, are already creating a new scientific and technological environment within which input-output relationships can be rearranged, existing products and processes redesigned and even whole new products and processes can be developed. Through the possibility of changing the "limits" within which biological processes interact with natural resources -soil, water, solar energy -the new technologies are opening up a whole new range of opportunities not only in food and fibre, but in almost every sector of the economy, including pharmaceuticals, energy production and industry in general. What is new in the "new bioeconomy" is not its conceptual basis; it is the scope of its reach. The consolidation of biotechnology as one of the key components of the technological substratum of today's and future societies, making it pervasive to all sectors of the economy, is probably a key strategy for an effective response to the challenge of building a more equitable society, while reducing humankind environmental footprint.

In its essence it's the coming together of biology, chemistry, materials sciences, genomics and information technology to better exploit natural resources both in agriculture and industry. As such, it implies to capture solar energy and transform it in energy and products, and eventually whole new value chains, with a reduced environmental impact, without sacrificing job creation, helping reduce poverty and continuing to improve the quality of life for a growing world population. In this context, the KBBE is about new ways of linking natural resources and processes to goods and services through increased knowledge intensity as a common denominator of the new value chains.

The transition to economies and societies based on this concept is already underway and there is clear evidence of its potential and impacts in a wide arch of application in different sectors, going from food and health to environment, transportation, construction and every sector of industry. Plant biotechnology -"green" biotech -through GMOs is rapidly becoming the standard rather than the exception -in the international agricultural commodities markets, but the applications are hardly confined to his controversial field. Biotech is today a strategic input into the breeding and propagation of crops, livestock and fish, through the use of genetic markers, and other techniques, such as embryo transfer, and they also play a key role in the development of more effective methods for veterinary and food safety -e.g. veterinary vaccines (pseudo-rabies or foot and mouth disease), detection of salmonella and BSE8 -and more efficient food production processes through enzyme design and engineering, and feed additives.

Applications in the health sector -"red" biotechnology -are probably the most developed ones, with biopharmaceuticals already representing a major share of turnover from all pharmaceuticals, with recombinant vaccines -e.g. hepatitis B -insulin and monoclonal antibodies for cancer treatment, and in vitro diagnostics (IVD), representing the main products but with other applications, such as genomic based diagnostics -e.g. diagnostic assays for detecting biomarkers associated with different diseases -rapidly gaining ground (Zilka et al, 2007).

Besides these applications for human medicine, there is also a rapidly growing interphase between human health and plant biotechnology, through the engineering of plants for the productions of pharmaceutical molecules or vaccines and also the development of new crops

The Knowledge Based Bio-Economy at work: from large scale experiences to instruments for rural and local development -G. Henry and E.J. Trigo for food/feed markets with safer and more-healthful products. This is a major opportunity for positive impact of agriculture on human health, with strong potential to contribute significantly to containment of escalating diseases such as obesity, diabetes, cardiovascular ailments, some cancers and possibly some neurodegenerative diseases. Although still in the early stages this field already has some concrete examples including oil-crops with more healthful oils -Omega 3 -functional foods -yogurt with high phytosterols -and Golden Rice with high vitamin A content for decreased blindness in developing countries, among others (NABC, 2007).

In industry and the environment -"white" and "blue" biotechnology -coverage is also expanding. The production of the new energy feedstocks, as well as in the design of new, more efficient, biofuel production processes, is a development that is just in its early stages, but which has already had a significant impact on the structure of the energy sector and it will probably continue to do so, as technologies evolve from the present generation -competitive with food production -into second and third generation applications aimed at the engineering of enzymes and micro-organisms to convert cellulose crops and waste into more advanced biofuels. Associated with this the production of biomaterials including biopolymers and bioplastics for the construction and engineering sectors is also rapidly developing, although it is still represents a very small fraction of the market. However, it is to be expected that this will change in the not so distant future as the combination of what has become to be known as "green" (plant) and "white" (industrial) biotechnology bringing together conventional plant breeding with genomics and genetic engineering tools allows the production of "high tech" plants designed to better perform under stress conditions, or with properties well suited for industrial processing and the biosynthetic production of specialty chemicals and more complex molecules.

Other examples of this emerging world can be found in mining and environmental remediation, where bioengineering bacteria to facilitate the liberation of valuable metals from mineral ore -e.g. copper from sulphite ore -or to capture and recycle metals form the waste produced from primary extraction processes, is already been used, not only increasing the efficiency of mining operations, but also significantly reducing its environmental impactreducing by 10 fold or more the concentration of heavy metals in refinery waste waters.

Beyond this there are already advanced experiences with the use of engineered "extremophyl" microorganisms to deal with different types of pollutants or fast growing plants and trees that could extract heavy metals and other contaminants form degraded soils (CBAC, 2006).

The above examples -by no means exhaustive -are clear cases of how KBBE processes work at integrating "green", "red" and "white" biotechnology, agricultural production, conversion technologies, materials sciences and other areas, to create new and more sustainable value chains.

Even though these developments are, in most cases, still at an early stage, and market wise they represent only small fractions of the different sectors, their feasibility is in most cases proven. The issues to be confronted are not whether the new concepts are feasible or not, but whether they represent a workable instrument to advance the situation in the developing world and can help improve poverty conditions and their insertion in the world economy. Most of existing experiences, both from science as well as from implementation of the KBBE concepts, refer to a relatively strong science basis and relatively complex applications (both in terms of process and scale). The questions and challenge is whether the new concepts are of relevance for the simpler and small-scale situations and what are the necessary policies to promote and guide capacity development and investments, as well as to ensure the safety of the new technologies and the equitable distribution of the benefits of the ensuing economic growth.

The Knowledge Based Bio-Economy at work: from large scale experiences to instruments for rural and local development -G. Henry and E.J. Trigo

ASSESSMENT OF PAST SMALL SCALE EXPERIENCES

So far, a macro picture was drawn on the concepts, aspects and general applications of the KBBE. This was principally driven by the KBBE diversity of possible products and processes, with the underlying notion of existing or potential demands and markets. In this section we will slice the KBBE cake from a user and/or actor perspective and within this realm especially look at the notions of scale, and foremost small scale. The latter notion first needs further specification. Since, while a developed country -be it US, Europe or Japan -definition of small scale or Micro, Small and Medium Enterprise (SME) has one specific definition1 , this not necessarily coincides with small scale in Developing Countries or Emerging Economies.

The Very Small Enterprises (VSE) in Europe could still be classified as medium enterprises in countries like Colombia or Vietnam. For example, [START_REF] Blank | Small and Medium Enterprises in Agricultural Biotechnology[END_REF] analyzing relations of SMEs in biotechnology (as part of OECD's KBBE analyses) notes that the average OECD countries' SME had an asset value of 40 million USD. Hence, caution is required on the correctness of terminology when comparisons are made between Less and Developed countries….

For our analysis and discussion, relevant small scale KBBE experiences are grouped in three principal classes: (i) Biotechnology crops, (ii) Bio-refineries value chain insertion, and (iii) Biodiversity prospecting and development.

-Biotechnology crops

The latest report2 on global commercialized biotech/GM crops, collected by the International Service for the Acquisition of Agri-Biotech Applications (James, 2010) states that global area of biotech crops continued to grow in 2009 and reached 134 million hectares, or 180 million "trait or virtual hectares". This translates to an "apparent growth" of 9 million hectares or 7% measured in hectares, whereas the "actual growth", measured in "trait or virtual hectares", was 14 million hectares or 8% year-on-year growth. Global growth in "trait or virtual hectares" increased from 166 million "trait or virtual hectares" in 2008 to approximately 180 million "trait or virtual hectares" in 2009. Recent growth over the last few years in the early-adopting countries has come largely from the deployment of "stacked traits" (as opposed to single traits in one variety or hybrid), as adoption rates measured in hectares reach optimal levels in the principal biotech crops of maize and cotton of the major biotech crop countries.

In [START_REF] James | Global Status of Commercialized Biotech / GM Crops[END_REF].

It has to be also noted that together with this evidence that so far biotech crops not only have not excluded small producers, but actually significantly benefited them (Tripp, 2009). It is also true, however, that when looking into the future that there is little in the research pipelines that could be of benefit for the small scale producer and the field is clearly dominated by the activity of the large multinational seed corporations. Furthermore there is also little evidence that developing countries -outside China, India and Brazil -are important players in today's biotechnology scene, an issue with important implications not only in relation to the GM crops, but with respect to many other applications of modern biotechnology which could be of significant impact for the welfare of the rural areas [START_REF] Stein | The Global pipeline of new GM CROPS. Implications of asynchronous approval for international trade[END_REF][START_REF] Trigo | Biotecnología agropecuaria para el desarrollo en América Latina: Oportunidades y Retos[END_REF].

In the same vein as GMOs, a biotech product example is noted by [START_REF] Ithaca | First the peasant? Reflections on modernity, technology and reconstruction[END_REF], that in Southern India, small scale farmers apply a Bt-based insecticide against infestations of their Castor bean plantations. The Bt component in powder form is fabricated in the same region and as such enables farmers to have easy and direct access. This is an example of a biotech product for which the farmer is the direct client and beneficiary.

A final note relates to the application of biotechnology tools to improve the levels of nutrients in grain crops. While for example, the research conducted in HarvestPlus or in GoldenRice [START_REF] Falck-Zepeda | Delivering Genetically Engineered Crops to Poor Farmers: Recommendations for improved biosafety regulations in developing countries[END_REF] to a large extent still rely on conventional plant breeding, the technological advances enable these kind of projects, that impact directly on poor (nutrient deficient) consumers in LDCs.

-Bio-refineries value chain insertion

KBBE is often (erroneously) interpreted as principally involving bio-based activities targeting the research and development of bio-fuel products and processes… Indeed, the latter do represent a large share of current KBBE activities. However, this study aims to look beyond the extensively published successes of large scale bio-refineries i.e. Brazil, US, Argentina…, and to identify tools and activities that are small scale, but also result in bio-energy and/or feed stocks. Furthermore, the objective is to look at cases where small scale farmers are not mere suppliers of raw materials for large scale bio-refineries, but rather to identify the opportunities for small scale farmers to add value on-farm. The documented experiences (mainly from LAC countries) fall into two groups:

3.2.1 Micro units fed with local biomass primary material and/or sub products:

• In Colombia, ethanol prepared from cassava by a pilot plant, has the capacity to produce an average of 300 liters of hydrated ethanol (containing 4%-5% water) per day. The processing plant is part of a major project to develop rural social bio-refineries-low-cost fuel production plants that use cassava, sugar sorghum, or sweet potato as feedstock. As well as powering vehicles, cassava biofuel can also be used to generate electricity-a lifeline for rural communities in developing countries lacking access to the national grid.

Waste products from fuel production can be used to make fertilizers and nutritional blocks for livestock. Any rural community that is not yet connected to electricity power can set aside 3-5 hectares to grow cassava as an energy crop and what they would produce would be sufficient to provide electric power for 6 hours a day, all year round. The biofuel requires the use of inedible high-starch cassava varieties, developed by CIAT researchers for industrial use, meaning there is no direct trade-off between fuel and food. The stems of sugar sorghum and sweet potato can also be used (CIAT, 2009).

• The same principal can also be applied to waste and by products, not only of cassava but also for example of banana. Besides plant waste products, the small-scale applications for on-farm biogas based on animal waste (pig, cattle) have been around for decades in both developed and developing regions. With current further advanced technologies, these applications can be further modified to become more cost-effective and applicable to additional biomass waste feed stocks (UNIDO, 2007a).

Rural processing for "intermediate" product supply to large scale factory

The logic of this small-scale actor model is based on the objective of reducing transportation costs for delivery to the principal refinery, which as a secondary objective generates added value to the raw material, by introducing a pre-processing step on-farm. [START_REF] Sanders | Presentation at the International ALCUE-Food Forum "Towards a Latin American KBBE: Prospects and Limitations[END_REF] argues for such "forward integration" model which, although not designed for the purpose of value adding at the farm,? offers the small scale farmer the following potential benefits :

• Pre-processing step generating additional demand for (family) labor and economic return,

• Possibility of spreading income stream beyond the principal harvest seasons,

• Adds further value through the on-farm utilization of the byproducts generated during the preprocessing step, • Shortens commercialization channels and hence improves/stabilizes prices, • Reduces transport costs.

A concrete application of Sanders' forward integration model is offered by DADTCO (2009) in Nigeria and further discussed by [START_REF] Sanders | Voorwaartse Integratie in de Akkerbouw[END_REF]: cassava is on farm processed by Autonomous Mobile Processing Units (AMPU), that are constructed in containers and driven to regions with a high concentration of cassava production. These units process the cassava into an intermediate product that significantly reduces its perishability, leaves by products on the fields, reduces farm-factory transport costs and increases farmers' bargaining power with starch factories. Currently there are 14 units operating together pre-processing 40 000 MT. A salient detail is that in 2009, the Bill and Melinda Gates Foundation, together with the Dutch DGIS, co-funded further replication of this model in African countries [START_REF] Sanders | [END_REF].

-Biodiversity prospection and development

Conventionally, biodiversity has been seen in the context of the conservation of the natural resources base, assuming that it deals with a public good and that no immediate economic value is attached to it. However, the KBBE paradigm turns the concept around from a defensive "conservation" strategy, to a pro-active "exploitation" strategy. Hence, the KBBE aims to fully target biodiversity to explore and capture the full economic potential by means of suitable economic, political and technological interventions. As stated by UNIDO (2010) "The recent phenomenal scientific and technological advances imbedded in biotechnologies, are enabling the construction of "win-win" scenarios. Today a wide range of technologies exist allowing to tap into the economic value of hitherto under-utilized or unexplored biological resources".

Cutting through the expanding literature on this subject with a small scale actor knife, suggests that perhaps an appropriate classification would be the following: (i) "known" biodiversity that needs further full exploitation and development, and (ii) "unknown" biodiversity that needs to pass the full cycle of exploration, discovery and development.

-Already prospected, hence "known"

but not yet (fully) exploited nor domesticated nor developed plant based feed-stocks typically targeting biofuel production i.e. Castor bean (Ricinis comunis), Purging nut tree (Jatropha curcas), Switch grass, ….

The Knowledge Based Bio-Economy at work: from large scale experiences to instruments for rural and local development -G. Henry and E.J. Trigo

• Probably the most frequently referenced "new" biodiesel feedstock is the purging nut tree (Jathropha c.). While not new, since experiments have been recorded for 80+ years, Jathropha c. needs much additional research, "domestication" and development. In the mean time, on the 3 continents, sizeable public and private investments are fast increasing planted areas. Current production systems range from large scale plantations to small family farms. So far production and processing costs remain relatively high and several research challenges remain regarding oil quality and detoxification. An expert consultation in Brazil (UNIDO, 2007b), concluded that Jatropha cultivation for biofuel, be it in Brazil or other parts of the world, can only be socio-economically beneficial to the rural poor and marginalized communities if done on small scale, retaining the small holder and family farm land use and ownership. However, recent data from pilot projects in Namibia (Bruntrup et al, 2009) show that economic and social benefits are positive and only vary according to different models (plantation, community, …). 2007b). Subsequent higher production costs in the semi-arid regions need to be offset by strong market demand from chemical industries. Nonetheless, principal bottlenecks are identified and could be resolved through RDT interventions.

• Non Timber Forest Products (NTFP) like andiroba, buriti, guarana, copaiba, unha de gato etc have been traditionally locally exploited for perhaps centuries. However, increased prospective research continually generates new (functional properties) opportunities for additional potential uses and markets [START_REF] Henry | Análise de Mercado de Produtos Extrativistas Florestais do Estado de Acre, Brasil[END_REF]. Small farmers and forest dwellers play an essential role in the collection, pre-processing and delivery of the raw materials. Subsequent economic returns are important. The construction of the value chain originating in native tropical forests and delivering to high-tech European or Asian based (cosmetic, pharmaceutical) laboratories has many challenges. Nonetheless, these traditional products that are inserted in new high-value international markets are an important part of the bio-based economy portfolio for small scale actors. Some of the well known examples are guarana for Brazilian soft drinks, and NTFP essential oils for NATURA cosmetics product line [START_REF] Henry | Análise de Mercado de Produtos Extrativistas Florestais do Estado de Acre, Brasil[END_REF]).

An important point in the aforementioned 3 experiences is that the basic relevance to small scale farmers is evident. However, the small scale actor model here of basic raw material supplier needs to evolve by introducing pre-processing steps on site (on farm), hence value adding and increasing economic returns. This is already the case for NTFPs.

-Unknown biodiversity needing prospection, domestication and development.

• Corporation Biotech in Colombia is advancing a program to indentify, explore and value add bioactive products in traditional tropical fruits produced by small scale farmers, for nutrition and industry uses incl. cosmetics (UNIDO, 2010). • In Peru a government funded project PNCC advances the prospection of camu-camu (Myrciaria dubia (HBK) Mc Vaugh), which is a native tree that grows in flooded areas in the Amazon region. The tree is an important component of riparian vegetation in Peru and Brazil, and is especially abundant in Peruvian black water river systems. It produces a fruit with the same name extremely rich in ascorbic acid (an average of 2.7-2.8g of AA and up to 4000mg AA per 100g of pulp). The tree is also highly productive, with some studies of wild populations estimating fruit production at 9000 to 12,000kg per hectare,

The Knowledge Based Bio-Economy at work: from large scale experiences to instruments for rural and local development -G. Henry and E.J. Trigo with the potential to generate exceptional economic returns for small local farmers and forest dwellers. Besides ascorbic acid, the camu camu fruits contain small quantities of calcium, iron, niacin, tiamin, riboflavin and other phytochemicals that gives to the fruit powerful oxidative, anti-inflammatory and anti-depressive properties. This, combined to the agreeable flavour, makes the camu-camu a good alternative to the nutraceutical industry (UNIDO, 2010). • The Industrial Biotechnology Center at INTI in Argentina is advancing a program to improve the nutraceutical properties of blueberries. The programme includes research for the determination and quantification of fisetin in blueberries and the process to obtain rich extracts of fisetin. In addition, fisetin rich varieties and blueberry appropriate agronomics characteristics are being studied. Results serve the pharmaceutical industry. Future blueberry varieties can be directly adopted by small and medium local producers [START_REF] Diaz | UNIDO Expert Group Consultation on the KBBE as Basis for Economic Development and Industrial Sustainability[END_REF]. • At the Pontificia Universidad Javeriana in Colombia research is well advanced in the identification of functional properties of Aloe barbadensis waste products that are subsequently being converted into lactic acids using thermophylic strains, producing feedstock for biofuels. This process is applicable to small and medium enterprises (Hodson, 2009). • Kew Gardens in the UK is among other research, focusing on the functional properties of oil seeds from annual and perennial crops and trees in moderate as well as tropical climates. One of its successful projects, in collaboration with Unilever, regards the Allanblackia floribunda which thrives in native African rain forest. Its seeds contain 70% oil for edible and cosmetic uses. In 2008 in Tanzania and Ghana, 100 000 seedlings were commercially planted. The technology is scale neutral and small farmers will be the principal beneficiaries [START_REF] Prittchard | New Technologies to ensure Seed Quality of Biodiverse Oilseeds[END_REF].

Mateo ( 2009) analyzed the lessons learnt and future prospects of biodiversity utilization in Latin America. He documents an experience in Costa Rica where the National Biodiversity Information System -CRBio (part of the Global Biodiversity Information Facility), has placed through electronic means, a total of 30,000 species of its principal taxonomic groups in the public domain. He also points out that bioprospecting not only is of principal interest to upmarket transnational companies, but also for national and regional markets. He further concludes that given the complexities attached to environmental policies, commercial use and IPRs, considerable additional discussion, analysis and agreement is needed between the principal (private and public) actors in the various sectors of society. In this discussion, small-scale actors and ethnic groups will deserve further intention to safeguard their appropriate share of potential future benefits.

-Relevance and bottlenecks for LDC small scale users and actors

The small-scale experiences so far discussed have been analyzed regarding their appropriateness for small scale farmers, benefits to rural development, bottlenecks and future R&D and policy needs (Table 1). What can be observed first of all is that there actually exists an interesting range of different options to connect small scale actors to the KBBE. As such, local small actor socio-economic, policy-institutional and agro-ecological conditions pre-destine the most viable options that could best intervene (once adapted appropriately). However, the issue of the lack -or not sufficiently developed -capacities in the biotech area, is a common limitations appearing in each case. Secondly, there seems to be a group of on farm (pre)processing technologies that offers a very wide variety of different applications using raw material ranging from traditional crops (cassava, banana, plantain, sorghum, maize, …), their by and/or waste products, and other materials such as grasses, wood products etc. Livestock can play an additional role in these scenarios. Thirdly, timelines are varying. Certain bio-based applications suitable to small scale actors are already (or close to) transfer/adoption i.e. biotechnology crops, small pre-processing units and some of the A recent paper by [START_REF] Langeveld | Towards a Bio-Based Economy: Food and beyond[END_REF] assesses the main development perspectives of bio-based products (for LDCs). While discussing bio-fuels, bio-refineries and bio-materials, they analyze in more depth the bio-products such as pharmaceuticals, chemicals, solvents, polymers, etc. Their results show that these mostly high value bio-products have only a low to moderate prospect of impact for local producers. This may serve as an additional set of information regarding KBBE prospects for development (though not per se small scale).

CONCLUSIONS AND RECOMMENDATIONS

In this paper the line of argument was that so far, typically the larger scale KBBE applications have driven and proven the model, thus raising the question regarding the possible benefits of the KBBE model, its concepts and applications for small-scale actors and rural / local development. The previous sections have shown evidence, that there exists a wide range of KBBE related innovations that can benefit small scale actors directly or indirectly through lower operational costs, improved markets, better product prices, expanded demand for labor, etc. It was also shown that most applications still need varying levels of RDT and/or policy interventions to optimize their suitability and subsequent diffusion, and the capacities to do so is lacking in many of the developing countries where the benefits of such work would be the greatest.

KBBE applications without policy interventions typically seek to increase scale economies for improved cost effectiveness and competitiveness. Equity considerations in the form of "propoor (public) policies" (in the form of targeted subsidies, investments, training, information, advice….) on the one side and "corporate social responsibility" as internal policies with the private sector, on the other side, will be needed to increase the probability of the degree of insertion of small-scale actors in the KBBE model. Furthermore, research and technology delivery agendas need to be adapted to include the wide array of (new topics and/or already existing) small-scale technologies, applications, systems etc that are part of the KBBE concepts and that target their benefit streams to small scale actors and/or rural development. This is in line with [START_REF] Langeveld | Towards a Bio-Based Economy: Food and beyond[END_REF] who argues that "one pathway to household food security and poverty reduction comprises household entitlements (income) associated with high value added products from biomass, through small scale local bio-refineries producing bio-energy or bio-products in poor, marginal and remote areas. Incentives for such decentralized investment in bio-refineries would require pro-poor institutional and policy environment".

Besides appropriate equity considerate policies, improved regulatory policies (IPR, bio safety) are needed to create the institutional and market conditions that allow for small actors to benefit from the KBBE RDT opportunities. As [START_REF] Louwaars | Tailoring rights regimes in biotechnology: Introducing DRIPS next to TRIPS[END_REF] argues "Opportunities exist to tailor intellectual property rights (IPR) to development objectives, and where policy space is reduced due to strong demands from trade negotiations, openings are forged through private and NGO initiatives. It appears that such openings are not being developed in developing countries in the field of genetic resources rights".

In moving forwards towards a more proactive environment in terms of taking advantage of the opportunities identified there is the need to increasing awareness among institutions and key actors regarding the opportunities and the needs that the KBBE represents for small scale and rural development, and consequently the need to mobilize resources significantly strengthening biotechnology capacities at the developing countries level. While this is done, there is also the need to start moving at the research level. Here there is a pressing lack of more detailed information about the opportunities themselves. As mentioned, there is amle evidence on large-scale cases, but little well substantiated prospective on what the new KBBE environment can bring to the small scale and the rural areas. As science applications in this field are enlarging, their scope (in many situations only on-paper potential), are becoming real and feasible, both scientifically and from the economic point of view. Additional and more quantitative studies are required to better select the possible win-win options to specific small scale actor, and also to support awareness rising at the policy level.

Beyond this, research should be focusing on to main fronts. One is the expansion of the small scale energy pilot experiences to other feedstocks, as to have a more varied menu of options that better reflects the diversity usually found in rural areas. The second is the development of more standardized methodologies for the evaluation of biodiversity and its potential use, that are able to fully integrate the empirical knowledge that local communities can offer with the advanced genomic technologies required for the proper valorization of the available diversity.

  2009, the number of farmers benefiting from biotech crops globally in 25 countries reached 14.0 million, an increase of 0.7 million over 2008. Of the global total of 14.0 million beneficiary biotech farmers in 2009, (up from 13.3 million in 2008), over 90% or 13.0 million (up from 12.3 million in 2008) were small and resource-poor farmers from developing countries; the balance of 1 million were large farmers from both industrial countries such as the USA and Canada, and emerging economies such as Argentina and Brazil. During the second decade of commercialization, 2006 to 2015, biotech crops have an enormous potential for contributing to the Millennium Development Goals (MDG) of reducing poverty by 50% by 2015. Initial research in China indicates that up to 10 million more small and resource-poor farmers may be secondary beneficiaries of Bt cotton in China

The Knowledge Based Bio-Economy at work: from large scale experiences to instruments for rural and local development -G. Henry and E.J. Trigo ISDA 2010, Montpellier 28-30 Juin 2010 6

  • Castor bean (Ricinis comunis) has traditionally been cultivated in Brazil and India. Castor bean meal is internationally traded animal feed ingredient. The crop is tolerant to the marginal climatic and soil conditions like the semi-arid Northeast of Brazil and as such it represents one of the few cropping options in the region (besides extensive cattle production, cassava and cashew). In NE Brazil virtually all production is done by small scale farmers. Current yields in this region are low (629 KH/ha), compared to those on the more fertile land in Minas Gerais or Sao Paulo state, that are double or triple (UNIDO,

Table 1 -

 1 The Knowledge Based Bio-Economy at work: from large scale experiences to instruments for rural and local development -G.Henry and E.J. Trigo "newer" feed stocks. Other applications at the earliest are "in the pipe" or need additional basic and/or applied research, i.e. biodiversity prospecting. This time aspect again is important to take into account when analyzing KBBE related innovations and their potential for small scale actors. Fourthly, while we argue that the discussed KBBE applications for small scale actors have varying degrees of suitability, they will all need additional RDT and policy interventions to become KBBE small scale champion examples for wider diffusion and adoption. Relevance and bottlenecks of small scale KBBE experiences and future needs

	KBBE type of	Applicability for	Beneficial for	Current	Necessary	Necessary RDT
	activity	small scale user	rural	principal	policy	needs
			development	bottlenecks	intervention
	Group 1 -Biotechnology crops
		Dependent on the	Important	Low levels of	Regulatory	Applications
		crop. Maize and	through	investments in	systems	emphasis on
		soybean	multiplier effects	non-commodity	harmonization	small farmers
		applications not	of increased	crops, by both	to facilitate	and "orphan"
		clear. But majority	activity in the	the public and	extension of	crops.
		of benefits in	non-farm sector	private sector.	available
		cotton have been	(estimated to be	Reduced	innovations to
		to small producers	50%-100% of	capacities and	small producers
			increases in the	regulatory	crops
			farm sector)	complexities
				limiting
				involvement
				from public
				research
				institutions
	Group 2 -Biorefinery value chain insertion
		Pilot experiences	Impacts are	Lack of biotech	Investment	Feedstock
		with high potential	through	capacities for	support both at	diversification
	On-farm micro	for small scale,	improved rural	exploring	farm level and	avoiding
	units i.e.	with feedstock not	communities	options and	for integrating	competition with
	CASSAVA IN	competing with	wellbeing	scaling up pilot	surplus energy	food use
	COLOMBIA	food production	through more	experiences in	production into
			reliable sources	terms of	the regional
			of energy to	equipment	/national energy
			support other	availability and	supply
			social and	diversity of
			economic	sources of
			activities	feedstock
		Pilot experiences	Potentially high	Lack of biotech	Market	Post harvest
	Intermediate	have proven	benefits from	capacities to	integrations	and value
	products pre-	viability at the	expanded	explore /	policies to	adding
	processed on-	small scale level	possibilities of	develop options	connect local	technologies
	farm i.e.		added value on	and product	and national	Impacts of
	CASSAVA IN		local production	standardization	markets	norms and
	AFRICA			and		standards
				limited
				experiences
				and small local
				markets
	Group 3 -Biodiversity prospection, domestication and development
		New crops have	Potentially	Not enough	Funding	Breeding and
		proven	important	knowledge on	mechanisms for	agronomy for
		competitive and	through	many aspects of	projects	productivity
	"Known" but	improved	expanded	the new crops	integrating on	improvement
	yet	alternatives over	income	and their	farm production	Market potential
	unexploited	traditional,	opportunities for	production	with value chain	and information
	biodiversity	particularly in poor	farmers and	systems and	development	for value chain
	i.e.	resources areas	other activities	value chains are		development.
	JATHROPHA		at the local level	not fully
				developed

See: http://ec.europa.eu/enterprise/policies/sme/facts-figures-analysis/sme-definition/index_en.htm

The following section is borrowed fromJames (2010)

The Knowledge Based Bio-Economy at work: from large scale experiences to instruments for rural and local development -G. Henry and E.J. Trigo


Langeveld

Catherine Macombe
email: catherine.macombe@cemagref.fr

Sophie Madelrieux
email: sophie.madelrieux@cemagref.fr

Benoît Dedieu
email: benoit.dedieu@clermont.inra.frd

Hans Langeveld

Family Farming Under Pressure: Reassessing options for liveability and permanence

Keywords: liveability, social impacts, farm ending, farm maintaining, work conditions, land use management, farming system shifts

Family farming under pressure: reassessing options for liveability and permanence -An overview

Presentation of the WS 2.3

Building sustainable future of rural areas requires more sustainable farming systems with better working conditions for farmers. Liveability is defined as the characteristics of the systems which satisfy farmers' expectations with respect to farming and relations with family and other activities. Farmers expectations of farming conditions are diverse, and deal with many issues: income, work load, options for vacations, nature of work, physical work, recognition by society, usefulness feeling etc. Until recently, farming systems appeared to be fairly stable entities, both in time and space, characterised by continuity and based on a steady set of productive and cultural activities. A range of societal changes are causing tensions which are exacerbated by the uncertainties stemming from globalization and demographic transitions.

Farming systems are affected by these developments just like other elements of society. New land use and production schemes are appearing. In this workshop, we address some of the social impacts caused by these shifts, with a special focus on work conditions and on continuity of farming. We deal with methods to analyse liveability and changes in farming systems, we ask how to improve farming systems to fulfil work duration and work patterns expectations, we get insights about integration of marginal people, job quality, interests for community, conditions and consequences of ending farm activities, etc.

A special session (session 1) "Social impacts of shifts in land use management" will be devoted to the social impacts caused by the farming systems shifts that affect land use management. We study three case studies from Portugal (Barroso et al.; Menezes at al.) and from the Sudano-Sahelian Africa zone (Dongmo et al.) where new organizations for plant biomass management are arising.

The workshop addresses the main social impacts of evolving farming systems through (session 2) and (session 3) "When the farming system evolves". What (Alexopoulos et al.) highlight the factors affecting farmers' decision to convert or to abandon organic farming, while (Hayashi and Sato) link interest for community practices with organic rice farming in Japan.

"From job and working conditions to permanence" is the title of the (session 4) and (session 5). The workshop produces a main piece of debate about the work conditions topic, wrapping up from job practices to the issue of farm permanence. A first paper (Crawford et al.) describes the special programme supporting liveability of dairy farms in Australia. Two papers aim to provide methods for the assessment of work conditions, either from the farmers (Fiorelli et al.) or as a way to characterize actual work features (Cournut and Chauvat). Liveability is at the core of the farms networks described in France, (by Kling-Eveillard et al.). It is the crucial issue when new supply chains (Petit et al.) too. (Schmitt and Inhetveen) deal with the assessment of the diversity of farm trajectories considering farming women coping with 30 years of agricultural restructuring in Germany, while (Rueff and Gibon) do the same by considering family dynamics during 60 years in South of France.

Sometimes, it is difficult to draw a clear line between social conditions and social impacts. This is demonstrated by the issue of farm continuation and termination. (Rossier) highlights conditions related to farm discontinuation for Switzerland, an issue discussed for Belgium by (Rogge and Kerselaers).

As usual, the strength of the International Farming System community is to deliver contrasted insights about the same topic, thanks to different backgrounds, countries and scopes. This workshop is a perfect illustration. The detailed contributions from this workshop are given below.

  happens when new crops are introduced (biofuel crops in Brazil, by Magalhaes; rubber in China, by Tang et al.; tea intercropping in China, by Leshem et al.)? In this ocean of uncertainties, preserving local resources may be worthwhile. Keeping local animal breeds may enhance food security (in Serbia, by Rodriguez et al.) and the Croatian smallholders pig farmers are the warranty for certain way of living, threatened by modernization (Wellbrock et al.). The workshop, further, tackles liveability of new farming systems, be it organic (in France, by Hellec and Blouet) or more conventional dairy farming (Madelrieux et al.). From a Greek case study,

th European IFSA Symposium, 4-7 July 2009, Vienna (Austria)


Graeme W Milton

Pierre Seppecher

Hybrid electromagnetic circuits

Keywords: Electromagnetic circuits, Electromagnetism, Circuits, Metamaterials PACS: 41.20.-q, 78.20.Bh, 84.30.Bv, 84.40.Dc

Electromagnetic circuits are the electromagnetic analog at fixed frequency of mass-spring networks in elastodynamics. By interchanging the roles of ε and µ in electromagnetic circuits one obtains magnetoelectric circuits. Here we show that by introducing tetrahedral connectors having ε = µ = 0 one can join electromagnetic and magnetoelectric circuits to obtain hybrid circuits. Their response is governed by a symmetric matrix with negative semidefinite imaginary part. Conversely given any such matrix a recipe is given for constructing a hybrid circuit which has that matrix as its response matrix.

At a given fixed frequency ω Maxwell's equations, which can be written in the form

∂ ∂x i C ijkℓ ∂E ℓ ∂x k + {iωj} j = -{ω 2 εE} j (1)
where C ijkℓ = e ijm e kℓn {µ -1 } mn [in which E is the electric field, j is the free current density ε the electric permittivity tensor, µ the magnetic permeability tensor, and e ijm = 1 (-1) if ijm is an even (odd) permutation of 123 and is zero otherwise] bear a close resemblance to the equations of continuum elastodynamics

∂ ∂x i C ijkℓ ∂u ℓ ∂x k + {f } j = -{ω 2 ρu} j (2)
[in which u is the displacement field, f is the bodyforce density ρ is the density tensor, and C is now the elasticity tensor]. It is therefore natural to ask: What is analogous in electrodynamics to a discrete system of springs and masses in elastodynamics? The answer is an electromagnetic circuit, introduced by us in [START_REF] Milton | Electromagnetic circuits, Networks and Heterogeneous Media Submitted[END_REF]. The idea of an electromagnetic circuit generalizes the idea of Engheta, Salandrino, and Alú [START_REF] Engheta | Circuit elements at optical frequencies: Nanoinductors, nanocapacitors, and nanoresistors[END_REF] and Engheta [START_REF] Engheta | Circuits with light at nanoscales: Optical nanocircuits inspired by metamaterials[END_REF] who realized that normal linear electrical circuits could be approximated, in the quasistatic limit (which does not imply the frequency is low, but only that the size of the network is small compared to the wavelength) by a connected network of thin cylinders each of material with a suitably scaled value of ε = εI surrounded by a cladding of zero-dielectric material (with ε = 0): a cylinder with a real positive value of ε approximates a capacitor, while a cylinder with a positive imaginary value of ε approximates a resistor, and a cylinder with an almost negative real value of ε approximates an inductor.

A system of springs and masses can be approximated by massless elastic bars (having ρ = 0 and C appropriately scaled; buckling of the bars is ignored as one working within the framework of linear elasticity) with rigid spherical masses (having C = ∞ and ρ appropriately scaled) at the junction nodes between bars and surrounded by void (having ρ = 0 and C = 0). A subset of nodes are chosen to be terminal nodes at which displacements are prescribed. Similarly an electromagnetic circuit, as illustrated in Figure 1, can be approximated by zero-dielectric diamagnetic thin triangular plates (of width h → 0 having ε = 0 and µ appropriately scaled) with zero-magnetic dielectric cylinders (having µ = 0 and ε appropriately scaled) at the junction edges between triangular plates and surrounded by a cladding (having ε = 0 and µ = ∞). A subset of edges are chosen to be terminal edges along which the electric field is prescribed. As expected from ( 1)-( 2) ε plays the role of ρ and µ -1 plays the role of C. What is perhaps not quite so expected is the different geometrical structure of the elements: triangular plates instead of bars and cylinders instead of spheres. However this is necessitated by the continuity of the fields. Indeed in the cladding one has H = 0, similarly to the way one has a stress σ = 0 in the void surrounding the elastodynamic network. A bar in such a medium cannot have a constant non-zero value of H inside it, by continuity of the tangential component of H across the boundary, while a triangular plate in the medium can have a constant value of H, directed normal to the surface. When j = 0 Maxwell's equations remain invariant when the roles of (E,D,ε,ω) are switched with those of (H,B,µ,-ω). Therefore for every electromagnetic circuit there is a corresponding magnetoelectric circuit, where the roles of ε and µ are interchanged.

Of course such extreme values of ε and µ are difficult to achieve, even with metamaterials over a narrow frequency range. So why introduce electromagnetic circuits when they are so difficult to realize? One answer is that they provide new ways of manipulating electromagnetic fields, that are relatively easy to analyze. Indeed, the possible responses of electromagnetic circuits in which no two terminal edges are connected have been completely characterized [START_REF] Milton | Electromagnetic circuits, Networks and Heterogeneous Media Submitted[END_REF]. If a desired manipulation is possible with an electromagnetic circuit this should motivate the search to achieve a similar manipulation with more realistic materials. Another answer is more fundamental. Shin, Shen and Fan [START_REF] Shin | Three-dimensional electromagnetic metamaterials that homogenize to uniform non-maxwellian media[END_REF] have shown that metamaterials can exhibit macroscopic electromagnetic behavior which is non-Maxwellian, even though they are governed by Maxwell's equations at the microscale. [See also Dubovik, Martsenyuk and Saha [START_REF] Dubovik | Material equations for electromagnetism with toroidal polarizations[END_REF] where other non-Maxwellian macroscopic equations are proposed]. So what sort of macroscopic electromagnetic equations can one obtain? The success of Camar-Eddine and Seppecher [START_REF] Camar-Eddine | Closure of the set of diffusion functionals with respect to the Moscoconvergence[END_REF][START_REF] Camar-Eddine | Determination of the closure of the set of elasticity functionals[END_REF] in addressing such questions in the context of three-dimensional (static) conductivity and elasticity, suggests one should try to characterize the continuum macroscopic behaviors of electromagnetic circuits and then try to prove that this encompasses all possible macroscopic behaviors. The continuum limits of electrical circuits can have interesting macroscopic behaviors as discussed in the book [START_REF] Caloz | Electromagnetic Metamaterials: Transmission line theory and microwave applications[END_REF] and references therein. The continuum limits of electromagnetic circuits should have an even richer span of macroscopic behaviors.

When discussing elastic networks it is quite usual to talk about applying a force to a terminal node. This could be a concentrated body force, or could be provided by say a medium external to the network which we do not need to precisely specify when talking about the applied force. In a similar way in an electromagnetic network it is convenient to talk about applying a free electrical current to a terminal edge. This could be a concentrated current caused by an electrochemical potential, or could be an intense H field provided by an external medium, acting across the terminal edge. To be more precise, at the interface between the external medium and the terminal edge we require that n • C∂E/∂x be continuous, which is nothing more than requiring the tangential component of H to be continuous. In elastodynamics the quantity n•C∂u/∂x is called 2 the surface force F. We adopt a similar terminology for electrodynamics and call (iω) -1 n • C∂E/∂x the surface free current J. (The additional factor of (iω) -1 is introduced because iωj in (1) plays the role of f in (2).) In a domain Ω which is divided in two subdomains Ω 1 , Ω 2 we can say that Ω 2 exerts on Ω 1 a surface free current J while Ω 1 exerts on Ω 2 the opposite surface free current -J. This formulation does not mean, in any way, that there exist actual free currents in the material, just like the Newtonian action-reaction law does not imply the existence of actual surface forces inside the domain. Similarly in magnetoelectric circuits it is convenient to talk about applying a free magnetic monopole current to a terminal edge. Again this in no way implies that there exist actual magnetic monopole currents, but rather that the equivalent effect is provided by intense E fields in an external medium acting across the terminal edge. But mathematically there is no barrier to thinking of magnetic monopole currents: in the presence of such currents the equation analogous to (1) is

∂ ∂x i L ijkℓ ∂H ℓ ∂x k -{iωg} j = -{ω 2 µH} j (3)
where L ijkℓ = e ijm e kℓn {ε -1 } mn and g is the free magnetic monopole current density.

The response of a electromagnetic circuit with n terminal edges is governed by a linear relation iωJ = WV between the variables J = (J 1 , J 2 , . . . , J n ) (not to be confused with the J in the previous paragraph) whose components now represent surface free currents acting along the terminal edges (the surface free current along terminal edge i is constant along the edge and directed along the edge, and the complex scalar J i represents the total current flow in that direction) and the variables V = (V 1 , V 2 , . . . , V n ) which are the line integrals of the electric field E along these edges. As shown in [START_REF] Milton | Electromagnetic circuits, Networks and Heterogeneous Media Submitted[END_REF], the matrix W is symmetric, with a negative semidefinite imaginary part. Furthermore if the terminal edges are disjoint, given any matrix W with these properties there exists a electromagnetic circuit (specifically an electromagnetic ladder network, as described in [START_REF] Milton | Electromagnetic circuits, Networks and Heterogeneous Media Submitted[END_REF]) which has W as its response matrix. Similarly, the response of a magnetoelectric circuit with m terminal edges is governed by a linear relation -iωG = YU, or equivalently ωG = Y(iU), between the variables G = (G 1 , G 2 , . . . , G m ) which represent surface free magnetic monopole currents acting on the terminal edges, and the variables U = (U 1 , U 2 , . . . , U m ) which are the line integrals of the magnetic field H along these edges. The matrix Y is symmetric, with a negative semidefinite imaginary part. Electromagnetic circuits and magnetoelectric circuits seem like completely different animals. Indeed, an electromagnetic circuit has a cladding with ε = 0 and µ = ∞, whereas a magnetoelectric circuit has a cladding with ε = ∞ and µ = 0. However they can be joined to create hybrid electromagnetic circuits at the cost of introducing an additional circuit element: a connector which joins a terminal edge of an electromagnetic circuit to a terminal edge of a magnetoelectric circuit. As illustrated in figure 2 the connector is comprised of a material with ε = µ = 0 in the shape of a tetrahedron with vertices ABCD, clad on the top and bottom faces ACD and BCD with a material having ε = 0 and µ = ∞ (in which H = 0 and D = 0) and on the two side faces ABC and ABD with a material having ε = ∞ and µ = 0 (in which E = 0 and B = 0). The edge CD and the edge AB are clipped to expose a surface of width h → 0 with ε = µ = 0. The edge CD may either be a terminal edge (what we will call an E-terminal), or may be connected to the terminal edge, of width h, of an electromagnetic circuit. The edge AB may either be a terminal edge (what we will call an Hterminal), or may be connected to a terminal edge, of width h, of a magnetoelectric circuit.

Inside the tetrahedron ε = µ = 0, and so both E and H are curl-free. Since H = 0 in the cladding on the top and bottom faces, and since the tangential component of H is continuous across these 3 interfaces, it follows that the line integral U v of H upwards along the edge BA must equal the line integral of H upwards across the edge CD of width h. Similarly, since E = 0 in the cladding on the sides, it follows that the line integral V h of E forwards along the edge DC must equal the line integral of E forwards across the edge AB of width h. If J h denotes the surface free electrical current acting along the edge CD in the forward direction and G v denotes the surface free magnetic monopole current acting along the edge AB in the upwards direction, then

J h = U v and G v = V h , or equivalently iωJ h ωG v = M V h iU v , (4)
where

M = 0 ω ω 0 . (5)
The terminal edges in hybrid electromagnetic circuits come in two varieties: those for which we can apply a free surface electrical current directed along the edge, which we call E-terminals, and those for which we can apply (in theory) a free surface magnetic monopole current directed along the edge, which we call H-terminals. The response of general hybrid electromagnetic circuit with n E-terminals and m H-terminals is governed by a linear relation A = MX between the variables A = (iωJ 1 , iωJ 2 , . . . , iωJ n , ωG 1 , ωG 2 , . . . , ωG m ) and X = (V 1 , V 2 , . . . , V n , iU 1 , iU 2 , . . . , iU m ), where the matrix M is symmetric, with a negative semidefinite imaginary part. These two facts are most easily verified if both opposing edges of all tetrahedral connectors are included among the terminal edges. Then the matrix M takes the form

M = W Z Z T Y ( 6
)
where W is the response matrix of the (possibly disconnected) part which is an electromagnetic circuit, Y is the response matrix of the (possibly disconnected) part which is a magnetoelectric circuit and Z ij is ω if there is a tetrahedral connector which connects E-terminal i with H-terminal j, and is zero otherwise. It then follows, for example from the arguments in section 5 of [START_REF] Milton | Electromagnetic circuits, Networks and Heterogeneous Media Submitted[END_REF], that these properties of M extend to hybrid electromagnetic circuits in which some or all of the tetrahedral connector edges are not terminal edges. Now the response matrix M can always be expressed in the form (6) if we allow more general ( possibly complex) matrices Z. Then by manipulating the relation A = MX we obtain the equivalent relation

iωJ iωU = S V -G (7)
where

S = W -ZY -1 Z T -ωZY -1 -ωY -1 Z T -ω 2 Y -1 (8)
is symmetric. Conversely it is easy to check that if S is symmetric so too is M. Also the fact that

-X ′ • M ′′ X ′ -X ′′ • M ′′ X ′′ = A ′ • X ′′ -A ′′ • X ′ = (iωJ) ′ • V ′′ -(iωJ) ′′ • V ′ -(iωU) ′ • G ′′ + (iωU) ′′ • G ′ = - V ′ -G ′ • S ′′ V ′ -G ′ - V ′′ -G ′′ • S ′′ V ′′ -G ′′ (9)
implies M ′′ is negative semidefinite if and only if S ′′ is negative semidefinite.

Given our hybrid circuit we can attach m tetrahedral connectors to the m H-terminals, and allow these H-terminals to be internal edges. At the (n + j)th E-terminal, which is connected to the former jth H-terminal we have J n+j = U j and V n+j = -G j where the minus sign arises, because while G j is the surface free magnetic monopole current acting on the hybrid circuit at the former jth H-terminal, -G j is the surface free magnetic monopole current acting on the tetrahedral connector. According to these relations and ( 7), S will be the response matrix of this new circuit, i.e. iωJ = SV. Thus from the hybrid circuit we have obtained a circuit which responds exactly like a pure electromagnetic circuit.

We can now establish that any given (n + m) × (n + m) symmetric matrix S with negative semidefinite imaginary part can be realized by a hybrid circuit with n E-terminals and m H-terminals which have no vertex in common. We first construct the n + m E-terminal electromagnetic ladder network which has S as its response matrix. Then to the edges n + 1, n + 2, . . . , n + m we attach tetrahedral connectors to convert these E-terminals into H-terminals (taking these former E-terminals to be internal edges in the new circuit). This leaves the response matrix S unchanged, which now governs the response [START_REF] Camar-Eddine | Determination of the closure of the set of elasticity functionals[END_REF] of our new hybrid circuit. The proof is complete. Note that in the ladder network (and more generally in other electromagnetic circuits), the internal zero-magnetic dielectric cylinders which join the terminal edge will generally carry some displacement current D. When we join a tetrahedral connector to the terminal edge this displacement current will flow into the side cladding which has ε = ∞ and µ = 0 and can support a nonzero value of D.

In summary, although hybrid electromagnetic circuits seem to be vastly more general than pure electromagnetic or magnetoelectric circuits, they are in a sense (modulo the addition of tetrahedral connectors to the terminal edges) all equivalent.

Figure 1 :

 1 Figure 1: An example of an electromagnetic circuit (EMcircuit) with six zero-dielectric diamagnetic thin triangular plates (having ε = 0 and µ = 0). It has small diameter zeromagnetic dielectric cylinders with ε = 0 and µ = 0 along a selection of edges including possibly the terminal edges.Here there are three terminal edges marked by thicker solid black lines. One internal edge, marked by the thin line, has a cylinder with ε = µ = 0 attached to it.

Figure 2 :

 2 Figure 2: The tetrahedral connector, which allows us to connect a terminal edge in an electromagnetic circuit (Eterminal) with a terminal edge in a magnetoelectric circuit (H-terminal).

Acknowledgements

G.W.M. is grateful for support from the National Science Foundation through grant DMS-0707978.