Thai V Hoang
email: vanthai.hoang@loria.fr

Salvatore Tabbone
email: tabbone@loria.fr

Invariant Pattern Recognition Using The RFM Descriptor

Keywords: Invariant pattern representation, Radon transform, Fourier-Mellin transform, feature extraction, noise robustness

A pattern descriptor invariant to rotation, scaling, translation (RST), and robust to additive noise is proposed by using the Radon, Fourier, and Mellin transforms. The Radon transform converts the RST transformations applied on a pattern image into transformations in the radial and angular coordinates of the pattern's Radon image. These beneficial properties of the Radon transform make it an useful intermediate representation for the extraction of invariant features from pattern images for the purpose of indexing/matching. In this paper, invariance to RST is obtained by applying the 1D Fourier-Mellin and discrete Fourier transforms on the radial and angular coordinates of the pattern's Radon image respectively. The implementation of the proposed descriptor is reasonably fast and correct, based mainly on the fusion of the Radon and Fourier transforms and on a modification of the Mellin transform. Theoretical arguments validate the robustness of the proposed descriptor to additive noise and empirical evidence on both occlusion/deformation and noisy datasets shows its effectiveness.

Introduction

The problem of recognition of image patterns that undergo geometric transformations like rotation, scaling, and translation is an important topic in pattern recognition and is the goal of many research works. Many approaches have been proposed for this problem and they can be classified into three main lines: brute force, image normalization, and invariant features. Brute force approaches are the most trivial ones, using "complete" training datasets; for each pattern category, the training dataset contains all its RST transformed versions. This line of approaches has inherent limitations in both storage requirement and time complexity that make them practically inapplicable. Image normalization is a solution for the reduction of the size of training datasets. The burden of the encoded RST transformation parameters in input pattern images is alleviated by normalizing them regarding their orientation, size, and position. However, despite its efficiency in the recognition stage, normalization of image patterns involves difficult inverse problems that are often ill-conditioned or ill-posed, leading to unreliable normalization results. Approaches using invariant features are based on the idea of describing image patterns by a set of measurable quantities that are insensitive to RST transformations while providing enough discriminatory power for their recognition. Mathematically speaking, if f (x, y) is an image and let g(x, y) be another image described as g = O(f ), where O is an RST transformation operator, then the invariant I is a functional satisfying I(f ) = I (O(f )).

Many pattern descriptors have been proposed in literature for the extraction of pattern's invariant features [START_REF] Wood | Invariant pattern recognition: a review[END_REF]2] using techniques that allow operator O to be rotation, scaling, translation, or their combinations. Translation and scaling invariance could be obtained by using the Fourier [3] and Mellin [START_REF] Bertrand | The Mellin transform[END_REF] transforms respectively; rotation invariance by computing the harmonic expansion [START_REF] Hsu | Rotation-invariant digital pattern recognition using circular harmonic expansion[END_REF] or performing the discrete Fourier transform on the circular coordinate of the pattern image in polar space [6], etc. However, the task of combining several techniques to make operator O full RST transformations while guaranteeing the discriminatory power of the invariant features is challenging and has attracted attention of many researchers. Most of the existing methods do not allow operator O to be full RST transformations, they usually require normalizations for the unavailability of any of RST transformations in operator O. For example, methods based on the theory of moments [START_REF] Teague | Image analysis via the general theory of moments[END_REF]8] usually normalize input pattern images regarding their centroid position and size: the pattern's centroid is required to coincide with the origin of the coordinate system and the longest distance from this centroid to a pattern point is set to a fixed value. These normalizations usually introduce errors, are sensitive to noise, and thus induce inaccuracy in the later recognition/matching process.

Radon-based methods are different from the others in the sense that Radon transform is used as an intermediate representation upon which invariant features are extracted from. By applying the Radon transform on a RST transformed pattern image, the transformation parameters are encoded in the radial (for translation and scaling) and angular (for rotation) coordinates of the obtained Radon image [START_REF] Hjouj | Identification of reflected, scaled, translated, and rotated objects from their Radon projections[END_REF]. Current techniques thus usually exploit this encoded information to define invariant features. A pioneer and notable work in this direction is the R-transform proposed in [10] to compute the R-signature. This approach uses an integral function and the discrete Fourier transform for the radial and angular coordinates of the Radon image respectively to get a 1D signature of the pattern image that is invariant to RST transformations. The strength of this approach is simplicity, however, even if an extension to a 2D signature has been proposed, the obtained signature still has low discriminatory power as there is a loss of information in the compression process from the Radon image to the 1D signature. Recently, there was an effort to apply the 2D Fourier-Mellin transform on the Radon image [START_REF] Wang | Scaling and rotation invariant analysis approach to object recognition based on Radon and Fourier-Mellin transforms[END_REF]. Similarly, Mellin transform and harmonic expansion are applied on the radial and angular coordinates of the Radon image respectively to get a pattern descriptor that is invariant to scaling and rotation. Similar efforts with different strategies to exploit the Radon image have been reported in [12][13][START_REF] Wang | Rotation invariant analysis and orientation estimation method for texture classification based on Radon transform and correlation analysis[END_REF]. These methods, however, do not allow operator O to be full RST transformations as in the R-signature, operator O could only be a combination of rotation and scaling. Moving the origin of the coordinate system to the pattern's centroid is a common solution to have translation invariance, however, this normalization step may introduce errors when the image is noisy or the pattern is slightly distorted.

Another direction in using the Radon transform for pattern description is to extract pattern features directly from the Radon image, similar to the way the Hough transform [15] is used. For example, pattern primitives in edge form are detected from the Radon image and represented analytically in [16]. Moreover, their spatial relations can be made explicit [17] and this leads to a taxonomy of pattern for its characterization [18]. This approach, however, is quite limited as it requires that the edge primitives should have analytical form. Recently, a set of spectral and structural features proposed in [START_REF] Chen | Invariant description and retrieval of planar shapes using Radon composite features[END_REF] is extracted from a slightly modified version of the Radon transform for pattern description. This set of features is not invariant to rotation and, consequently, in the matching step, these features need to be rotated to all possible angles corresponding to potential pattern's orientations in order to compute patterns' similarity. Long matching time may prevent the application of this approach in real systems. A generalization of the Radon transform, called the trace transform, has also been proposed and used for image description [20, 21] but its application is restricted due to high computational complexity.

This paper presents a novel RST invariant pattern descriptor which has briefly presented in [22], called the RFM descriptor, defined based on the Radon, Fourier, and Mellin transforms that is invariant not only to rotation, and scaling but also to translation by definition. The proposed descriptor is a global one, its computation is essentially different from that of another type of descriptors, local descriptors [23,[START_REF] Liu | Image retrieval based on multi-texton histogram[END_REF], which can work only on textured patterns and fail to describe smooth or non-textured ones. The main idea here is to apply the 1D Fourier-Mellin transform, which is invariant to translation and scaling, on the radial coordinate of the pattern's Radon image. The result of this transform is a representation of pattern that is shifted horizontally according to the pattern's rotating angle. A further application of the discrete Fourier transform on this representation guarantees a pattern descriptor totally invariant to RST transformations. The proposed descriptor is fast in implementation and is more robust to additive noise than existing descriptors. Experimental results on both occlusion/deformation and noisy databases show the usefulness of the proposed descriptor.

The main contribution of this paper is to adopt and use the three basic transformations (Radon, Fourier, and Mellin) to transform an input pattern image into a descriptor that is totally invariant to RST transformations. Moreover, we apply the 1D Fourier-Mellin transform on the radial coordinate of the pattern's Radon image. We use an implementation of the Mellin transform that is fast, accurate, and can preserve the discriminatory information of the input signal. Additionally, we analyzed and provided theoretical evidence for the robustness of the proposed descriptor to additive noise. And finally, we illustrate the usefulness of the proposed descriptor in the presence of occlusion/deformation and additive noise using several datasets.

The remainder of this paper is organized as follows. Section 2 gives some background on the Radon, Fourier, and Mellin transforms and their beneficial properties. Section 3 presents the combined 1D Fourier-Mellin transform and then defines the RFM descriptor. A reasonable fast and accurate implementation of the proposed descriptor is provided in Section 4 and then an analysis on the robustness of the proposed descriptor to additive noise is carried out in Section 5. Experimental results are given in Section 6 and finally conclusions are drawn in Section 7.

Basic material

This section reviews the three transformations used in the proposed descriptor, that are the Radon, Fourier, and Mellin transforms. It includes their definition and their derived beneficial properties.

The Radon transform

Let f (x, y) ∈ R 2 be a 2D function, L(θ, ρ) be a straight line in R 2 represented by

L = {(x, y) ∈ R 2 : x cos θ + y sin θ = ρ},
where θ is the angle L makes with the y axis and ρ is the distance from the origin to L. The Radon transform [25] of f , denoted by R f , is a function defined on the space of lines L(θ, ρ) by the line integral along each line:

R f (L) = R f (θ, ρ) = L f (x, y) dxdy = ∞ -∞ ∞ -∞ f (x, y) δ(ρ -x cos θ -y sin θ) dxdy. (1)
In the field of shape analysis and recognition, the function f (x, y) is constrained to the following particular definition: where D is the domain of the binary shape represented by f (x, y). In the illustration of the Radon transform given in Fig. 1, the shaded region represents the region D. The value of the line integral in Eq. ( 1) equals the length of the intersection between the line L and the shaded region.

f (x, y) = 1 if x ∈ D 0 otherwise,
The Radon transform has some properties that are beneficial for pattern recognition problems as outlined below: P 1 linearity: The Radon transform is linear.

R (f +g) (θ, ρ) = R f (θ, ρ) + R g (θ, ρ).
P 2 periodicity: The Radon transform of f (x, y) is periodic in the variable θ with period 2π.

R f (θ, ρ) = R f (θ + 2kπ, ρ), ∀k ∈ Z. P 3 semi-symmetry: The Radon transform of f (x, y) is semi-symmetric. R f (θ, ρ) = R f (θ ± π, -ρ).
P 4 translation: A translation of f (x, y) by a vector u = (x 0 , y 0 ) results in a shift in the variable ρ of R f (θ, ρ) by a distance d = x 0 cos θ + y 0 sin θ, equal to the length of the projection of u onto the line x cos θ + y sin θ = ρ.

R f (θ, ρ) → R f (θ, ρ -x 0 cos θ -y 0 sin θ).
P 5 rotation: A rotation of f (x, y) by an angle θ 0 implies a circular shift in the variable

θ of R f (θ, ρ) by a distance θ 0 . R f (θ, ρ) → R f (θ + θ 0 , ρ).
P 6 scaling: A scaling of f (x, y) by a factor α results in a scaling in the variable ρ and the amplitude of R f (θ, ρ) by the factors α and 1 α respectively.

R f (θ, ρ) → 1 α R f (θ, αρ).

The Fourier transform

Let f (x) ∈ R 1 be a 1D function. The Fourier transform [3] of f , denoted by F f , is a function defined for every real number ξ by:

F f (ξ) = ∞ -∞ f (x)e -i2πξx dx. (2)
It is well known that the Fourier transform possesses a shift or translation invariant property. Consider a function g(x) = f (xx 0 ), a version of f (x) shifted by x 0 , then:

F g (ξ) = ∞ -∞ g(x)e -i2πξx dx = e -i2πξx 0 F f (ξ).
(3)

Taking the magnitude of the two sides of Eq. (3) results in:

|F g (ξ)| = |F f (ξ)|.
or, in other words, the magnitude of the Fourier transform of a 1D function is invariant to translation. Discrete version of Eq. (2) defined for a sequence of numbers {f (n) : n = 0, . . . , N -1} has the following definition:

DF f (k) = N -1 n=0 f (n) e -2πi N kn , k = 0, . . . , N -1.
Similarly, the discrete Fourier transform possesses a circular shift invariant property: a circular shift of the input {f (n)} corresponds to multiplying the output DF f (k) by a linear phase. Let {g(n)} be a sequence obtained by circular shifting {f (n)} by a distance m, then:

DF g (k) = e -2πi N km DF f (k). (4)
Taking the magnitude of the two sides of Eq. ( 4) results in:

|DF g (k)| = |DF f (k)|,
or the magnitude of the discrete Fourier transform of a 1D function is invariant to circular shifting.

The Mellin transform

Let f (x) ∈ R 1 be a 1D function. The Mellin transform [START_REF] Bertrand | The Mellin transform[END_REF] of f , denoted by M f , is a function defined by:

M f (s) = ∞ 0 f (x)x s-1 dx, (5)
where s = σ + iτ . The real part of s, σ, is a constant chosen such that the integral in Eq. ( 5) converges.

The imaginary part of s, τ , is the transform variable. Consider a function g(x) = f (αx), a scaling of f by a factor α (α > 0), then:

M g (s) = ∞ 0 g(x)x s-1 dx = α -s M f (s). ( 6
)
Taking the magnitude of the two sides of Eq. ( 6) results in:

|M g (s)| = α -σ |M f (s)|.
Thus, except for a constant multiplicative factor α -σ , the Mellin transform is scale invariant. However, this scaling factor can be easily eliminated by normalization or it can be used to find the relative scale between two functions.

3 An invariant pattern descriptor based on the Radon, Fourier, and

Mellin transforms

This section presents a combination of the Fourier and Mellin transforms for 1D signals and its application on the radial coordinate of the pattern's Radon image to have a pattern descriptor invariant to RST transformations. A measure of similarity for the proposed pattern descriptor is also defined, allowing the comparison/matching between any pair of images.

The combined Fourier-Mellin transform for 1D signals

Combinations of the Fourier and Mellin transforms have been proposed in literature to have signal representations that do not vary with rotation/scaling or translation/scaling. For 2D signals, they are first converted from Cartesian coordinates into polar coordinates then Fourier and Mellin transforms are performed independently on the circular and radial coordinates of the converted signals respectively [START_REF] Casasent | New optical transforms for pattern recognition[END_REF][START_REF] Sheng | Circular Fourier-radial Mellin descriptors for pattern recognition[END_REF]. In this way, the obtained representation is invariant to rotation and scaling. For 1D signals, Fourier and Mellin transforms are performed in sequence directly on the signals [START_REF] Altes | The Fourier-Mellin transform and mammalian hearing[END_REF][START_REF] Zwicke | A new implementation of the Mellin transform and its application to radar classification of ships[END_REF] to have signal representations invariant to translation and scaling. Consider the Fourier transform of a function g(x) = f (αxx 0 ), a scaled and translated version of f (x) by parameters α and x 0 :

F g (ξ) = ∞ -∞ f (αx -x 0 )e -i2πξx dx. (7)
Denoting y = αxx 0 , Eq. ( 7) becomes

F g (ξ) = 1 α e -i2π ξ α x 0 ∞ -∞ f (y)e -i2π ξ α y dy = 1 α e -i2π ξ α x 0 F f ξ α . (8)
Taking the magnitude of the two sides of Eq. ( 8) results in:

|F g (ξ)| = 1 α F f ξ α . (9)
The translation parameter x 0 has disappeared in Eq. ( 9); the remaining scaling parameter α could be eliminated by applying the Mellin transform on both sides of Eq. ( 9):

M |Fg| (s) = ∞ 0 1 α F f ξ α ξ s-1 dξ = α s-1 M |Ff | (s),
or:

M |Fg| (s) = α σ-1 M |Ff | (s) . (10)
Therefore, by defining:

MF f (s) = M |Fg| (s) = ∞ 0 ∞ -∞ f (x)e -i2πξx dx ξ s-1 dξ , (11)
as the combined 1D Fourier-Mellin transform of a function f (x) and from Eq. (10), MF f (s) is invariant to translation and scaling, except for a constant multiplicative factor.

The proposed RFM descriptor

Attractive properties of the Radon and 1D Fourier-Mellin transforms lead to the proposal of a novel region-based pattern descriptor, called the RFM descriptor. The proposed descriptor of an image I is computed by RFM(I) =

DF MF R I (k,s)
DF MF R I (0,s) , namely:

Step 1: The Radon transform performed on the image I, R I (θ, ρ).

Step 2: The 1D Fourier-Mellin transform performed on the radial coordinate of the obtained Radon image, MF R I (θ, s).

Step 3: The magnitude of discrete Fourier transform performed on the angular coordinate of the obtained, discretized Fourier-Mellin image normalized by the DC component,

DF MF R I (k,s) DF MF R I (0,s) .
Invariant properties of the proposed descriptor described above are proved as follows.

Properties. The proposed RFM descriptor is invariant to translation, rotation, and scaling.

Proof. Let J be the image obtained after scaling, rotating, and translating a image I using transformation parameters α, θ 0 , and u = (x 0 , y 0 ). Properties P 4-6 of the Radon transform imply:

R J (θ, ρ) = 1 α R I (θ + θ 0 , αρ -d), (12)
where d = x 0 cos(θ + θ 0 ) + y 0 sin(θ + θ 0 ) is the shift distance depending on θ. Eq. ( 12) indicates that, except for a constant multiplicative factor 1 α , each column constant-θ slice of the Radon image of J, R J (θ, •), can be obtained by scaling and translating the constant-θ + θ 0 slice of the Radon image of I, R I (θ + θ 0 , •) by a factor α and a distance d.

Applying the 1D Fourier-Mellin transform on R J (θ, •) and R I (θ + θ 0 , •) and using its invariant property and Eq. (10), Eq. ( 12) becomes:

MF R J (θ,•) (θ, s) = α σ-2 MF R I (θ+θ 0 ,•) (θ + θ 0 , s).
By varying the value of θ and s, the two images MF R J (θ, s) and MF R I (θ, s) are obtained. Moreover, except for a constant multiplicative factor α σ-2 , MF R J (θ, s) can be directly obtained by shifting MF R I (θ, s) along the θ axis by a quantity -θ 0 . Applying the discrete Fourier transform on the angular coordinate of the Fourier-Mellin images MF R J (θ, s) and MF R I (θ, s) then ignoring the phase information in the coefficients, we have

DF MF R J (k, s) = α σ-2 DF MF R I (ξ, s) . ( 13
)
This demonstrates that, except for a constant multiplicative factor α σ-2 , the proposed descriptor computed on a scaled, rotated, and translated version J of a image I is exactly the same as the descriptor computed on the original image I. The factor α σ-2 , however, can be easily eliminated by a normalization step using the DC component as

DF MF R I (k,s)
DF MF R I (0,s) or it can be used to determine the relative scale between any two pattern images of the same category.

Calculating the proposed RFM descriptor, as described above, does not require any normalization regarding the size, position, or orientation of the patterns, it only requires a normalization in the intensity of the computed descriptor. As a consequence, the proposed descriptor is totally invariant to RST transformations.

Similarity measure

For any two images I and J represented by RFM(I) and RFM(J) respectively, their measure of similarity is defined as the ℓ 2 -norm distance between their descriptors as

sim(I, J) = RFM(I) -RFM(J) 2 . ( 14
)
The computation of sim(I, J) is simple and fast, permitting the RFM descriptor to be used in pattern matching problems with large-size datasets. More sophisticated distances like the weighted Euclidean distance [START_REF] Khotanzad | Invariant image recognition by Zernike moments[END_REF] could be used to reduce the dominance of some of the coefficients in the RFM descriptor. However, as small value coefficients usually correspond to high frequency components, meaning that they are more sensitive to additive noise and sampling/quantization effect, balancing the coefficient contributions thus reduces the performance of the descriptor in noisy environment. Moreover, due to the orthogonality in the basis of the discrete Fourier transform, there is no correlation among coefficients of the RFM descriptor and thus the Mahalanobis distance [START_REF] Mahalanobis | On the generalised distance in statistics[END_REF], if employed, reduces to the weighted Euclidean distance.

Implementation of the RFM descriptor

From the definition of the RFM descriptor in Subsection 3.2, its calculation could be separated into three steps: Radon, 1D Fourier-Mellin, and discrete Fourier transforms. However, due to the interpretation of the Radon transform by means of the Fourier transform, a computational reduction is possible by fusing these two transforms. Moreover, in order to have fast implementation and high discriminatory power, a modified version of the Mellin transform has been used.

Possible fusion of Radon and Fourier transforms

The Radon transform could be computed based on recursively defined digital straight lines [START_REF] Götz | A fast digital Radon transform -an efficient means for evaluating the Hough transform[END_REF][START_REF] Brady | A fast discrete approximation algorithm for the Radon transform[END_REF] requiring O(N log N ) operations for an images of N = n × n pixels. It could also be computed through 2D Fourier transform with the same complexity by means of the projection slice theorem [START_REF] Averbuch | Fast Slant Stack: A Notion of Radon Transform for Data on A Cartesian Grid Which Is Rapidly Computable, Algebraically Exact, Geometrically Faithful, and Invertible[END_REF], which states that the 1D constant-θ slice of the Radon image R f (•, θ) and the 1D radial slice of the 2D Fourier transform F f (ξ cos θ, ξ sin θ) make a 1D Fourier transform pair:

F R f (•,θ) (ξ) = ∞ -∞ ∞ -∞ ∞ -∞ f (x, y) δ(ρ -x cos θ -y sin θ) dxdy e -iρξ dρ = ∞ -∞ ∞ -∞ f (x, y) e -i(x cos θ+y sin θ)ξ dxdy = F f (ξ cos θ, ξ sin θ).
Thus, if one chooses to implement the Radon transform through 2D Fourier transform, due to the successive application of inverse and forward Fourier transforms, performing the 1D Fourier-Mellin transform on each constant-θ slice of the Radon image is equivalent to directly performing the Mellin transform on the 1D radial slice of the 2D Fourier transform. This equivalence results in a computational reduction with no change in complexity. However, for notation convenience and for the clarity of presentation, the RFM descriptor in this paper sticks with the definition in Subsection 3.2. This choice of presentation has no impact on the remaining part. It is only when the RFM descriptor is chosen to be implemented through 2D Fourier transform that the readers should pay attention to this fusion for their own computational benefit.

The modified direct Mellin transform

The Mellin transform, as defined in Eq. ( 5), has a very attractive property of scaling invariance and it can be implemented optically by using an optical scale invariant correlator [START_REF] Casasent | New optical transforms for pattern recognition[END_REF]. However, there are reported problems with its digital implementation in today's digital systems [START_REF] Derrode | Robust and efficient Fourier-Mellin transform approximations for gray-level image reconstruction and complete invariant description[END_REF]. Traditionally, the Mellin transform is implemented by changing the variable x = e y :

M f (s) = ∞ 0 f (x)x s-1 dx = ∞ 0 f (x)x σ-1 e iτ ln x dx = ∞ -∞
[f (e y )e σy ] e iτ y dy. This is, by definition, the Fourier transform of the distorted function f (e y ) weighted by e σy . For sampled data, the discrete (fast) Mellin transform is implemented through FFT by re-sampling the data exponentially. Exponential sampling means interpolating the data to make them uniformly sampled in the y domain [START_REF] Sena | A fast Mellin and scale transform[END_REF]. This process introduces errors into the transform and accentuates the low frequency components. Additionally, if f (x) is sampled at the Nyquist rate and N is the number of data samples in the x domain then the number of the required data samples in the y domain will be M = N ln N [START_REF] Robbins | Inverse filtering for linear shift-variant imaging systems[END_REF]. Likewise, when f (x) is nonzero at x = 0, f (e y ) will be nonzero at y = -∞ and the implementation is not realizable. In this case, the Mellin transform can be approximated by using a correction term defined based on the value of f (x = 0) [START_REF] Robbins | Inverse filtering for linear shift-variant imaging systems[END_REF].

Another problem is on the application of the combined Fourier-Mellin transform in Subsection 3.1 for feature extraction despite it is invariant to scaling and translation. The problem is the obscurity of the discriminatory information contained in the input function [START_REF] Johnson | The Shift and Scale Invariant Fourier-Mellin Transform for Radar Applications[END_REF]. Possible reasons are the discard of the phase information from the output of the Fourier and Mellin transforms and the accentuation of low frequency components. Moreover, as FFT is applied on the radial coordinate of the pattern's Radon image, the DC component is always nonzero and it, in turn, is the value of the Mellin transform's input function f (x) at x = 0.

To avoid these problems, an alternative implementation of the Mellin transform proposed in [START_REF] Zwicke | A new implementation of the Mellin transform and its application to radar classification of ships[END_REF], which is called the direct Mellin transform, is adopted for this work. Assuming f (x) is in the form of sampled data with sampling period T , expanding Eq. ( 5) gives:

M f (s) = T 0 f (x)x s-1 dx + 2T T f (x)x s-1 dx + • • • + N T (N -1)T f (x)x s-1 dx. ( 15
)
The value of f (x) is assumed to be piecewise constant in any interval T , then:

sM f (s) = f (0)x s | 1 0 + f (T )x s | 2 1 + • • • + f ((N -1)T )x s | N N -1 . (16)
Denoting

f (iT ) = f i+1 , then ∆ k = f k -f k-1
and, without loss of generality, assuming T = 1 and f N = 0, Eq. ( 16) becomes:

sM f (s) = f 1 x s | 1 0 + f 2 x s | 2 1 + • • • + f N x s | N N -1 = N -1 k=1 k s (f k -f k+1 ) = N -1 k=1 k s ∆ k . ( 17
)
Substituting s = σ + iτ into Eq. ( 17):

(σ + iτ )M f (σ + iτ ) = N -1 k=1 k σ+iτ ∆ k . (18)
Because k σ+iτ = e (σ+iτ ) ln k is bounded for any fixed constant value of σ, the right hand side of Eq. ( 18) is bounded, which means |M f (s)| converges to zero when |s| increases. This indicates the low-pass filtering of the Mellin transform.

For a set of s i = σ + iτ i with i = 1, . . . , m, τ i is the arbitrary spectral component, and denoting k s i = φ i,k , Eq. ( 17) can be rewritten in matrix form as

      s 1 M f (s 1 ) s 2 M f (s 2 ) . . . s m M f (s m )       =       φ 1,1 φ 1,2 • • • φ 1,N -1 φ 2,1 φ 2,2 • • • φ 2,N -1 . . . . . . . . . . . . φ m,1 φ m,2 • • • φ m,N -1             ∆ 1 ∆ 2 . . . ∆ N -1       . ( 19
)
The direct Mellin transform, as defined in Eq. ( 15), is an exact implementation of the Mellin transform for sampled data and has the following properties:

• It does not require exponential re-sampling and there is no requirement for a correction term.

• Only the differences in the value of adjacent data points are used for computing the transform.

• The coefficients φ i,k in Eq. ( 19) can be computed off-line and stored. For each specific value of s i , the number of stored coefficients for direct Mellin transform is N whereas the number of stored coefficients for fast Mellin transform is N ln N .

• Computing s i M f (s i ) consists of only an inner product of two vectors, one of which has been pre-computed and stored and the other vector could be obtained from the input data by simple subtraction operations. Thus, Eq. ( 19) is very fast in digital implementation.

In order to eliminate the low-pass filtering characteristic of the direct Mellin transform, a modification is carried out by removing the s = (σ + iτ ) factor in Eq. ( 18). The resulting transform is called the modified direct Mellin transform, denoted by MM, as follows:

MM f (s) = sM f (s) = N -1 k=1 k s ∆ k .
In the time domain, the modified direct Mellin transform is defined as

MM f (s) = s ∞ 0 f (x)x s-1 dx.
It is easy to verify that the modified direct Mellin transform maintains the scaling invariance property and the combined Fourier-modified direct Mellin transform, defined as

MF f (s) = MM |Ff | (s) = s ∞ 0 |F f | x s-1 dx. ( 20
)
is invariant to both translation and scaling. Therefore, the 1D Fourier-Mellin transform used in the definition of the RFM descriptor in Subsection 3.2 is henceforth defined as in Eq. ( 20), instead of Eq. ( 11), and implemented through Eq. ( 19).

To qualitatively illustrate the invariant properties of MF R I (θ, s), Fig. 2 provides some example pattern images and their corresponding Radon and Fourier-Mellin images. The first row contains two original pattern images (Fig. 2(a)-2(b)) and the RST transformed versions (Fig. 2(c)-2(e)) of the pattern image in Fig. 2(b). The second row shows the Radon transform of these pattern images and the third row shows the images obtained after performing the 1D Fourier-Mellin transform on the Radon images in the second row using 150 values of τ ranging from 2.0 to 16.9 with increment of 0.1. It is noteworthy here that, due to the periodicity and semi-symmetry properties of the Radon transform (P 2-3), the effective range of θ used in the computation is 0-π (rad) or 0-180 (degree). A quantitative evaluation of the invariant properties of the Fourier-Mellin transform is given in Fig. 3 using the normalized cross correlation between all possible pairs of Fourier-Mellin images MF R I i (θ, s) and MF R I j (θ, s) from the third row of Fig. 2. Normalized cross correlation is selected for the sake of overcoming the constant multiplicative factor α σ-2 in Eq. (13). To overcome the remaining rotating parameter, the correlation is calculated for all possible rotating angle, meaning that one of the two Fourier-Mellin images can be circular shifted along its θ direction by 180 possible values from 0 to 179 with increment of 1. Denoting ϕ as the shifting distance, the correlation between image I i and I j at ϕ is defined as where corr2(A, B) is the normalized cross correlation function between two 2D input data A and B of size m × n calculated using the following formula:

f) RI 1 (θ, ρ) (g) RI 2 (θ, ρ) (h) RI 3 (θ, ρ) (i) RI 4 (θ, ρ) (j) RI 5 (θ, ρ) (k) MFR I 1 (θ, s) (l) MFR I 2 (θ, s) (m) MFR I 3 (θ, s) (n) MFR I 4 (θ, s) (o) MFR I 5 (θ, s) (
C ij (ϕ) = corr2 MF R I i (θ, s), MF R I j (θ + ϕ, s) ,
m i=1 n j=1 (A ij -Ā)(B ij -B) m i=1 n j=1 (A ij -Ā) 2 m i=1 n j=1 (B ij -B) 2
, where Ā and B are the mean values of A and B respectively. The 10 curves C ij (ϕ) corresponding to the 10 possible pairs of five Fourier-Mellin images have two different patterns, C ij (ϕ) is peaky only when the two pattern images I i and I j are similar (i, j = 2, 3, 4, 5). The maximum value of C ij (ϕ) are 0.6200, 0.6236, 0.6239, 0.6239, 0.9962, 0.9943, 0.9943, 0.9970, 0.9970, 1.0000 respectively from left to right, top to bottom. The non-peaky and peaky maxima exhibit the discriminatory power of the proposed descriptor and the maximum of nearly 1 means that the Fourier-Mellin image is invariant to translation and scaling. The value of ϕ * corresponding to the peak in C ij (ϕ) denotes the difference in orientation (in degree) between the patterns in I i and I j .

Robustness to noise

Pattern descriptor defined based on the Radon transform has the advantage of being robust to additive noise [START_REF] Jafari-Khouzani | Rotation-invariant multiresolution texture analysis using Radon and wavelet transforms[END_REF]. Suppose the image f (x, y) is corrupted by additive white noise η(x, y) with zero mean and variance σ 2 to be f (x, y) = f (x, y) + η(x, y), the Radon transform of the noisy image f (x, y) is obtained by applying the linearity property of the Radon transform (P 1):

R f (θ, ρ) = R f (θ, ρ) + R η (θ, ρ).
Recall that the Radon transform, as illustrated in Fig. 1, is line integrals of f (x, y). In the continuous domain, the Radon transform of additive white noise R η (θ, ρ) is proportional to the mean value of the noise, which means R η (θ, ρ) = 0, or The ideal additive white noise, therefore, has no effect on the Radon transform of the image in the continuous domain. However, in practice, the image represented and processed in digital computer is not a continuous function but its sampled and quantized version and Eq. ( 21) does not hold. Suppose f (x, y) is in the form of a sampled 2D signal of size m × n (0 ≤ x ≤ m, 0 ≤ y ≤ n) whose values are random variables with mean µ and variance σ 2 . The computation of the Radon transform of f (x, y) is assumed to follow the definition, that is the values of f (x, y) along the lines L(θ, ρ) are summed up, as shown in Fig. 4. The contribution of each pixel i to the sum is equal to the length of its intersection with L(θ, ρ). The sum of all the pixels' contribution corresponding to L(θ, ρ) is therefore equal to the length of the segment AB, of f y) with L(θ, The sum of f (x, y) along all the lines L(θ, ρ) having the same direction θ can also be interpreted as the projection of f (x, y) onto an axis ρ perpendicular to L(θ, •). This projection is actually the line R f (θ, •) in the Radon image of f (x, y).

R f (θ, ρ) = R f (θ, ρ). (21)
To study this projection, let θ = const and denoting n ρ = AB, then the sum of the pixel values p ρ = R f (θ, ρ) for each line L(θ, ρ) has mean n ρ µ and variance n ρ σ 2 . The average of the expected value of p 2 ρ is:

E p = 1 2N ρ Nρ -Nρ+1 E{p 2 ρ } dρ = 1 2N ρ Nρ -Nρ+1 n ρ σ 2 dρ + 1 2N ρ Nρ -Nρ+1 n 2 ρ µ 2 dρ. (22)
In the above equation, the integral Nρ -Nρ+1 n ρ dρ represents all the area of f (x, y) and is equal to the number of pixels in f (x, y), which is mn. Then, by denoting A(θ) = Nρ -Nρ+1 n 2 ρ dρ, Eq. ( 22) is simplified as

E p = mnσ 2 2N ρ + A(θ)µ 2 2N ρ . (23)
For the image f (x, y) corrupted by white noise η(x, y), assuming f (x, y) has mean µ s , variance σ 2 s and η(x, y) has mean µ = 0, variance σ 2 n , then Figure 5: The values of the multiplicative factor A(θ) mn in Eq. ( 24) at different values of θ computed on an image f (x, y) of different sizes. It gets its maximum at θ = 90 (degree) corresponding to the direction of the longer side of f (x, y) and its minimum near θ = 0 (degree). The values of A(θ) mn relates directly to the noise-suppressing ability of the Radon transform. ratio (SNR) of f (x, y) and its projection R f (θ, •) is:

E s = mnσ 2 s 2Nρ + A(θ)µ 2
SNR image = σ 2 s + µ 2 s σ 2 n , SNR proj(θ) = E s E n = mnσ 2 s + A(θ)µ 2 s mnσ 2 n = σ 2 s + A(θ) mn µ 2 s σ 2 n , or: SNR proj(θ) = SNR image + A(θ) mn -1 µ 2 s σ 2 n . ( 24
)
The SNR is increased by a quantity A(θ) mn -1 µ 2 s σ 2 n after projecting f (x, y) along the direction θ. As the value of A(θ) depends both on θ, m and n, the multiplicative factor A(θ) mn -1 is not constant. Moreover, the value of A(θ) mn is "big" because A(θ) = Nρ -Nρ+1 n 2 ρ dρ is one order higher than mn = Nρ -Nρ+1 n ρ dρ. Eq. ( 24) can be rewritten as

SNR proj(θ) ≃ SNR image + A(θ) mn × µ 2 s σ 2 n . As A(θ) mn × µ 2 s σ 2 n
is positive and has high value, corresponding to a high increase in the value of SNR after projection, this means that the Radon transform is very robust to additive white noise. Fig. 5 depicts the values of A(θ) mn for a range of θ from 0 to 180 (degree) using input images of different sizes. We can notice that the value of A(θ) mn depends on the actual size of f (x, y) and it gets its maximum in the direction of the longer side and its minimum near the direction of the shorter side of f (x, y), therefore:

min θ A(θ) mn ≃ min(m, n), max θ A(θ) mn = max(m, n).
In the field of shape analysis and recognition, f (x, y) is constrained to have binary values of 0 or 1 and the additive noise to f (x, y) is in the form of "salt & pepper" noise, instead of white noise. To model this type of noise, let D and d be the percentage of pixels in f (x, y) occupied by the shape region and flipped by the noise respectively, then:

µ s = D, σ 2 s = D -D 2 , µ n = d(1 -2D), σ 2 n = d -d 2 (1 -2D) 2 .
Using Eq. ( 23), the signal-to-noise ratio (SNR) of f (x, y) and its projection R f (θ, •) is:

SNR image = σ 2 s + µ 2 s σ 2 n + µ 2 n = D d , SNR proj(θ) = mnσ 2 s + A(θ)µ 2 s mnσ 2 n + A(θ)µ 2 n = D -D 2 + A(θ) mn D 2 d -d 2 (1 -2D) 2 + A(θ) mn d 2 (1 -2D) 2 = D d × 1 + D A(θ) mn -1 1 + d(1 -2D) 2 A(θ) mn -1 , then: SNR proj(θ) SNR image = 1 + D A(θ) mn -1 1 + d(1 -2D) 2 A(θ) mn -1 .
It is clear that

SNR proj(θ)
SNR image depends on the size of the input noisy image f (x, y), the projection direction θ, the percentage of shape region D, and the level of noise d. For explicit minimum value of SNR image , at a specific value of A(θ) mn , is reached at min D = 0.3. The depiction of the value of

SNR proj(θ)
SNR image for the case A(θ) mn = 100 over the domain D ∈ [0.3, 0.7] and d ∈ [0, 0.2] is given in Fig. 6(a).

Fixing D = 0.3 and d = 0.2, the dependance of

SNR proj(θ)
SNR image on A(θ) mn is further given in Fig. 6(b). It is evident that

SNR proj(θ)
SNR image > 1, meaning the projection in the Radon transform has the property of suppressing "salt & pepper" noise. Additionally,

SNR proj(θ)
SNR image increases with the increase in A(θ) mn from 4.1667 at A(θ) mn = 20 (a very small image) to its limit at: lim

A(θ) mn →∞ SNR proj(θ) SNR image = D d(1 -2D) 2 = 9.375,
implying a better suppression of noise in the projection of bigger-size images.

Experimental results

In order to demonstrate the effectiveness of the proposed RFM descriptor, three experiments on grayscale and binary image datasets have been carried out. The robustness of the proposed descriptor to additive white noise is first demonstrated by using two sets of datasets generated from images of 26 Latin characters and from the COIL-20 dataset [40] by adding white noise of different levels to them. Secondly, the proposed descriptor is computed on a set of datasets generated from the UMD Logo dataset [START_REF] Doermann | Applying algebraic and differential invariants for logo recognition[END_REF] by adding "salt & pepper" noise of different levels to its images. The aim of these experiments is to demonstrate the robustness to "salt & pepper" noise of the proposed descriptor. Finally, the proposed descriptor is computed and compared with other descriptors on the reference Shapes216 dataset [START_REF] Sebastian | Recognition of shapes by editing their shock graphs[END_REF], which is used to evaluate its robustness to occlusion and deformation. Thus, the first experiment deals with grayscale patterns and the last two ones with binary patterns. The proposed RFM descriptor is compared with angular radial transform (ART) [START_REF] Bober | Shape descriptors[END_REF], generic Fourier descriptor (GFD) [6], Zernike moments [START_REF] Khotanzad | Invariant image recognition by Zernike moments[END_REF], R-signature [10], and Radon 2D Fourier-Mellin transforms (R2DFM) [START_REF] Wang | Scaling and rotation invariant analysis approach to object recognition based on Radon and Fourier-Mellin transforms[END_REF]. Except for the R-signature, all other comparing descriptors need normalizations in order to be invariant to RST transformations. Moreover, the R-signature and R2DFM descriptors are also defined on the Radon transform. These descriptors are selected because they are commonly used and have good reported performance. The measure used for comparison among descriptors is the precision-recall curve defined in information retrieval context.

In computing the precision-recall curve for each dataset, in the experiment, each of the images in the dataset is used as a model to which all the images in the dataset are compared/matched with. The matching is realized using the similarity measure defined in Eq. ( 14). The obtained matching results are then sorted, or ranked, for the determination of the n th nearest matches for each model.

Grayscale pattern recognition

The performance of the proposed descriptor has been first tested on grayscale noisy images to demonstrate its robustness to additive white noise. Two experiments have been carried out on two different sets of datasets:

• ExpA: The first set of six alphabet datasets has been generated from images of 26 Latin characters as shown in Fig. 7(a). Each of these six datasets has 260 images of 26 categories, each category contains 10 images.

• ExpB : The second set of six object datasets has been generated from 20 object images from the The main characteristic that differentiates ExpA and ExpB, besides the semantic content of their images, is the number of intensity levels in the original images: character images have only two levels of intensity whereas object images have multi-level intensity. Noisy images are generated from the corresponding noise-free images by randomly scaling, rotating, translating and then adding white noise to them. Let SNR be the signal-to-noise ratio defined as SNR =

x,y f 2 (x, y)

x,y η 2 (x, y)

where f (x, y) is the noise-free image and η(x, y) is the added white noise, the value of SNR for each dataset is kept constant and, for each experiment, SNR has six possible values {0.5, 1, 2, 4, 8, 16}, corresponding to the six datasets. Some example images from the six datasets in the two experiments are given in Figs. 7(b), 8(b).

Comparison of the proposed RFM descriptor with ART, GFD, Zernike, R-signature, and R2DFM descriptors on these noisy datasets have been performed and the obtained results on ExpA and ExpB are given in Figs. 9 and 10 respectively. It is observed from these set of figures that:

• ART, GFD, Zernike, and R2DFM descriptors are not robust to additive white noise at all, their performance is similarly poor for different levels of noise.

• R-signature and RFM descriptors have reasonably good performance when the noise is weak (SNR = 16, 8, 4), demonstrating its resistance to additive noise.

• As SNR decreases (the images get noisier), the precision-recall curves of the R-signature and RFM descriptors generally move downwards. However, the curves of the RFM descriptor is always above that of the R-signature, meaning a more robustness of the RFM descriptor to noise than the R-signature. It is thus can be concluded that the proposed RFM descriptor is more robust to additive white noise than all other comparing descriptors on grayscale noisy datasets and this provides an empirical evidence for the analytical results developed in Section 5. Poor performance of ART, GFD, Zernike, and R2DFM descriptors has its root in the required normalizations in their computation and can be explained as • To have invariance to translation, the origin of the polar coordinate system needs to be placed at the centroid of the pattern. In the presence of noise, position of the centroid is shifted arbitrarily according to the actual noise.

• To have invariance to scaling, the radial axis is normalized by the distance from the origin of the polar coordinate system to the furthest pattern's point. In the presence of noise, this furthest point might not belong to the actual pattern but the noise.

Furthermore, it is also evident from the two set of experiments that the performance of R-signature and RFM descriptors is better on ExpB than on ExpA at each value of SNR, leading to a conclusion that the proposed descriptor performs better on multi-level than two levels grayscale images. Possible explanation for this come from the effect of re-sampling and re-quantization when an image is rotated, scaled, and/or translated. Multi-level images have "smooth" intensity surface which suffers less deformation due to re-sampling and re-quantization than the deformation occurred at sharp edge in two levels images.

Binary pattern recognition

Noisy datasets

The robustness of the proposed RFM descriptor to additive "salt & pepper" noise is demonstrated using six logo datasets generated from the first 25 logo images of the UMD Logo dataset as shown in Fig. 11(b). Each of these six logo datasets has 275 images of 25 categories, each category contains 11 images generated by randomly scaling, rotating, translating the original corresponding logo image and then adding "salt & pepper" noise to it. Let d be the percentage of pixels flipped by the noise, the The proposed RFM descriptor is again compared with ART, GFD, Zernike, R-signature, and R2DFM descriptors using these six datasets and the computed precision-recall curves of these descriptors are depicted in Fig. 12. For the noiseless dataset with d = 0 (Fig. 12(a)), all shape descriptors have ideal performance. This demonstrates that the proposed descriptor is totally invariant to both translation, rotation, and scaling. When d = 0 (Fig. 12(b)-12(f)), deterioration appears in the performance of all descriptors and their precision-recall curve moves downwards. However, the impact of d on those curves differs from one descriptor to another. Among the other descriptors, the proposed descriptor has the best performance for all the five noisy datasets while ART and Zernike descriptors have similarly worse performance. It is also observed that:

• As d increases, the curve of all the descriptors generally move downwards.

• ART and Zernike descriptors are not robust to noise at all, their performance is similarly poor for different levels of noise.

• GFD have more resistance to noise than ART and Zernike because its curves are pushed away from the ideal curve (when d = 0) with distance which increases along with the increase in d.

However, the resistance of GFD is weaker than that of descriptors defined on the Radon transform.

• Among the three Radon-based descriptors, the shift in the curve of R-signature and RFM is more regular than that of R2DFM. In addition, and more importantly, the RFM descriptor has the

Figure 1 :

 1 Figure1: Graphical illustration of the Radon transform of a function f (x, y). The Radon transform is a mapping from the image space (x, y) to the parameter space (θ, ρ) and can be mathematically represented by a line integral of f (x, y) along all the lines L parameterized by (θ, ρ) represented in the image space (x, y).

Figure 2 :

 2 Figure 2: Illustration of the properties of the Radon and 1D Fourier-Mellin transforms performed on pattern images. The first row contains two original pattern images (a)-(b) and the RST transformed versions (c)-(e) of the pattern image in (b). The second row shows the Radon transform of these pattern images and the third row shows the images obtained after performing the 1D Fourier-Mellin transform on the Radon images in the second row using 150 values of τ ranging from 2.0 to 16.9 with increment of 0.1. The intensity of these images has been rescaled to fit the display range.

Figure 3 :

 3 Figure 3: The normalized cross correlation between all possible pairs of Fourier-Mellin images from the third row of Fig. 2. For each pair of images, 180 correlation values are calculated by circular shifting one of the two Fourier-Mellin images along its θ direction by 180 possible values from 0 to 179 with increment of 1.

Figure 4 :

 4 Figure 4: Illustration of the computation of the Radon transform by definition: for each value of θ, the image function f (x, y) is projected onto an axis ρ which is perpendicular to the line L and makes with the x axis an angle of θ. The projection makes itself a line R f (θ, •) in the Radon image.

s

  2Nρ and E n = mnσ 2 n 2Nρ . The signal-to-noise

  SNR proj(θ)SNR image , let's say D ∈ [0.3, 0.7] and d ∈ [0, 0.2]. This is a practically reasonable assumption as the shape usually occupies around half of the image region (D = 0.5) and the shape image is not too noisy (the highest level of noise used in experimentation is d = 0.1, meaning 10% of the image pixels is flipped, corresponding to the dataset in the bottom row of Fig.11(b)). Due to the inverse proportion of SNR proj(θ) SNR image to d, SNR proj(θ) SNR image gets its minimum value at max d = 0.2. Moreover at d = 0.2, as SNR proj(θ) SNR image decreases when D is going away from the point D = 0.5, the minimum value of SNR proj(θ)

  Figure 6: (a) The values of SNR proj SNR image over the domain D ∈ [0.3, 0.7] and d ∈ [0, 0.2] for the case A(θ) mn = 100,

Figure 7 :

 7 Figure 7: (a) Images of 26 Latin characters of size 64 × 64 pixels in Arial bold font used to generate the six alphabet datasets. (b) Sample images from the six alphabet datasets generated from the first four character images with six possible values of SNR = {16, 8, 4, 2, 1, 0.5}, corresponding to the six datasets.

  Figure 8: (a) Twenty object images from the COIL-20 dataset [40] used to generate the six object datasets. (b) Sample images from the six object datasets generated from the four object images with six possible values of SNR = {16, 8, 4, 2, 1, 0.5}, corresponding to the six datasets.

Figure 11 :

 11 Figure 11: (a) Twenty-five logo images from the UMD Logo dataset [41] used to generate the six logo datasets. (b) Sample images from the six logo datasets generated from the first four logo images with six possible values of d = {0, 0.02, 0.04, 0.06, 0.08, 0.1}, corresponding to the six datasets.

Acknowledgments

The authors would like to thank the anonymous reviewers for their valuable and insightful comments and suggestions that help to improve the quality of the paper.

As SNR decreases (the images get noisier), the precision-recall curves of the RMF and R-signature descriptors generally move downwards but the curves of the RMF descriptor is always above that of the R-signature, meaning its superiority. As SNR decreases (the images get noisier), the precision-recall curves of the RMF and R-signature descriptors generally move downwards but the curves of the RMF descriptor is always above that of the R-signature, meaning its superiority. The above observations lead to a conclusion that the proposed RFM descriptor is more robust to additive "salt & pepper" noise than all other comparing descriptors and this provides an empirical evidence for the analytical results developed in Section 5. Poor performance of ART, GFD, Zernike, and R2DFM descriptors in binary noisy datasets has similar explanations as in grayscale noisy datasets given in the previous subsection.

Occlusion and deformation dataset

Fig. 13 shows some shapes from the Shapes216 dataset. This dataset is composed of 18 shape categories with 12 samples per category, each of these cannot be obtained by RST transforming any other shape from the same category. The overall resulting nearest matches by category using the proposed RFM descriptor is given in Table 1. The value of each cell in the center region of this table represents the number of times the n th nearest matches for each category are in the appropriate category. The precision of the first 12 nearest matches for each category is given in the last column and the value varies according to category. The proposed descriptor obtains:

• 100% precision in the first 12 nearest matches on Car, Children, Face, and Fountain.

• High performance on Bone, Brick, and Classic.

• And low performance on Elephant, Fork. The last row of Table 1 provides the precision of each nearest match for all categories. It is clear that the categorization is not as good as that given in [START_REF] Sebastian | Recognition of shapes by editing their shock graphs[END_REF] where the precision of each nearest match for all categories are reported as (100, 100, 100, 100, 99, 97, 99, 96, 96, 95, 91, 80). However, it should be noted that the proposed RFM descriptor is not intended nor designed to work solely with binary patterns as in [START_REF] Sebastian | Recognition of shapes by editing their shock graphs[END_REF]. It is designed, instead, to work also with grayscale patterns under RST transformations allowing a certain level of additive noise. Furthermore, methods such as [START_REF] Sebastian | Recognition of shapes by editing their shock graphs[END_REF] cannot work with noisy patterns and need "clean" images for feature extraction, which the proposed one does not. Comparison of the proposed RFM descriptor with commonly used descriptors in this direction is given in Fig. 14. The performance of the RFM descriptor outperforms the performance of R-signature and is comparable to the performance of ART, GFD, Zernike, and R2DFM descriptors.

A better insight into the performance of proposed descriptor is by using the confusion matrix plotted as an image Fig. 14(b). The value in each table cell of row i and column j corresponds to the number of shape images of actual category j being predicted as shape images of category i. Thus, it is easy to see if the system is confusing two classes, i.e., commonly mislabeling one as another. In the case of the Shapes216 dataset, it is observed that: • Low performance on Fork is due to its high confusion with Hammer and Key, the value in the cells of column Fork and rows Hammer, Key is high. This is understandable as the shapes of Hammer and Key are quite similar to the shapes of Fork.

• There is no high confusion between Elephant and any other category. However, low performance on Elephant is due to the diffusion of the values in the column Elephant. Noticeable values are in the cells of column Elephant and rows Misk, Bird, Camel, Car, Ray, and Turtle.

Conclusions

This paper presents a novel pattern descriptor defined based on the Radon, Fourier, and Mellin transforms. The proposed descriptor is obtained by applying 1D Fourier-Mellin and Fourier transforms on the radial and angular coordinates of the pattern's Radon image respectively and has been proved to be invariant to rotation, scaling, and translation, without the need of any normalization step. The proposed descriptor's computation is reasonably fast and correct, based mainly on the fusion of the Radon and Fourier transforms and on a modification of the Mellin transform. It is proved to be robust to additive noise both theoretically and experimentally. Experimental results show that, when compared to commonly used pattern descriptors, the proposed RFM descriptor has comparable performance on a occlusion/deformation dataset and outperforms on noisy datasets. Future work will investigate possible combinations of the RFM descriptor with local descriptors. In our perspective, due to the orthogonality between these two type of descriptors, the combination may potentially boost the performance of pattern recognition and image retrieval systems.


Eric Ferrandis

Sarbendra L Pradhananga

Caroline Touvay

Carol Kinoshita

Ian R Wilkinson

Kevin Stafford

Zida Wu

Christian J Strasburger

Jon R Sayers

Peter J Artymiuk

Professor Richard J M Ross
email: r.j.ross@sheffield.ac.uk

Keywords: Immunogenicity, toxicology, pharmacokinetics and pharmacodynamics of growth hormone ligandreceptor fusions Growth Hormone, GH, ligand, receptor, fusion, immunogenicity, toxicity, pharmacokinetics, pharmacodynamics

A fundamental concern for all new biological therapeutics is the possibility of inducing an immune response. We have recently demonstrated that a ligand-receptor fusion (LR-fusion) of growth hormone generates a potent long-acting agonist, however the immunogenicity and toxicity of these molecules have not been tested. To address these issues we have designed molecules with low potential as immunogens and undertaken immunogenicity and toxicology studies in Macaca fascicularis and pharmacokinetic and pharmacodynamic studies in rats. Two variants of the LRfusion, one with a flexible linker (GH-LRv2) and the other without (GH-LRv3), were tested. Comparison was made with native human GH.

GH-LRv2 and GH-LRv3 demonstrated similar pharmacokinetics in rats, showing reduced clearance compared to native GH and potent agonist activity with respect to body weight gain in a hypophysectomised rat model. In M. fascicularis a low level of antibodies to GH-LRv2 was found in one sample but there was no other evidence of any immunogenic response to the other fusion protein. There were no toxic effects and specifically no changes in histology at injection sites after two repeated administrations. The pharmacokinetic profiles in monkeys confirmed long half lives for both GH-LRv2 and GH-LRv3 representing exceptionally delayed clearance over rhGH. The results suggest that repeated administration of a GH LR-fusion is safe, non-toxic and the pharmacokinetic profile suggests that two to three weekly administration is a potential therapeutic regimen for humans.

INTRODUCTION

A major challenge for recombinant DNA-based protein drugs is the generation of long-acting molecules that are non-toxic, non-immunogenic and provide appropriate pharmacokinetics for the drug in question. We have recently described a technology to make long-acting cytokines by fusing cytokine to its cognate extracellular domain receptor at the DNA level; a ligand-receptor fusion (LRfusion) [START_REF] Wilkinson | A ligand-receptor fusion of growth hormone forms a dimer and is a potent longacting agonist[END_REF]. The advantages of this technology include fusion at the DNA level, no requirement for chemical modification, and delayed clearance. We have made a growth hormone (GH, to growth hormone extracellular domain (GHRext, 1-238) LR-fusion that has a long circulating half-life, greatly reduced clearance and potent agonist activity [START_REF] Wilkinson | A ligand-receptor fusion of growth hormone forms a dimer and is a potent longacting agonist[END_REF]. However, a key question in the development of this new therapeutic technology is whether such a LR-fusion is immunogenic or toxic.

The original LR-fusion was generated using a peptide linker, (Gly 4 Ser) 4, between the C-terminus of GH and the N-terminus of GHRext. This long linker, with a predicted length of 80 Å, was chosen as a relatively flexible tether between GH and GHRext such that the GH moiety could still interact with cell surface GHR. Similar Gly 4 Ser linkers have been used in recombinant single chain Fv antibody production because of their stability and lack of immunogenicity [START_REF] Huston | Antigen recognition and targeted delivery by the single-chain Fv[END_REF]. In silico screening of the GH LR-fusion showed no new T-cell epitopes and the only non-native sequence was the (Gly 4 Ser) 4 linker. As the N-terminal residues of the GHR molecule are disordered in the crystal structure [START_REF] De Vos | Human growth hormone and extracellular domain of its receptor: crystal structure of the complex[END_REF] and therefore likely to be relatively flexible we questioned whether there was a need for the (Gly 4 Ser) 4 linker in this particular case. We have therefore generated a new variant GH LR-fusion which is a direct fusion between the GH C-terminus and the N-terminus of the GHRext.

Antibodies develop to varying degrees during treatment with most human proteins, including insulin, growth hormone, granulocyte-macrophage colony-stimulating factor, factor VIII, erythropoietin, and interferons [START_REF] Schellekens | Immunogenicity of recombinant proteins: causes and consequences[END_REF]. These antibodies may reduce clinical efficacy. In 30-60% of patients antibodies against pituitary-derived human growth hormone were detected and in 5% of treated patients these antibodies blocked the therapeutic activity of GH [START_REF] Frasier | Human pituitary growth hormone (hGH) therapy in growth hormone deficiency[END_REF]. Antibody formation to GH has been reported in 1-75% of patients treated with recombinant human GH [START_REF] Gunnarsson | Clinical experience with Genotropin worldwide: an update March 1987[END_REF][START_REF] Job | 1-year treatment with recombinant somatropin in prepubertal and pubertal growth hormone deficient patients: results from a French multicentre trial[END_REF][START_REF] Takano | Treatment of hypopituitarism with recombinant somatropin for 1 year[END_REF][START_REF] Buzi | Antigenicity and efficacy of authentic sequence recombinant human growth hormone (somatropin): first-year experience in the United Kingdom[END_REF][START_REF] Chang | Recombinant DNA produced somatropin in the treatment of prepubertal growth hormone deficient children[END_REF][START_REF] Rougeot | Comparative study of biosynthetic human growth hormone immunogenicity in growth hormone deficient children[END_REF][START_REF] Lundin | Development of anti-GH antibodies during therapy with authentic human growth hormone[END_REF][START_REF] Massa | Five-year follow-up of growth hormone antibodies in growth hormone deficient children treated with recombinant human growth hormone[END_REF][START_REF] Shih | Linear growth response to recombinant human growth hormone in children with growth hormone deficiency[END_REF][START_REF] Pirazzoli | Follow-up of antibodies to growth hormone in 210 growth hormone-deficient children treated with different commercial preparations[END_REF][START_REF] Li | The immunogenicity and effect of domestic recombinant human growth hormone: serum GH-antibody determination and evaluation[END_REF], but these antibodies have not been inhibitory or harmful. Antibodies are found whether GH is produced in E. coli or mammalian cells [START_REF] Pirazzoli | Follow-up of antibodies to growth hormone in 210 growth hormone-deficient children treated with different commercial preparations[END_REF]. Antibody formation probably relates to a small amount of denatured GH in the preparation [START_REF] Jones | The use of an animal immunogenicity model in the development of Protropin somatrem (methionyl human growth hormone)[END_REF], although antibody production is generally greater with pituitary derived GH and recombinant GH with an N-terminal methionine; recent studies suggest similar immunogenicity when GH is presented in a depot preparation [START_REF] Lee | In vivo characterization of sustained-release formulations of human growth hormone[END_REF].

There is no established format or single model for defining immunogenicity for new biological molecules. It is generally accepted that the only final proof is use of the new molecule in man. However, low immunogenicity and lack of toxicity in non-human primates provides strong reassurance for a non-toxic effect in man. In Rhesus monkeys a comparison of pituitary derived GH with recombinant methionyl GH and natural sequence GH showed GH antibodies occurring in 69, 81 and 23% of animals respectively [START_REF] Zwickl | Comparison of the immunogenicity of recombinant and pituitary human growth hormone in Rhesus monkeys[END_REF]. In most animals maximal antibody titers were detected before 40 days of treatment. There was no dose effect of GH on antibody production.

To address the question of immunogenicity with LR-fusions we have designed two LR-fusion molecules, tested them for biological activity, and undertaken an immunogenicity study in M. fascicularis. A c c e p t e d M a n u s c r i p t Licenced copy. Copying is not permitted, except with prior permission and as allowed by law.

MATERIALS AND METHODS

Materials

All the materials were purchased from Sigma (Poole, UK) unless otherwise stated. E. coli derived Recombinant GH, Nutropin AQ (5mg/ml in 2ml diluent), was purchased from Ipsen (Slough, UK).

LR-fusions and recombinant hGH

The methodology used has been previously described [START_REF] Wilkinson | A ligand-receptor fusion of growth hormone forms a dimer and is a potent longacting agonist[END_REF]. Two LR-fusions were designed one with a flexible (Gly 4 Ser) 4 linker, GH-LRv2, and the other a direct fusion of the C-terminus GH to the Nterminus of the GHR extracellular domain, GH-LRv3. Stable clones were made in CHO Flp-In cells (Invitrogen, Paisley, UK), adapted to protein free media and grown in suspension culture. LR-fusion expression was confirmed by an in-house ELISA. Affinity purification was performed using a GH mAb column. Recombinant hGH was also made under the same conditions.

In vivo bioactivity

The rat experiments were conducted in compliance with French laws (Council Directive N° 86/609/EEC of 24 November 1986). Seven weeks old normal Sprague Dawley rats from Janvier (Le Genest Saint Isle, France) were used for pharmacokinetic studies. Subcutaneous (sc) administration and blood withdrawal were conducted under isoflurane anaesthesia. The rats (n=4-6/group) were injected subcutaneously with the test molecule or control molecule (rhGH). Blood samples were collected from the retro-orbital plexus. Serum was harvested and stored at -70°C until assayed using an ELISA against GH. The growth studies used hypophysectomized rats and were performed on Sprague Dawley rats from Charles River Laboratories (Larbresle, France). Rats were hypophysectomized under isoflurane anaesthesia at 4 weeks of age by the breeder. The injection solutions of excipient, rhGH and LR-fusion never exceeded 2 ml/kg. The rats were given a single dose of the LR-fusions or excipient on day 1 and then retained without further dosing for 10 days; the rats given rhGH were dosed daily over the 10 day period. The rats were weighed daily. Terminal bleeds were collected and these assayed using the GH ELISA and an IGF-1 ELISA (Octeia Rat/Mouse IGF-1 Assay Kit, iDS).

In vivo immunogenicity and extended pharmacokinetic study

The monkey study was conducted in a Good Laboratory Practice (GLP) compliant facility [United Kingdom GLP Regulations 1999 (S.I. 1999 No. 3106), as amended by the S.I. 2004 No. 994, which are in accordance with the Organisation for Economic Co-operation and Development (OECD) Principles of GLP (ENV/MC/CHEM (98) [START_REF] Jones | The use of an animal immunogenicity model in the development of Protropin somatrem (methionyl human growth hormone)[END_REF], and which are accepted by the US FDA and Japanese authorities]. Male M. fascicularis (n=19) aged 2 to 3 years weighing approximately 2.5 kg were acclimatized for a minimum of 2 weeks prior to study (Bioculture, Mauritius Ltd). Animals were assigned to 5 treatment groups (Group 1: vehicle, Group 2: GH-LRv2 test group, Group 3: GH-LRv3, Group 4: growth hormone control and Group 5: Nutropin control), comprising 3 males in the vehicle group and 4 males per group in the remaining 4 groups. All animals from the vehicle control were dosed on two occasions (Days 0 and 14), by subcutaneous injection. GH-LRv2, and GH-LRv3 treated groups were dosed on two occasions (Days 0 and 14), by subcutaneous injection with 1 mg/kg protein, and then retained without dosing for a further 4 week period. All animals from the rhGH and Nutropin treated groups were dosed daily, for 27 days, by subcutaneous injection with 0.1 mg/kg/day, and then retained without dosing for a further 2 week period. Blood samples were collected and serum was harvested and stored at -70°C. The sampling times for groups 1-3 were at pre-dose, 1 hour postdose, then at approximately [START_REF] Gunnarsson | Clinical experience with Genotropin worldwide: an update March 1987[END_REF][START_REF] Lundin | Development of anti-GH antibodies during therapy with authentic human growth hormone[END_REF][START_REF] Cook | The pharmacokinetic and pharmacodynamic charateristics of a lon-acting growth hormone (GH) preparation (Nutropin Depot) in GH-Deficient Adults[END_REF]48,72,144,192,288 and 336 hours (for Day 1, pre-dose for Day 15) and 1 hour post-dose, then at approximately [START_REF] Gunnarsson | Clinical experience with Genotropin worldwide: an update March 1987[END_REF][START_REF] Lundin | Development of anti-GH antibodies during therapy with authentic human growth hormone[END_REF][START_REF] Cook | The pharmacokinetic and pharmacodynamic charateristics of a lon-acting growth hormone (GH) preparation (Nutropin Depot) in GH-Deficient Adults[END_REF]48,72,144,192,288,360, 648 and 696 hours after administration. The sampling times for groups 4 and 5 were at pre-dose and 1 hour post-dose, then at approximately [START_REF] De Vos | Human growth hormone and extracellular domain of its receptor: crystal structure of the complex[END_REF][START_REF] Gunnarsson | Clinical experience with Genotropin worldwide: an update March 1987[END_REF][START_REF] Lundin | Development of anti-GH antibodies during therapy with authentic human growth hormone[END_REF][START_REF] Cook | The pharmacokinetic and pharmacodynamic charateristics of a lon-acting growth hormone (GH) preparation (Nutropin Depot) in GH-Deficient Adults[END_REF]48,72,144,192,288 and 336 hours after dose administration, then pre-dose on Day 27 and 1 hour post-dose, then at approximately [START_REF] De Vos | Human growth hormone and extracellular domain of its receptor: crystal structure of the complex[END_REF][START_REF] Gunnarsson | Clinical experience with Genotropin worldwide: an update March 1987[END_REF][START_REF] Lundin | Development of anti-GH antibodies during therapy with authentic human growth hormone[END_REF][START_REF] Cook | The pharmacokinetic and pharmacodynamic charateristics of a lon-acting growth hormone (GH) preparation (Nutropin Depot) in GH-Deficient Adults[END_REF]48,72,144,192,288 and 360 hours after administration. The serum samples were analysed by GH ELISA and immunogenicity ELISA. Blood samples, for haematology, coagulation assays and blood chemistry, were obtained from all animals before the start of treatment, then during Week 6 of treatment. Urine samples were obtained for analysis before the start of treatment, then during Week 6 of treatment. After necropsy external and internal examinations of the monkeys were performed; which included examinations of the major organs and the sites of injection.

Quantitative assay of GH-containing molecules

An in-house GH and LR-fusion ELISA has been established based on the sandwich ELISA format. In the assay, standards (GH or LR-fusion), controls and unknowns are incubated with biotin-labeled mouse antibody to human GH (mAb 7F8) in wells pre-coated with a mouse antibody to human GH antibody (mAb 10A7). The detection limit for the assay is 2.5 pg/ml and the intra and inter assay CV is <10%.

Assessment of immunological response to GH-containing molecules

An ELISA based method was used to analyze serum samples from the M. fascicularis for the presence of antibodies against the respective injected molecule. The serum samples were diluted in phosphate buffered saline and incubated on plates coated with the relevant antigen. The presence of bound antibodies was then analyzed by the addition of protein-A peroxidase followed by 3, 3', 5, 5'tetramethylbenzidine (TMB) solution. The positive control for the assays was a mAb to hGH (mAb 10A7). Samples were assayed at a 1 in 400 dilution, at which dilution the negative control showed acceptable background levels and there was clear differentiation from the positive control. The cutoff for a positive result was the mean absorbance measured spectrophotometrically at 450nm (A 450 ) for the negative control in the individual assay plus 3 standard deviations. Serum samples giving a positive reading were further analyzed using an inhibition assay to determine if the result was a false positive or a true positive.

Statistics and pharmacokinetics

Comparisons of groups of treatment regimes were done using one-way ANOVA analysis, if P < 0.5 then the Dunnett's test was performed. Pharmacokinetic parameters were estimated by fitting values of hormone concentration versus time to compartmental models using non-linear least-squares regression analysis. Clearance values were normalized to animal weight. Clearance rate per animal weight and terminal half lives (t 1/2 ) were calculated using the coefficient and exponents obtained from the intravenous bolus model fits.

RESULTS

Pharmacokinetic analysis

The GH LR-fusions showed greatly delayed clearance compared to rhGH in both rats and monkeys (Table 1 and Figure 1). Clearance was even more delayed in monkeys than in rats. The monkeys received two injections of the GH LR-fusions within an interval of 14 days. The kinetics of GH-LRv2 and GH-LRv3 showed no appreciable accumulation or reduction in systemic exposure over the dosing period. Systemic exposure (AUC0-τ) to GH-LRv3 was, on average, approximately 1.7-fold greater than GH-LRv2 on Days 0 and 14. Both molecules gave continuous exposure appropriate for the subsequent immunogenicity study.

LR-fusions and growth promotion

To test biological activity, the GH LR-fusions and GH were administered to hypophysectomised (GHdeficient) rats. Daily administration of rhGH induced continuous growth over 10 days. The LR-fusion was then compared to rhGH with a single injection. For all experiments equimolar doses of rhGH and LR-fusion were used with the same total dose being given over the 10 day period: 220 µg/kg/day, approximately 10 nmol over 10 days similar to the dose previously used to obtain a maximal growth response [START_REF] Massa | Five-year follow-up of growth hormone antibodies in growth hormone deficient children treated with recombinant human growth hormone[END_REF]. The LR-fusions promoted an increase in weight gain which was similar to that seen after daily GH injections (Figure 2). There was no difference in weight gain between GH-LRv2 and GH-LRv3. IGF-I levels were all elevated in the treatment groups compared to controls (P<0.05) but not different between the treatment groups (Figure 3).

Immunogenicity and toxicology

No anti-test substance antibodies were detected in the serum of any animals, with the exception of anti-GH-LRv2 antibodies which were detected in the serum obtained from one animal on Day 44 (Table 2). The single positive result observed (A 450 =0.210) was only just above the cut-off (A 450 =0.157). One animal treated with GH-LRv3 had high background reading before drug administration and this appeared to be non-specific and did not change after drug administration. The serum GH-LRv2 concentration in the animal with a single positive did not appear to be appreciably different compared to concentration data from the other animals in the same group. There were no treatment-related effects on clinical signs, bodyweights or bodyweight gains during the course of the study. There were no treatment-related effects on haematology, blood chemistry or urinalysis observed during the course of the study. No treatment-related effects were noted on organ weights or necropsy in any of the groups tested. Both on inspection and histology. There was no reaction at the injection sites and no evidence of lipoatrophy.

DISCUSSION

The use of LR-fusions as a therapeutic is an attractive target for a number of reasons. Since the therapeutic molecule is produced as a single amino acid chain and does not require further modification both the heterogeneity of the final product and the costs required for production are greatly reduced compared to pegylated or depot preparations. The efficacy of the GH LR-fusion has already been reported [START_REF] Wilkinson | A ligand-receptor fusion of growth hormone forms a dimer and is a potent longacting agonist[END_REF], and has been shown to have a superior pharmacokinetic profile compared to GH; with a terminal half-life 100 times that of GH when both are given intravenously. However, a concern with any protein-based therapy is the generation of antibodies against the therapeutic protein. Such a response may reduce the efficacy of the treatment or in extremis cause inhibition of the native protein.

The immunogenicity of protein therapeutics can be influenced by many factors including; the amino acid sequence, the glycosylation patterns, degradation of the protein, aggregation or aberrant folding of the protein, chemical modification (such as oxidation), the processing of the protein, product formulation, and even the route of administration [START_REF] Covic | Biosimilars: recent developments[END_REF]. The GH LR-fusion proteins were designed to minimize the immunogenic proclivity of the therapeutic protein. GH and GH binding protein (GHBP), the extracellular domains of the GH receptor, form a natural non-covalent complex in the blood circulation. This "natural depot" is thought to prolong the circulation time and stability of GH. GH-LRv2 consists of GH and the extracellular domain of its receptor with the only non-natural sequence being the (Gly 4 Ser) 4 linker, this was further modified to remove the linker all together in GH-LRv3. Additionally, the LR-fusions were expressed in CHO cells to provide suitable glycosylation of the receptor domain.

The lack of an immunogenic response in M. fascicularis is not definitive proof that antibodies against the protein therapeutic will not occur in man. However the fact that the LR-fusion proteins, given at ten times the human pharmacological dose, do not initiate an immune response in these animals, gives a strong reassurance of its non-immunogenicity when used as a biopharmaceutical in man. One animal injected with GH-LRv2 gave a low level immunological response just above the established cut-off point at day 43 (Table 2). The group injected with GH-LRv3 did not show any immunological response at all; although one individual gave an apparent positive response in the pre-dose sample which remained high throughout the course of the study, the response for this individual was regarded as a false positive. These results compare favorably with results previously obtained for treatment with natural sequence recombinant GH which showed a 23% immunogenic response in 22 treated Rhesus monkeys [START_REF] Zwickl | Comparison of the immunogenicity of recombinant and pituitary human growth hormone in Rhesus monkeys[END_REF].

The duration and dose of treatment may influence the probability of inducing an immunogenic response. Either too low a dose or too high a dose may reduce immunogenic potential. In human treatment in the clinic the GH dose ranges from 0.01 mg/kg/day for adult replacement to 0.055 mg/kg/day in children with renal failure. For our immunogenicity study we chose to use a GH dose 10 times greater than the lower dose used in human adults. This equated to 0.1 mg/kg/day given daily for 28 days in the GH treated control group. For LR-fusions it was expected that the duration of action was 10 days or greater and therefore we gave a single total dose 10 times that of GH daily dose which equated to 1 mg/kg/dose given on days 1 and 14. In a study of methionyl-GH immunogenicity in monkeys, where up to 81% of animals developed detectable antibody titers, the majority of animals who developed an immunogenic response had done so by 35 days [START_REF] Zwickl | Comparison of the immunogenicity of recombinant and pituitary human growth hormone in Rhesus monkeys[END_REF]. Based on this study it is unlikely that we have missed a major immunogenic reaction to the LR-fusion proteins.

In man, GH therapy has been variably reported to initiate an immune response in between 1 to 75% of patients [START_REF] Gunnarsson | Clinical experience with Genotropin worldwide: an update March 1987[END_REF][START_REF] Job | 1-year treatment with recombinant somatropin in prepubertal and pubertal growth hormone deficient patients: results from a French multicentre trial[END_REF][START_REF] Takano | Treatment of hypopituitarism with recombinant somatropin for 1 year[END_REF][START_REF] Buzi | Antigenicity and efficacy of authentic sequence recombinant human growth hormone (somatropin): first-year experience in the United Kingdom[END_REF][START_REF] Chang | Recombinant DNA produced somatropin in the treatment of prepubertal growth hormone deficient children[END_REF][START_REF] Rougeot | Comparative study of biosynthetic human growth hormone immunogenicity in growth hormone deficient children[END_REF][START_REF] Lundin | Development of anti-GH antibodies during therapy with authentic human growth hormone[END_REF][START_REF] Massa | Five-year follow-up of growth hormone antibodies in growth hormone deficient children treated with recombinant human growth hormone[END_REF][START_REF] Shih | Linear growth response to recombinant human growth hormone in children with growth hormone deficiency[END_REF][START_REF] Pirazzoli | Follow-up of antibodies to growth hormone in 210 growth hormone-deficient children treated with different commercial preparations[END_REF][START_REF] Li | The immunogenicity and effect of domestic recombinant human growth hormone: serum GH-antibody determination and evaluation[END_REF] and the degree of immune response is due to the source of the GH, its purity and the formulation [START_REF] Schellekens | Immunogenicity of recombinant proteins: causes and consequences[END_REF]. The presence of anti-GH antibodies did not appear to have a detrimental effect on the therapeutic efficacy of GH in most of these cases. Pegylation has been used as a means of reducing the immunogenic profile of protein molecules, this has the additional advantage of extending the halflife of the molecule in vivo. In a short 12-week study, antibodies against the GH receptor antagonist Pegvisomant, a pegylated form of mutated GH, low titers of anti-GH antibodies were detected in 10% of patients [START_REF] Trainer | Treatment of acromegaly with the growth hormone-receptor antagonist Pegvisomant[END_REF].

Sustained-release formulations of GH, where the protein is encapsulated in a microsphere, are also associated with an immunogenic response. This is dependant on the formulation, GH encapsulated in poly(lactic-co-glycolic acid) (PLGA) microspheres has been shown to have an immunogenic response in rhesus monkeys when delivered as microspheres, in contrast none of animals receiving GH as a protein solution gave an immunogenic response [START_REF] Lee | In vivo characterization of sustained-release formulations of human growth hormone[END_REF].

The GH LR-fusions did not show any toxicological effects, there were no treatment related changes in the blood analysis, urine analysis and clinical signs; on inspection and histology there was no reaction at the injection sites nor evidence of lipoatrophy. Bruising and pain at the injection site is associated with other long-acting GH therapies such as pegylated GH (Pegvisomant) [START_REF] Hodish | Long-term effects of pegvisomant in patients with acromegaly[END_REF] and Nutropin Depot [START_REF] Silverman | A long-acting growth hormone (Nutropin Depot): efficiency and safety following two years of treatment in children with growth hormone deficiency[END_REF], the latter is also associated with small erythematous nodules at the injection site [START_REF] Silverman | A long-acting growth hormone (Nutropin Depot): efficiency and safety following two years of treatment in children with growth hormone deficiency[END_REF][START_REF] Cook | The pharmacokinetic and pharmacodynamic charateristics of a lon-acting growth hormone (GH) preparation (Nutropin Depot) in GH-Deficient Adults[END_REF]. Lipoatrophy has been of concern with long acting GH molecules and has recently been reported with pegylated GH [START_REF] Touraine | and the GH Lipoatrophy Study Group (2009) Lipoatrophy in GH deficient patients treated with a long-acting pegylated GH[END_REF].

A more serious side-effect of Pegvisomant therapy is that of hepatic dysfunction due to the hepatotoxic effects of the pegylated protein [START_REF] Hodish | Long-term effects of pegvisomant in patients with acromegaly[END_REF][START_REF] Bernabeu | Pegvisomantinduced cholestatic hepatitis with jaundice in a patient with Gilbert's syndrome[END_REF]. It has been postulated that pegylated proteins can accumulate to toxic levels due to their unsuitability for renal clearance [START_REF] Lou | Growing a PEG-like alternative[END_REF]. Theoretically, high levels of GH in a fat depot for a prolonged period might directly affect fat metabolism. This has not been reported with other long acting formulations of GH. The LR-fusion appears to be well absorbed and its delayed clearance is related to reduced clearance from the intravascular space which may be an advantage in preventing local reactions.

The LR-fusions showed prolonged pharmacokinetic profiles in comparison with rhGH in both rats and monkeys, however as the main focus of these studies was on immunogenicity no comparison with i.v. dosing was made so at this stage it is not possible to comment on bioavailability or rate of absorption. In M. fascicularis the half-lives of the fusions are approximately 35 times and 45 times greater than GH alone for GH-LRv2 and GH-LRv3, respectively. In comparison, Pegvisomant shows a similar pharmacokinetic profile in man with a half-life of 64.7 +/-10.9 hours [START_REF] Yin | Clinical pharmacodynamic effects of the growth hormone receptor antagonist pegvisomant: Implications for cancer therapy[END_REF]. Another putative GH-fusion therapeutic, Albutropin, a fusion of GH with albumin, has a half-life 6 times that of GH when administered subcutaneously in monkeys [START_REF] Osborn | Albutropin: a growth hormone-albumin fusion with improved pharmacokinetics and pharmacodynamics in rats and monkeys[END_REF]. Based on previous studies of allometric scaling the clearance of hGH is 4 fold less in monkeys compared to rats and 2 fold less in humans compared to monkeys [START_REF] Mordenti | Interspecies scaling of clearance and volume of distribution data for five therapeutic proteins[END_REF]. We found a similar difference for the LR-fusions with 1.6-2.8 fold reduced clearance in monkeys compared to rats and if there is a similar reduction of clearance in humans, the LR-fusions could provide a two to three weekly GH replacement therapy.

The removal of the linker does not seem to affect the activity of the LR-fusion, as the pharmacokinetics and the pharmacodynamics of GH-LRv2 and GH-LRv3 are all very similar. The clearance profile of GH-LRv2 and GH-LRv3 in rat is similar and is also comparable to the clearance profile of GH-LRv1, the GH LR-fusion reported previously [START_REF] Wilkinson | A ligand-receptor fusion of growth hormone forms a dimer and is a potent longacting agonist[END_REF]. The IGF-1 levels post-treatment with the LR-fusions are also similar for both the LR-fusions tested and the previously reported value of 198 +/-6 ng/ml [START_REF] Wilkinson | A ligand-receptor fusion of growth hormone forms a dimer and is a potent longacting agonist[END_REF]. The pharmacokinetic profile in monkeys is also alike with values of 60 +/-5 hours and 78 +/-17 hours for GH-LRv2 and GH-LRv3, respectively. The single injection of the LRfusions gave a different growth profile to daily rhGH, with greater growth over the first five days and lesser growth over the second five days. Over the 10 days the LR-fusions produced a weight gain lower but not significantly different to a daily dose of rhGH. Neither GH-LRv2 nor GH-LRv3 gave an immunogenic response or adverse side-effects during this study.

The GH LR-fusions demonstrate a long-acting GH therapy which is simple to manufacture and shows good pharmacokinetic and pharmacodynamic profiles with low immunogenicity and no treatmentrelated effects on haematology, blood chemistry or urinalysis . Other long-acting GH therapies do not have the all advantages of the GH LR-fusions. Pegylation is effective at delaying the clearance of the proteins, but requires chemical modification and reduces the affinity of the ligand for its receptor [START_REF] Clark | Long acting growth hormones produced by conjugation with polyethylene glycol[END_REF]. Sustained-release formulations or depot formulations require encapsulation of the GH into microspheres, this may require excess hormone and may result in some protein instability due to the inefficiencies of the encapsulation process [START_REF] Pai | Poly(ethylene glycol)-modified proteins: Implications for ply(lactide-co-glycolide)-based microsphere delivery[END_REF]. Microsphere delivery of GH has also been associated with an initial high peak of GH followed by a slow decline in GH concentration over the dosing period; the initial release of GH was associated with dissociation of GH from the surface of the microsphere and the prolonged release phase associated with GH release as the microsphere disintegrated [START_REF] Kemp | Pharmacokinetic and pharmacodynamic charateristics of long-acting growth hormone (GH) preparation (Nutropin Depot) in GH-deficient children[END_REF].

Fusion of GH with albumin (Albutropin) [START_REF] Osborn | Albutropin: a growth hormone-albumin fusion with improved pharmacokinetics and pharmacodynamics in rats and monkeys[END_REF] has similar advantages to the GH LR-fusions in that they do not need to be further processed after purification. Albutropin has a six-fold longer terminal half-life than GH when given subcutaneously to M. fascicularis [START_REF] Mordenti | Interspecies scaling of clearance and volume of distribution data for five therapeutic proteins[END_REF]; in comparison GH-LRv2 and GH-LRv3 have 35-fold and 45-fold longer terminal half-lives, respectively, compared to native GH when given subcutaneously to M. fascicularis. Additionally the GH LR-fusions should produce a more native-like protein, since GH is naturally found bound to GHBP, thus one would expect the GH LR-fusions to be less immunogenic than the GH-albumin fusion.

The LR-fusion technology has been shown to be efficacious and to have the characteristics required for a good therapeutic agent, i.e. low immunogenicity and toxicity. A clear benefit of this technology is that it can be applied to other ligand-receptor systems for the therapy of a wide range of diseases. Licenced copy. Copying is not permitted, except with prior permission and as allowed by law.

Clinical

  Science Immediate Publication. Published on 02 Jul 2010 as manuscript CS20100241 THIS IS NOT THE VERSION OF RECORD -see doi:10.1042/CS20100241A c c e p t e d M a n u s c r i p t Licenced copy. Copying is not permitted, except with prior permission and as allowed by law.

Clinical

  Science Immediate Publication. Published on 02 Jul 2010 as manuscript CS20100241 THIS IS NOT THE VERSION OF RECORD -see doi:10.1042/CS20100241A c c e p t e d M a n u s c r i p t Licenced copy. Copying is not permitted, except with prior permission and as allowed by law.

Clinical

  Science Immediate Publication. Published on 02 Jul 2010 as manuscript CS20100241 THIS IS NOT THE VERSION OF RECORD -see doi:10.1042/CS20100241 A c c e p t e d M a n u s c r i p t

Figure 1 :

 1 Figure 1: (a) Delayed clearance of rhGH, GH-LRv2 and GH-LRv3 in rats, following subcutaneous delivery of protein at time = 0 days; the inset shows the clearance for rhGH alone. [N.B. Units for protein concentration given in nM to enable direct comparison of GH with the GH-LR proteins]. (b) Delayed clearance of GH-LRv2 and GH-LRv3 in M. fascicularis, following subcutaneous delivery of protein at time = 0 days and 14 days (vertical arrows); the inset shows the clearance for rhGH and Nutropin.

Figure 2 :

 2 Figure 2: Weight gain in hypophysectomized rats due to LR-fusions compared with rhGH. The LRfusions and excipient were injected subcutaneously only on day 1 and the rhGH was given daily.

Figure 3 :

 3 Figure 3: IGF-1 measurement in the terminal bleeds of rats (n=6) given LR fusions compared to rhGH. The LR-fusions and excipient were injected subcutaneously only on day 1 and the rhGH was given daily. ** = P < 0.05 versus Excipient.

  Clinical Science Immediate Publication. Published on 02 Jul 2010 as manuscript CS20100241 THIS IS NOT THE VERSION OF RECORD -see doi:10.1042/CS20100241

1. kg -1 ) T 1/2 s.c. (hrs)

	hGH	820 +/-94 *	0.66 +/-0.1 *	154 +/-4.4 *	1.7 +/-0.4 *
	GH-LRv2	2.8 +/-0.2	25 +/-6.4	1.7 +/-0.2	54 +/-6.8
	GH-LRv3	2.7 +/-0.2	26 +/-6.8	0.96 +/-0.2	76 +/-4.9
	* = From Osborn et al. (28)

Table 1 :

 1 Pharmacokinetic parameters in rats and M. fascicularis given a single administration of hGH and GH LR-fusion. The table shows the A 450 for the sera from M. fascicularis given GH-LRv2, GH-LRv3 or hGH. Positive immunogenic results are shown in bold, false positives are shown in brackets. The cut-offs for positive reactions were ≥0.157 for GH-LRv2, ≥0.156 for GH-LRv3 and ≥0.210 for hGH, these cut off values were determined during the qualification of the respective assays.

	Clinical Science Immediate Publication. Published on 02 Jul 2010 as manuscript CS20100241

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Table 2 :

 2 Immunogenicity results for GH-LRv2, GH-LRv3 and hGH * . Science Immediate Publication. Published on 02 Jul 2010 as manuscript CS20100241 THIS IS NOT THE VERSION OF RECORD -see doi:10.1042/CS20100241 A c c e p t e d M a n u s c r i p t

	Clinical

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© 2010 The Authors Journal compilation © 2010 Portland Press Limited

Licenced copy. Copying is not permitted, except with prior permission and as allowed by law.© 2010 The Authors Journal compilation © 2010 Portland Press Limited