Thin Nets and Crest Lines: Application to Satellite Data ... - CiteSeerX

Thin Nets and Crest Lines: Application to Satellite Data and Medical Images OLIVIER MONGA, NASSER ARMANDEy AND PHILIPPE MONTESINOSy INRIA, Domaine de Voluceau -

Rocquencourt BP. 105, 78153 Le Chesnay Cedex, France y LGI2P, Parc Scienti que G. Besse 30000 N^mes, France

In this paper, we describe a new approach for extracting thin nets in grey level images. The key point of our approach is to model thin nets as crest lines of the image surface. Crest lines are lines where the magnitude maximum curvature is a local maximum in the corresponding principal direction. We de ne these lines using rst, second, and third derivatives of the image. The image derivatives are computed using recursive lters approximating the Gaussian lter and its derivatives. Using an adaptive scale factor, we apply this approach to the extraction of roads in satellite data, blood vessels in medical images, and actual crest lines in depth maps of human faces.

1

INTRODUCTION

In the elds of image processing and computer vision, a large amount of research has focused on the broader area of \ridge nding" [5, 14, 15]. This work is related to previous approaches and we use dierential properties of the image surface to characterize the thin structures in the grey level images. In many images, thin nets (TN) correspond to important features [1, 10, 11]. For instance, in aerial and medical images, TN are attached respectively to roads and blood vessels. TN are formed by the points where the grey-level is locally extremum in a given direction. This direction is the normal to the curve traced by the TN at this point. Classical edge detection algorithms [4, 6, 13, 23] fail to detect TN. In this paper, we propose a new method to detect TN using dierential geometry. An important point of our approach is its ability to identify TN as crest lines of the surface de ned by the image [22]. We use the de nition of crest lines proposed in [18] for 3D volumetric images, i.e. the points where 1

the magnitude of the maximum curvature is maximum along the maximum curvature direction at a point. In the present case, we come up with dierent expressions of the surface dierential properties, using partial derivatives of the 2D images. The principal curvatures and principal curvature directions of the surface de ned by the image are expressed using rst and second order partial derivatives of this image. These are the same as those given in [22]. We propose a new criterion which uses an extra partial derivative for characterizing the crest lines. We show that this criterion is dierent from the one proposed in [18] for 3D volumetric images, although it characterizes the same dierential property. To compute the partial derivatives, we use an extension to the third order of the recursive lters approximating the Gaussian and its derivatives [7, 19]. We come up with a three-stage algorithm which extracts the TN: 1. Computation of the rst second and third order partial derivatives of the image using recursive Gaussian lters. 2. Computation of the principal curvatures, the principal curvature directions and the directional Derivative of the Maximum Curvature (DMC) along the corresponding principal direction. 3. Extraction of the zero-crossings of the DMC which form crest lines or TN. The paper is organized as follows: We present in section 2 the partial derivative computations. In section 3, we give the de nition of TN and we compute the DMC and the zero-crossings of the DMC. The algorithm for the extraction of TN is described in section 4. Before concluding, experimental results are given in section 5. The computation of image derivatives is detailed in the appendix. We have tested this method on satellite data and medical images in which TN correspond to roads and blood vessels respectively. We have also applied this method on depth maps of human faces provided with a stereo vision process [8] to extract the crest lines. The quality of the results obtained clearly shows the exibility and the pertinence of our approach.

2

PARTIAL DERIVATIVES OF A 2D IMAGE

Let I (x; y ) be a 2D continuous grey level image. We look for the partial derivatives of I (x; y ) using the following formula: n y)) ; n = m + p: (1) Ixm yp = @ @x(Im(x; @y p

Here, we use the subscript notation Ixm yp to describe the derivative orders. When g (x; y ) = g (x)g (y ) is the impulse response of a smoothing lter, the restored image Ir is equal to I  g , where  is the 2

convolution product. Given the properties of the convolution product, we can write: @ nIr = @ n(I  g) = I  ( @ ng ); n

@xm @y p

@xm@y p

@xm@y p

(2)

g de nes the impulse response of the lter which computes I m p . We use the Gaussian where @x@m @y x y p smoothing lter and its derivatives up to the third order (see Appendix) to compute the derivatives of a 2D image. We use these lters because all the results of the forthcoming algorithms strongly depend on the way the partial derivatives are computed. We come up with the following algorithm ng for computing @x@m @y p , m + p  3 [19]:

for (m; p) such2that(m + p)  3 do 64 R = I  gm(x) Ixm yp = R  gp(y) where the convolution products are implemented using the recursive implementation of the lters. In the next section, we use rst, second, and third order partial derivatives which estimate using this algorithm.

3 3.1

FROM THE IMAGE SURFACE TO THE EXTRACTION OF TN TN and crest lines of an image surface

Thin nets of grey level images correspond to the points where the luminance is locally extremum in a given direction. This direction is the normal to the curve traced by the TN at this point. Classical edge models [4, 6, 13, 23] fail to characterize TN. One way to tackle this problem is to de ne TN, using the dierential properties of the image surface, i.e. the surface de ned by (x; y; I (x; y )). Some work on corner and vertex detection [12, 21] use the same paradigm. We de ne TN as crest lines of the image surface for an adaptive scale factor. We use the de nition of crest lines proposed in [18] for 3D volumetric images, i.e. the points where the maximum curvature is locally maximum in the corresponding principal direction. These lines can be characterized as the zero-crossings of a coecient called extremality and denoted by em . It is the directional Derivative of the Maximum principal Curvature (DMC) in the corresponding principal direction ~tmax :

~ kmax : ~tmax : em = r

(3)

We use expression em =0 in order to extract TN or crest lines according to scale factor, but our explicit equation of em is completely dierent because it must be obtained from a 3D surface de ned by a 2D image. 3

We can also extract the TN by the zero-crossings of the following expression:

~ I (x; y) : ~tmax ; gm = r

(4)

where I (x; y ) is the 2D grey level image function. Equation 4 shows that gm can be computed directly from the rst and second derivatives of the image. This is an advantage over methods requiring the computation of third order derivatives. We have tested this expression on a number of images provided with satellite data, medical images and depth maps but unfortunately, the quality of the results obtained is not acceptable for various reasons including noise. In particular, for the depth maps it is evident that this expression is not a solution of our problem. Moreover conceptually speaking, this method does not stem from the dierential geometry properties of the surfaces. This is why we have chosen em = 0 using the third derivative of the image. 3.2

3.2.1

From partial derivatives to crest lines of an image surface Computation of the directional Derivative of the Maximum principal Curvature (DMC)

Let us consider the surface S~ (x; y ) associated to the grey-level intensity of a 2D image I (x; y ) described by the equation: S~ (x; y) = (x; y; I (x; y))T : (5) At each point P of this surface, there is an in nite number of curvatures attached to each direction ~t in the tangent plane at P [3]. The principal directions denoted by t~1 and t~2 correspond to the two extremal values of the curvatures k1 and k2, except for the umbilic and at points (see Appendix), where we cannot de ne two principal directions. One of these two principal curvatures is maximum in absolute value and is called the maximum curvature. The other is the minimum curvature. We de ne the tangent plane TS (x; y ) of surface S~ (x; y ) described in Equation 5, at each point P by:

~ T S~x = ( @S @x ) = (1; 0; Ix) ; ~ T S~y = ( @S @y ) = (0; 1; Iy) ; TS (x; y) = fS~x; S~y g:

(6)

where Ix and Iy represent respectively the partial derivatives of the image function along x and y . The rst and second Fundamental Forms can be used to compute the principal curvatures and principal directions at each point P of the surface S~ (x; y ) [3]. The coecients of the rst Fundamental Form, 4

in the basis fS~x ; S~y g, form a matrix F1 de ned by:

1 0 e fC F =B A; @ 1

where

f g

e = kS~xk2; f = S~x :S~y ; g = kS~y k2 :

(7)

Using the partial derivatives of the image I (x; y ), for derivative of S~ (x; y ), we have:

e = 1 + Ix2 ; f = Ix:Iy ; g = 1 + Iy2:

(8)

Similarly, we obtain for the second Fundamental Form, in the same basis, a matrix denoted by F2 :

1 0 l m CA : F =B @ 2

m n

The elements l; m, and n can be written as:

~ S~xx; m = N: ~ S~xy ; n = N: ~ S~yy ; l = N:

(9)

where S~xx ; S~xy ; S~yy are respectively the second derivatives of S~ (x; y ) along the corresponding axes and N~ is the surface unit normal given by: x^ y = (10) k x ^ yk Using partial derivatives, we obtain: ~ N

~ S

~ S

~ S

~ S

:

l = p(1+IxxIx +Iy ) ; m = p(1+IxyIx +Iy ) ; n = p(1+IyyIx +Iy ) : 2

2

2

(11)

2

2

2

The principal curvatures and principal directions correspond respectively to the eigenvalues (k1 and k2 ) and the eigenvectors (t~1 and t~2) of the Weingarten endomorphism [3], whose matrix is:

0 w W = F? : F = B @ 1

1

11

2

w21

1

w12 C A: w22

The coecients of W are computed as follows:

w12 = H1 (gm ? fn); w11 = H1 (gl ? fm); w21 = H1 (em ? fl); w22 = H1 (en ? fm): (where H = eg ? f 2 ): 5

(12)

By de nition, K=k1 k2 is the Gaussian curvature and S= 12 (k1 + k2 ) is the average curvature at each point of the surface S~ (x; y ). Moreover, K and S are also respectively the determinant and half the trace of the matrix W. Therefore, the expressions of K and S are given by:

K = w11w22 ? w12w21 = H1 (ln ? m2 ); S = 12 (w11w22) = 21H (en ? 2fm + gl):

(13)

Using Equations 8 and 11, K and S are expressed by: K S

= =

d (Ixx Iyy ? Ixy ); 2 2 1 2d3=2 (Ixx ? 2Ix Ixy Iy + Ixx Iy + Iyy + Iy yIx ); 2

1

(where

d

=1+

I

x

2

(14)

2

+ Iy ):

Finally, the principal curvatures k1 and k2 are the solutions of a second-order equation given as follows: p 2 ? 1= + (15) p 2 ? 2= ? The explicit expressions of k1 or k2 (see Appendix) show that the principal curvatures of a surface can be computed directly from the rst and second derivatives of the image. Once k1 and k2 are computed, the principal directions t~1 and t~2 , which are the two eigenvectors of the matrix W, may be represented in the basis fS~x; S~y g with t~i = (ui vi )T and thus satisfy the system of: k

S

S

K;

k

S

S

K:

(w11 ? ki )ui + w12vi = 0; w21ui + (w22 ? ki )vi = 0 i 2 1; 2:

(16)

For each i 2 1; 2, the two equations are dependent, and thus the solution set for (ui vi )T will lie along a line collinear with ~ti . We can compute ~ti with either of the two equations presented in 16, which gives us two vectors: 0 1 0 1 t~1

=

@

A

w12

t~2

=

@

w12

A

(17)

:

? 11 2 ? 11 Let kmax = k1 be the curvature with greater magnitude (jk1j  jk2j), with associated direction ~tmax = t~1 . The gradient of kmax in the associated direction ~tmax , called the directional Derivative of the Maximum Curvature (DMC), is given by: k1

w

k

DM C

where

0 r max = @ ~ k

@kmax @x @kmax @y

=

1 A

r

~ k

max

w

max ;

0 max = @

w12

~ t

The expression of the DMC can also be written as:

(18)

: ~ t

k

max ? w11

1 A

max )(w ) + ( @kmax )(k DMC = ( @k@x 12 max ? w11): @y

6

:

(19)

(20)

Equation 20 shows that the directional derivative of the principal maximum curvature at each point of a surface can be directly computed from the rst, second and third derivative of the image. Once the DMC is computed, we extract the points where DMC=0 (the zero-crossings of the DMC).

3.2.2

Extraction of the zero-crossings of the DMC

Here, we present how to extract the zero-crossings of the DMC, computed in previous section. Theoretically, the absolute value of the DMC is invariant, but its sign depends on the orientation of the vectors t~1 or t~2 (see Equation 17). This is one reason why Thirion introduced the Gaussian extremality, an Euclidean invariant but having a dierent geometric meaning than crest lines [24, 25, 26]. However, in our case, by assuming that (~n; t~1 ; t~2) is always a direct orthogonal frame, where ~n is the surface unit normal, we can ensure the coherence of the orientation of ~tmax (see Equation 19). Therefore, the zero-crossings of the DMC can be characterized, in most cases, as the points where the sign of the DMC changes. Using Equation 20 and separating the at and umbilic (see Appendix) points, we compute the DMC image. The sign image (Sdmc ) can be built from the DMC image as follows:

8 > < Sdmc = 1 if > : Sdmc = 0 if

; 0:

DM C > DM C

0

(21)

The zero-crossings of the DMC are the transitions 0%1 or 1&0. We select the zero-crossings of the DMC such that the maximum curvature kmax is greater than a given positive threshold. This thresholding image allows us to remove the valley points having one principal curvature which is negative with a high absolute value and the other having a small value near zero. Moreover, this thresholding stage yields to hold only the points where the maximum curvature is locally maximum (the zero-crossings of the DMC correspond to both the maximum and the minimum of the maximum curvature).

4

ALGORITHM

Our TN extraction algorithm can be split into 4 steps: 1. Computation of rst, second and third order partial derivatives of the image which we denoted by: Ix , Ixx , Ixxx , Iy , Iyy , Iyyy , Ixy , Ixyy , Ixxy . We estimate these derivatives using recursive Gaussian lter and its derivatives (see Appendix). 7

2. Computation of the two principal curvatures k1 , k2 , as well as the two principal curvature directions t~1 , t~2 . 3. Using the direct derivatives up to the third order of the image, computation of the directional Derivative of the Maximum Curvature along the corresponding principal direction as follows:

 Separating the at and umbilic points (see Appendix).  Computing the DMC on the other points, i.e. elliptic, hyperbolic and parabolic points (see Appendix).

8 > if jk j > jk j do kmax = k ~tmax = t~ ; > > > < else do kmax = k ~tmax = t~ ; > end if > > > : do ~ kmax : ~tmax : DMC = r 1

2

1

1

2

2

4. For each point extraction of the zero-crossings of the DMC expression where kmax, is greater than a given positive threshold (see Section 3.2.2).

5

EXPERIMENTAL RESULTS

In this section, we discuss some experimental results obtained on synthetic and real data. All the lowlevel processes (smoothing and deriving) required, are performed using an extension of the development method for implementing the Gaussian lter and its derivatives (see Appendix). The detection and extraction of kmax , DMC and TN are performed at pixel precision. We illustrate the results on synthetic data in Figures 1 to 5. In Figures 1 and 4, we have added a Gaussian noise with a standard deviation n , to pictures representing a circle and a square which has a slope of 45 degrees in the grey-level. This is done in order to make complicated noisy data. The Signal to Noise Ratio (SNR) on each image can be computed as follows: SN R

PP i j dB = 10 log P P i

I (i; j )

2

j N (i; j )2

:

(22)

where I (i; j ) denotes the pure image and N (i; j ) the noise image. We have taken n =10 and SNR=14dB for these images. Figure 2 presents the results of the edge detection with a Gaussian lter ( = 1) [6] of the original images. These images show that a classical edge detection algorithm fails to detect these pictures. Figure 3 is the results of our extraction algorithm of TN in which we illustrate the extraction of these thin nets. We have also compared in Figures 4 and 5 the response delity of our method using two pictures in which step edge and TN are presented. The ability to successfully extract the thin net yet exclude the step edge shows the pertinence and delity of the TN extraction algorithm. Figures 6 to 8 present the results of our algorithm on satellite data in order to extract TN 8

in these images. Figure 9 is a DSA (Digital Substraction Angiography) image of the cerebral vessels. It comes from a time sequence of 19 images showing dierent phases of the contrast agent distribution in the vessel tree following a single bolus injection. We have chosen this section and extracted blood vessels represented in Figure 10. Finally, we have also applied this method on depth maps of human faces as illustrated in Figures 11 to 13. Figure 11(left) is a human face grey-level image. From this image, we can obtain the depth maps (Figure 11 right), using a stereo vision process. We have applied our method on the depth maps obtained and for two values of  , we illustrate the results in Figures 12 and 13.

6

CONCLUSION

We have presented a new approach for extracting crest lines and thin nets in grey level images, using dierential properties of the image surface. This work clearly shows that even third order dierential features can be robustly extracted from grey level images. A key element in achieving this is, of course, the lters used to compute the partial derivatives of the image. The quality of the experimental results shows the pertinence of our approach but also illustrates that eorts still need to be done for extracting the regions of intersecting crest lines, our technique failing to register those pixels at the intersections of crest lines. This result was expected since the principal curvatures will both have large magnitude and the direction normal to the crest lines is not de ned. We are currently developing new constraints on the dierential properties of the image surface to solve this problem.

ACKNOWLEDGMENTS We wish to thank R. Deriche and J.P. Fabre for stimulating discussions about the dierential geometry of surfaces. Thanks also to G. Szekely and P. Fua who provided us with the medical images and depth maps. We wish also to thank H. Chabbi and P. Soille for their advice on this paper.

References [1] J.E. Boggess. Identi cation of roads in satellite imagery using arti cial neural networks: a contextual approach. Technical Report, Mississippi St. Univ. August. 1993. [2] J.M. Braemer and Y. Kerbrat. Geometrie des courbes et des surfaces. Hermann Paris, 1976. 9

[3] M.P. Do Carmo. Dierential Geometry of Curves and Surfaces. Prentice Hall, 1976. [4] J.F. Canny. Finding Edges and Lines in Images. Technical Report 720. MIT, Arti cial Intelligence Laboratory, Cambridge, Massachusetts, June 1983. [5] C. David and S.W. Zucker. Potentials, valleys and dynamic global coverings. In IJCV 5(3), pp. 219-238, 1990. [6] R. Deriche. Optimal edge detection using recursive ltering. In Proceedings of the First International Conference on Computer Vision, London, 8-12 June 1987. [7] R. Deriche. Recursively implementing the Gaussian and its derivatives. In ICIP92 Proceedings of the 2nd Singapore International Conference on Image Processing, Singapore, pp. 263-267, Sept 1992. Also INRIA Research Report 1893, 1993. [8] P. Fua and Y.G. Leclerc. Representation without Correspondences. IEEE Conference on Pattern Recognition, Seattle, USA, June 1994. [9] M. Fischler, J. Tenenbaum and H. Wolf. Detection of roads and linear structures in low-resolution aerial imagery using a multi-source knowledge integration technique. In Computer Graphics and Image Processing, 15, pp. 201-223, 1981. [10] D. Geman and B. Jedynak. Shape Recognition and Twenty Questions. INRIA Rocquencourt, Research Report 2155, 1993. [11] D. Geman and B. Jedynak. Detection of roads in SPOT satellite images. In Proceedings of the IGRASS 91, Helsinski 1991. [12] G. Giraudon and R. Deriche. On Corner and Vertex Detection. In Conference on Computer Vision and Pattern Recognition, Maui (HI), USA, pp. 650-655, June 1991. [13] R.M. Haralick. Digital step edges from zero-crossing of second directional derivatives. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6(1), pp. 58-68, January 1984. [14] R.M. Haralick. Cubic facet model edge detector and ridge-valley detector: implementation details. In E.S. Gelsema & L.N. Kanal, eds, Pattern Recognition in Practice II, pp. 81-90, Oxford, Elsevier Science, 1986. [15] M. Kass and A. Witkin. Analyzing oriented patterns. In CVJIP, 37(3), pp. 362-385, 1987. [16] C. Lanczos. Applied Analysis. London Sir Isaac Pitman and Sons Ltd, 1956. 10

[17] O. Monga and S. Benayoun. Using dierential geometry in R4 to extract typical surface features. In IEEE Conference on Computer Vision and Pattern Recognition, pp. 684-685, New York, June 1993. [18] O. Monga, S. Benayoun and O.D. Faugeras. Using partial derivatives of 3D images to extract typical surface features. In IEEE Conference on Vision and Pattern Recognition, Urbana Champaign, Illinois, pp. 379-382, July 1992. [19] O. Monga, R. Lengagne, R. Deriche. Extraction of the zero-crossings of the curvature derivative in volumic 3D medical images : a multi-scale approach. In IEEE Conference on Computer Vision and Pattern Recognition, Seattle, USA, pp. 852-855, June 1994. [20] O. Monga, R. Lengagne, R. Deriche. Crest lines extraction in volumic 3D medical images: a multi-scale approach. In International Conference on Pattern Recognition (ICPR), Jerusalem, Israel, pp. 553-555, October 1994. [21] J.A. Noble. Finding Corners. In Image and Vision Computing, Vol 6, pp. 121-128, May 1988. [22] J. Ponce and M. Brady. Towards a surface primal sketch. In Proceedings of the IJCAI 85, 1985. [23] V. Torre and T.A. Poggio. On edge detection. IEEE Transaction on Pattern Analysis and Machine Intelligence, 8(2), pp. 187-163, March 1986. [24] J.P. Thirion and A. Gourdon. The 3D marching lines algorithm and its application to crest lines extraction. INRIA Technical Report 1672, 1992. [25] J.P. Thirion and A. Gourdon. The 3D marching lines algorithm: new results and proofs. INRIA Technical Report 1881, 1993. [26] J.P. Thirion. The Extremal Mesh and Understanding of 3D Surfaces. INRIA Research Report 2149, December 1993.

APPENDIX 1

1.1

Recursive lters to approximate the Gaussian lter and its derivatives Approximation of the Gaussian lter and its derivatives

Here, we review the main results of [7] and we show how to approximate the Gaussian lter and its derivatives up to the third order, using Prony's method [16]. 11

The 1D Gaussian smoothing lter is given by: x

g0 (x) = 0e?  : 2 2 2

(23)

Its rst, second and third derivatives are respectively as follows: x

g1 (x) = 1xe?  ; 2 2 2

x

g2(x) = 2 (1 ? 3x2)e?  ; 2 2 2

(24)

g3(x) = 12 ( 4x + 5x3 )e?  : x

2 2 2

where i , i=0, ..., 5 are the normalization coecients, detailed in section 1.2. The lters g0(x), g1(x), g2 (x) and g3 (x) can be approximated in a mean square error sense by a function h(x) such that [7]:

h(x) =

m X i=1

x

ie? i ;

(25)

where i and i are the coecients that we nd in the following (in the present case, m is set equal to 4). This approximation problem can be solved using Prony's approximation method. This method usx es an equidistant sampling of a function y = f (x) (in our case, f (x) = e?  ) at coordinates (x0 ; y0); (x1; y1); :::; (xn; yn ). We can set xr = x0 + rh, for r = 0; :::; n, n  m and de ne a function as follows: m m X X (26) f (xr ) = ak ek xr = ak ek x ek rh (with 0 = ?  ): 2 2 2

0

Let ck = k ek x and vk = follows: 0

0

k=1 ehk . 0

0

0

0

k=1

Consequently, f (xr ) forms a system of n equations (r = 0; :::n) as

8 > c + c +


+ cm = y > > > < c v + c v +


+ c m vm = y .. > . > > : c vn + c vn +


+ cmvmn = yn

(27)

(v) = (v ? v1)(v ? v2 ) . . .(v ? vm);

(28)

(v) = v n + s1 vn?1 + . . . + sm ;

(29)

1

2

0

1 1

2 2

1 1

2 2

1

To solve this equation system, we use a polynomial de ned by:

which can also be written as:

12

. where s0 = 1; s1 = v1 + v2 + ..+ vm ; ::: : We now multiply, in turn, each equation belonging to system 27 by sm ; sm?1 ; :::; s1; s0 (with s0 = 1). By adding together all the terms of the equation system obtained, we get the following equation:

(v1 )c1 + (v2 )c2 +


+ (vm )cm = sm y0 + sm?1 y1 + . . . + s1 ym?1 + s0 ym = 0:

(30)

Using Equation 30 and since (vk ) = 0 for k = 1; :::; m, if we shift the origin by a distance h = 1 to the right and repeat this operation, we obtain a new equation system as follows:

8 > ym? s + ym? s + . . . + y sm = ?ym ; > > > < ym s + ym? s + . . . + y sm = ?ym ; .. > . > > : yn? s + yn? s + . . . + yn?m sm = ?yn : 1 1

1

2 2

0

1 2

1 1

1

(31)

+1

2 2

These n ? m + 1 equations with m unknowns sk can then be solved by a least-square method. Afterwards, we obtain the unknowns vk by solving Equation 30. Finally, we obtain the coecients 0k x with the formula 0k = lnhvk , the ck from Equation 27, and the ak from ak = ck vk? h . The coecients i = ai and i = ?0i, i = 1; ::; 4, may be complex but conjugate (i.e. m?i = i and m?i = i ) in order to deal with real coecients. The advantage of this method is that we need not set  to a particular value to compute the coecients i and i. 0

1.2

Computation of the normalization coecients

For the lters g0, g1 , g2, g3 of the previous section, the normalization coecients in the continuous case are chosen so that: Z +1 g0(x)dx = 1;

Z ?11 +

xg1(x)dx = 1; Z +1 Z +1 x2 g (x)dx = 0 and g (x)dx = 1; ?1

2 Z?1+1 x3 Z ?1 +1 g3(x)dx = 1: xg3(x)dx = 0 and ?1 6 ?1 2

2

These conditions ensure that the product convolution by these lters yields the correct derivatives when applied to polynomials. When the lters above are to be applied in a discrete setting, the normalization constants are expressed as follows : ?

1 = P1 k=0 e  k 4 ?

3 = P1 k=0 k e 

2 ?

2 = P1 k=0 k e  k 6 ?

4 = P1 k=0 k e 

k

k

2 2 2

2 2 2 2 2 2

2 2 2

13

(32)

P

k

 ? We can show that a function F ( ) = 1 k=1 k e  ,  > 0, can be approximated with a very low R t error by a polynomial function Fa ( ) = I () +1, with I () = 01 t e? dt. q We know that I (0) = 2 ; I (1) = 1; and I ( + 2) = ( + 1)I (). If  = 0, we simply approximate F ( ) with Fa ( ) = 1:25 ? 0:5 for   0:5. Therefore, we obtain the following approximations: 2 (2 2 )

2

2

q

2   3 2 ; q

4  15 7 2 :

1  0:5 + 1:25; q

3  3 5 2 ;

(33)

Finally, the normalization coecients are equal to: 1 = ? 21  ? 0:34 ; 0 = 2 1? 1  0:4 ; 1 2 2

0 : 4 2

2 1?1 2 = ? ?2 2 + 2 ?  ? 3 ; 3 = 2  12 ; 3 1 3 2 2 3 1 : 2 4 = ?  3 ; 5 = ( 3 2? )  0:54 : 3 

   4 2 2

3

3 2

4 2 2

3

(34)

3 2

In this way, g0 and its derivatives g1 , g2, g3 can be approximated by a 4th recursive operator h(x): b b h(x) = (a0 cos( !0 x) + a1 sin( !0 x))e?  x + (c0 cos( !1 x) + c1 sin( !1 x))e?  x : 0

1

(35)

where the coecients a0 , a1 , b0 , b1, c0 , c1, w0, w1 are derived from i, i and i . We have the following results for these coecients:

a0 a1 c0 c1 b0 b1 !0 !1 2

g0(x) 0.657/ 1.979/ -0.258/ -0.239/

g1(x) -0.173/ 2 -2.004/ 2 0.173/ 2 0.444/ 2

g2(x) -0.724/ 3 1.689/ 3 0.32 0 k1:k2 < 0

16

at point; umbilic point; parabolic point; elliptic point; hyperbolic point: