Fractional dierentiation for edge detection

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1 Signal Processing 83 (2003) Fractional dierentiation for edge detection B. Mathieu, P. Melchior, A. Oustaloup, Ch. Ceyral LAP-UMR 5131 CNRS, Universite Bordeaux 1-ENSEIRB, 351 cours de la Liberation, F33405 Talence cedex, France Received 2 December 2002 Abstract In image processing, edge detection often makes use of integer-order dierentiation operators, especially order 1 used by the gradient and order 2 by the Laplacian. This paper demonstrates how introducing an edge detector based on non-integer (fractional) dierentiation can improve the criterion ofthin detection, or detection selectivity in the case ofparabolic luminance transitions, and the criterion ofimmunity to noise, which can be interpreted in term ofrobustness to noise in general.? 2003 Elsevier B.V. All rights reserved. Keywords: Image processing; Edge detection; Robust edge detector; Noise immunity; Fractional dierentiation 1. Introduction Fractional dierentiation, also called non-integer dierentiation, is not a new concept: it dates back to Cauchy, Riemann, Liouville and Letnikov in the 19th century. Since then, several theoretical physicists and mathematicians have studied fractional dierential equations, especially fractional-order linear dierential equations. See for example the now classic [18], the more formalized [17,19,27] for a thorough mathematical study ofthe equations. In the last two decades, fractional dierentiation has played a very important role in various physical sciences elds, such as mechanics, electricity, chemistry, biology, economics, modelling, time and frequency domains system identication and notably control theory, mechatronics Corresponding author. Tel.: ; fax: address: melchior@lap.u-bordeaux.fr (P. Melchior). URL: and robotics [19,29]. Fractal theory is already used in fractal image compression [1,3,8,9,12,14,15,30]. In image processing, edge detection often makes use ofinteger-order dierentiation operators, especially order 1 used by the gradient and order 2 by the Laplacian [2,4,5,10,13,16,25,26,28,31]. In [20 23], the principles ofnon-integer order dierentiation operators in edge detection is introduced. This paper demonstrates with details how using an edge detector based on fractional dierentiation can improve the criterion ofthin detection, or detection selectivity in the case ofparabolic luminance transitions, and the criterion ofimmunity to noise, which can be interpreted in term ofrobustness to noise in general. This paper also evaluates the improvement ofthe proposed method over Prewitt s gradient. After some denitions in image processing in Section 2 ofthe paper, Section 3 deals with the detection power offractional dierentiation, particularly through an analysis ofthe non-integer derivative function ofa parabolic step-type luminance transition. The derivative order n is considered between 0 and 1, then between 0 and 2. Between 0 and 1 the maximum ofthe /$ - see front matter? 2003 Elsevier B.V. All rights reserved. doi: /s (03)

2 2422 B. Mathieu et al. / Signal Processing 83 (2003) derivative does not appear at the inexion point ofthe transition. However, between 1 and 2, not only does it appear at the inexion point, but it also displays a vertical slope ofthe derivative to the right ofthe inexion point abscissa. To increase selectivity even more, a methodical approach, through the generation ofa cusp ofthe derivative at the inexion point abscissa, leads to the synthesis ofa fractional-order robust edge detector. This is referred to CRONE detector (French abbreviation of Contour Robuste d Ordre Non Entier ). Its robustness is to noise, which ensures its ltering function, particularly of high frequencies when using a negative derivative order. Section 3 ofthis paper deals with the spatial description of the CRONE detector. Section 4 analyses the detection selectivity. The compromise between selectivity and immunity to noise is improved in Section 5 by extending the CRONE detector, designed for n ]1; 2[, to a derivative order between 1 and 1. Section 6 deals with the spatial description ofthe CRONE detector, and Section 7 presents the extension ofthe CRONE detector to two dimensions and the comparison ofits performance with respect to Prewitt s gradient. Performance is evaluated using a wide variety ofimages: articial images with and without noise, and real images ofvarious origins (biomedical and industrial) where the noise level is suciently high for their processing to be signicant. 2. Denitions An image can be interpreted as a function of two variables dened within a bounded area, the function s value also being within a bounded interval. One ofthe rst operations to be carried out on an image is its discretization. This is done by spatial sampling ofthe image, and then by quantication of luminous intensity. An image is thus represented by a matrix ofbounded positive integer values. The gradient image ofimage f(i; j), is the matrix ofpositive real values whose elements a ij are the modulus ofthe gradient at point M(i; j) within the denition area ofimage f(i; j). A point ofcontour is generally dened by the in- exion point ofa luminance transition. A contour Fig. 1. The maximum ofthe derivative f (1) (x) determines the abscissa x 0 ofthe inexion point oftransition f(x). can thus be dened as the position ofthe luminance inexion points. The extraction of contour consists in detecting all inexion points ofluminance transitions. This is to reduce the quantity ofinformation contained in an image to a minimum for recognizing shapes. Given that an inexion point is the point where the slope is maximal, its detection can consist in nding the abscissa ofthe maximum ofthe rst derivative (Fig. 1). The contour image ofimage f(i; j) is the positive real value matrix where elements a ij are obtained by thresholding the gradient image. 3. Analysis of the detection power of fractional dierentiation The object ofthis paragraph is to study the fractional derivative applied to a parabolic step-type transition, and in particular to compare the abscissa ofits maximum to that ofthe inexion point ofthe transition. In this study, the derivative order n is considered initially between 0 and 1, and then between 1 and 2. Each ofthese intervals determines a specic shape of the derivative curves. The parabolic transition considered provides an inexion point at abscissa x 0 = 200 (Fig. 2). It is

3 B. Mathieu et al. / Signal Processing 83 (2003) Fig. 2. Parabolic step-type transition where a =1=(2x 2 0 ). described analytically by f(x) with f(x)=0 forx60; (1) f(x)=ax 2 for 0 6 x 6 x 0 ; (2) f(x)= ax 2 +4ax 0 x 2ax 2 0 for x 0 6 x 6 2x 0 ; (3) f(x)=2ax0 2 for x 2x 0 : (4) By applying the Cauchy formalism [6,7], the fractional derivative of f(x), being f n (x), can be written as follows: f (n) (x)=0 forx60; (5) f (n) (x)= f (n) (x)= 2a ( n +3) x n+2 for 0 6 x 6 x 0 ; (6) 4a ( n +3) (x x 0) n+2 + 2a ( n +3) x n+2 for x 0 6 x 6 2x 0 ; (7) f (n) (x)= 4a ( n +3) (x x 0) n a ( n +3) x n+2 2a ( n +3) (x 2x 0) n+2 for x 2x 0 : (8) Fig. 3. Normalized modulus of f (n) (x) for various derivative orders between 0 and 1: the maximum ofthe fractional derivative f (n) (x) no longer determines the abscissa ofthe transition inexion point Derivative order between 0 and 1 In the case where n ]0; 1[, the abscissa ofthe fractional derivative maximum no longer corresponds to that ofthe transition inexion point. The farther n is from 1, the greater the shift (n) to the right ofthe inexion point (Fig. 3). The shift (n) thus dened is determined analytically as follows. If x 1 is the abscissa ofthe maximum of f (n) (x), the rst derivative of f (n) (x) is zero at x 1, i.e. [ ] d dx f(n) (x) =0: (9) x=x 1 In order to show that x 1 is greater than x 0, i.e. that maximum of f (n) (x) is clearly oset to the right of x 0, it suces to express the rst derivative f (n) (x) atx 0 [ d dx f(n) (x) ] x=x 0 = 2a ( n +2) x n+1 0 (10) then to verify that this is positive. This is the case as indicated clearly by relation (10). Thus having shown that x 1 is greater than x 0 and putting forward the hypothesis that x 1 remains inferior to 2x 0, and with f (n) (x) given by (7), Eq. (9) becomes 4a ( n +2) (x 1 x 0 ) n+1 2a + ( n +2) x n+1 1 =0 (11)

4 2424 B. Mathieu et al. / Signal Processing 83 (2003) Fig. 4. Variation ofnormalized shift (n)=(n)=x 0 for n ]0; 1[. Fig. 5. Normalized modulus of f (n) (x) for various derivative orders between 1 and 2: the maximum ofthe derivative off (n) (x) once again determines the abscissa ofthe transition inexion point. from which can be extracted the expression of the shift (n)=x 1 x 0 as a function of x 0, i.e. 1 (n)= 2 1=(1 n) 1 x 0: (12) This result shows that (n) remains inferior to x 0 for n ]0; 1[ and thus validates x 1 ]x 0 ; 2x 0 [. The graph of (n) variation (Fig. 4) shows that the shift increases as n decreases from 1 to 0, thus conrming the interpretation offig. 3. and [ ] d dx f(n) (x) x=x 0 + 4a = lim x x + 0 ( n +2) (x x 0) n+1 2a = + ( n +2) x n+1 (14) 3.2. Derivative order between 1 and 2 In the case where n ]1; 2[, the abscissa ofthe fractional derivative maximum corresponds to that ofthe transition inexion point. As can be seen in Fig. 5, there is no shift (n) as there is for n ]0; 1[. Another remarkable property is the innite slope ofthe derivative to the right ofthe inexion point abscissa (Fig. 5). Thus, for 1 n 2, fractional dierentiation ensures, not only that detection ofthe transition inexion point is achieved (1st property), but also that is highly selective (2nd property). The discontinuity ofthe slope off (n) (x) at x 0 is demonstrated by expressing successively the rst derivative of f (n) (x) on the left and on the right of x 0 i.e., using relation (6) and (7) [ d dx f(n) (x) ] x=x 0 = 2a ( n +2) x n+1 0 (13) the expression thus obtained show the slope discontinuity found in Fig. 5. In order to improve further the detection selectivity ofthe inexion point, the following paragraph shows how to bring in an operator where the response to a parabolic step-type transition presents an innite slope both on the right and on the left ofthe abscissa ofthe inexion point. 4. Detection selectivity 4.1. Denition The direction selectivity ofan inexion point, S N, can be dened as the inverse ofthe bandwidth at N% ofthe detector response maximum, x, (Fig. 6), i.e. S N = 1 x : (15)

5 B. Mathieu et al. / Signal Processing 83 (2003) Fig. 6. The ratio 1=x denes detection selectivity. Fig. 7. Selectivity improvement by creating a cusp. Fig. 8. Stages ofthe adopted approach for 1 n 2: (a) obtained f (n) (x) for increasing x, (b) expected f (n) (x) for decreasing x, (c) obtained f (n) (x) for decreasing x Strategy for increasing selectivity Principle The aim is to improve inexion point detection selectivity. The strategy for achieving such an improvement consists in designing an operator where the response (in this case to a parabolic step) shows a cusp on the inexion point abscissa, i.e., an innite slope on each side ofthis abscissa (Fig. 7) Methodicalapproach to achieve a CRONE detector using a derivative order between 1 and 2 For a derivative order between 1 and 2, we have shown that the slope ofthe derivative is innite on the right ofthe inexion point abscissa when calculated with increasing x (Fig. 8a). Calculating the derivative function with decreasing x could then be expected to provide an innite slope on the left (Fig. 8b). Although the resulting slope is in fact innite, the curve, Fig. 8c, is the opposite offig. 8b. Thus, an adequate solution can be (Fig. 9): rst, to take the opposite ofthe derivative function calculated with decreasing x, second, to add the derivative function calculated with increasing x. This provides cusp. Fig. 9. Construction ofthe response ofthe CRONE detector for 1 n 2: (a) f (n) (x) calculated with increasing x; (b) opposite f (n) (x) calculated with decreasing x; (c) detector response. The procedure thus dened is that used by the CRONE detector, (French abbreviation of Contour Robuste d Ordre Non Entier ). The robustness involved is a measure ofthe immunity to noise and is studied below. The spatial operator which denes this detector, applied to transition f(x), thus subtracts f (n) (x) calculated using decreasing x, from f (n) (x) calculated using increasing x. It is written as D n, the double arrow indicating increasing and decreasing x for dierentiation at order n. Fig. 10 illustrated the selectivity variation as a function of the derivative order by giving the

6 2426 B. Mathieu et al. / Signal Processing 83 (2003) Fig. 10. Normalized modulus of D n f(x) for various derivative orders between 1 and 2 (the f(x) transition is that offig. 2). response ofthe CRONE detector to the parabolic transition offig. 2 for various derivative orders between 1 and 2. It can be seen that selectivity increases with the derivative order and tends toward innity when the orders tends towards 2. Fig. 11. Normalized response ofthe CRONE detector to the parabolic transition offig. 2 for a greater calculation range than the transition width, and for each of two derivative orders between 1 and Improvement of the selectivity/immunity-to-noise compromise 5.1. The selectivity/immunity-to-noise dilemma The selectivity/immunity-to-noise dilemma stems from two contradictory requirements: the detection selectivity ofan inexion point and its insensitivity to transition noise. Increasing one ofthese performances causes the other to decrease. For n ]1; 2[, we have shown that selectivity increases along with the derivative order. However, immunity to noise diminishes along with the derivative order, thus a derivative order as low as possible is preferable, i.e. less than 1, and even less than 0 for generalized (any order) integration Extension of the CRONE detector to a derivative order between 1 and 1 In order to improve the selectivity/immunityto-noise compromise, notably by increasing immunity to noise, an extension ofthe CRONE detector (designed for n ]1; 2[) to derivative orders between 0 and 1, and between 1 and 0, can be achieved by Fig. 12. Normalized response ofthe CRONE detector to the parabolic transition offig. 2 for a narrower calculation range than the transition width, and for each of two derivative orders between 1 and 1. considering that the (n) shift inherent to such orders is cancelled out due to the symmetric character represented spatially by the D n operator. Figs. 11 and 12 illustrate this consideration to be valid. The application ofthe D n operator to the parabolic transition ofthe Fig. 2 for two derivative orders between 1 and 1, shows that the (n) shift seen above (Fig. 3) forn ]0; 1[ is present longer Selectivity IfFig. 12, which corresponds to a mask (41) whose width is less than that ofthe transition, shows that the (n) shift is null, it also illustrates another

7 B. Mathieu et al. / Signal Processing 83 (2003) Fig. 13. Normalized derivative function of the parabolic transition ofthe Fig. 2. property quite as remarkable, that is, for n ] 1; 1[, the CRONE detector conserves comparable selectivity to that achieved using integer order 1 (Fig. 13) Immunity to noise The gain ofthe symbolic operator offractional differentiation s n decreases by 6n db=oct for n ] 1; 0[ (low-pass ltering) whereas it increases by 6 db=oct for n = 1 (high-pass ltering). Thus, for a comparable selectivity of Figs. 12 and 13, a CRONE detector with an order between 1 and 0 can ensure a much better immunity to noise than an order 1 derivative and improves thus the selectivity/immunity-to-noise compromise. This is conrmed by the response ofan order 0:75 CRONE detector to a noise parabolic step (Fig. 14). The gure shows that the detector response peak determines the abscissa ofthe inexion point ofthe transition independently ofthe noise. The detector is therefore robust to noise (cf. Section 4.2.2). Fig. 14. (a) Noisy parabolic step, (b) normalized response ofthe order 0:75 CRONE detector. 6. Spatial characterisation of the crone detector 6.1. Impulse response of D n Fig. 15. Impulse response (causal) ofthe spatial operator D n for Re[n] R. f (n) (x), the fractional derivative of f(x), calculated with increasing x, can be written as D n f(x)=f(x) D n (x) (16) in which D n (x) represents the nth derivative for increasing x ofthe unitary Dirac s impulse (x), i.e. for Re[n] R (Fig. 15) x n 1 for x 0; D n (x)= ( n) 0 f or x 6 0: (17)

8 2428 B. Mathieu et al. / Signal Processing 83 (2003) Fig. 16. Impulse response (anticausal) ofthe spatial operator D n for Re[n] R. f (n) (x), the fractional derivative of f(x), calculated for decreasing x can be written as D n f(x)=f(x) D n (x) (18) in which D n (x) represents the nth derivative for decreasing x ofthe unitary Dirac s impulse (x), i.e. for Re[n] R (Fig. 16) 0 f or x 0; D n (x)= x n 1 (19) for x 0: ( n) The spatial operator D n, as dened in Section 4.2.2, is applied to transition f(x), i.e. D n f(x)=d n f(x) D n f(x) (20) or D n f(x)=f(x) D n (x) f(x) D n (x) (21) or, given the distributivity ofthe convolution product D n f(x)=f(x) (D n (x) D n (x)) (22) or, given, (20) D n f(x)=f(x) D n (x) (23) with D n (x)=d n (x) D n (x) (24) in which D n (x) is the impulse response ofthe spatial operator D n which denes the CRONE detector. Fig. 17. Impulse response (non causal) ofthe CRONE detector for Re[n] R. By including (17) and (19), in relation (24) the analytical expression of D n (x) forre[n] R is obtained D n (x)= x n 1 ( n) x n 1 ( n) or, more concisely for x 0; for x 0 (25) D n (x)= x n 1 sign (x): (26) ( n) Also, the combination offigs. 15 and 16 in accordance with relation (24) provides, still for Re[n] R, the representative graph ofthe impulse variations of the spatial operator D n (Fig. 17). This impulse response has two inherent properties ofany linear edge detector: it is odd, it tends to zero at ±. The rst permits the determination ofan absence ofan inexion point in an area ofconstant levels ofgrey. The latter ensures a reasonable enough level ofthe complexity ofthe calculation. We have now formulated the impulse response of D n with a condition on the derivative order n. The following deals with the more general formulation of D n with no condition on the derivative order n.

9 B. Mathieu et al. / Signal Processing 83 (2003) Formulation of D n f (1) (x), the rst derivative of f(x), calculated with increasing x can be written as D n f(x) f(x h) f(x)= h and f (1) (x) with decreasing x (27) D n f(x) f(x + h) f(x)= (28) h h being innitely small. The introduction ofshift operator q applicable to a concrete function and dened by qf(x)=f(x + h) or q 1 f(x)=f(x h) (29) gives q 1 f(x)=1 f(x) (30) h and D D f(x)=1 q f(x) (31) h and thus D = 1 q 1 (32) h and D = 1 q (33) h so, at order n: ( ) 1 q D n 1 n = (34) h and ( ) n 1 q D n = : (35) h Thus, the spatial operator D n dened by (20), i.e. D n = D n D n gives (36) D n = 1 h n [(1 q 1 ) n (1 q) n ] (37) and, by expanding respectively (1 q 1 ) n and (1 q) n using Newton s binomial formula D n = 1 h n k=0 k n(n 1) (n k +1) ( 1) k! (q k q k ): (38) The application of D n thus formulated at a transition f(x) gives D n f(x)= 1 h n where a k =( 1) k ( n k k=0 ) a k [f(x kh) f(x + kh)]; (39) k n(n 1) (n k +1) =( 1) : (40) k! 7. The CRONE detector in two dimensions 7.1. Extension of the CRONE detector to 2D In order to extract the contours ofan image, one can extend the CRONE detector, designed for one dimension, to two dimensions, then verify the preservation ofits properties. We shall therefore consider the vectorial operator with two independent components, respectively on x and y. Each component is provided by a single dimension CRONE detector where its dening D n operator is spatially truncated, as permitted in practice with unlimited series (38). If N N designates the dimension ofthe image, and 2m + 1 the size ofthe components ofthe mask, the calculation complexity ofeach component is mn 2 multiplications and (2m 1)N 2 additions. The mask corresponding to the horizontal component can be written as [+a m + a 1 0 a 1 a m ] (41)

10 2430 B. Mathieu et al. / Signal Processing 83 (2003) and for the vertical component +a m. +a 1 0 a 1. a m (42) the a k elements being the coecients ofthe Newton s binomial (40). Fig. 18. Non-noisy circular image with parabolic luminance transitions (top: 3D view; bottom: grey level view) Comparison of CRONE detector and Prewitt s gradient performances To verify whether the properties of selectivity and immunity to noise presented in Section 5.2 are maintained in 2D, a wide variety ofimage types need to be tested, including: articial images, on the one hand non-noisy and using a positive derivative order (fractional dierentiation), and on the other hand noisy, using a negative derivative order (fractional integration), real images, where positive or negative derivative orders are chosen depending on the type ofimage treated. Fig. 19. Non-noisy circular image with Gaussian luminance transitions (top: 3D view; bottom: grey level view) Articialimages Simulated images (100 2 pixels and 256 levels of grey) are circular and the luminance transitions are parabolic or Gaussian. Non-noisy images are given in Figs. 18 and 19, and noisy in Figs. 20 and 21. Performance evaluation can be carried out by analyzing the quality ofthe extracted contour, as compared to that obtained using Prewitt s gradient. For each type ofimage, the gures give both a 3D and a level ofgrey version for: luminance L=f(i; j), in which i and j are the pixel coordinates (1 i 100; 1 j 100), gradient image obtained with the Prewitt s gradient, gradient image obtained with the CRONE detector. Fig. 20. Noisy circular image with parabolic luminance transitions (top: 3D view; bottom: grey level view). For non-noisy images where a positive derivative order (between 1 and 2) is used, the gradient image in Fig. 18, obtained from parabolic transition images, displays better selectivity with the CRONE detector.

11 B. Mathieu et al. / Signal Processing 83 (2003) Fig. 22. Biomedical image: echography ofa kidney. Fig. 21. Noisy circular image with Gaussian luminance transitions (top: 3D view; bottom: grey level view). For the Gaussian transition image, Fig. 19, no selectivity dierence can be noted. This is to be expected as CRONE detector selectivity has no particular robustness to transition type. For noisy images where a negative derivative order (between 1 and 0) is used, the gradient images of Figs. 20 and 21 show the CRONE detector to have not only a much better immunity to noise, but also robustness ofthis immunity to transition type. As well as the qualitative analysis providing these comparative performances, the use of Peli and Malah s criteria [24] permits quantitative analysis [11] ofthese performances which are signicantly conrmed, notably through avoiding any visual subjectivity Realimages The real images considered are ofdiverse origins (biomedical and industrial) and are suciently noisy for processing to be signicant. They are taken from the CNRS Image Department data base. As for articial images, performance evaluation can be carried out by analysing the quality ofthe extracted contour, as compared to that obtained using Prewitt s gradient. Also, for each type of image, the gradient image is given for each of the two extraction techniques (Prewitt s and CRONE). The contours ofnoisy images in Figs. 22 and 23 are extracted using a derivative order between 1 and 0. The gradient images displays a much better immunity to noise ofthe CRONE detector. Fig. 23. Industrial image: X-ray revealing ssuring. 8. Conclusion In order to increase the selectivity ofthe edge detection for parabolic step-type luminance transition, a cusp is created on the inexion point abscissa. This is made possible by the synthesis ofa new detector, the CRONE detector, obtained though the generalization ofthe derivative order to include fractional order, and dened by relation (38). The particularity ofthe CRONE detector is that a derivative order between 1 and 2 favours detection selectivity, whereas a derivative order between 1 and 1 favours robustness of immunity to noise; with a selectivity comparable to that ofa rst derivative, the CRONE detector provides improved immunity to noise. The comparison ofits performance with respect to Prewitt s gradient was presented. Performance was evaluated using a wide variety ofimages: articial images with and without noise, and real images ofvarious origins. Performance levels obtained reveal in all cases a better immunity to noise ofthe CRONE detector. As prospects, next works consist in the comparison ofthe CRONE detector with respect to the others modern and robust edge detectors. References [1] M.F. Barnsley, L.P. Hurd, Fractal Image Compression, AK Peters Ltd, Wellesley, MA, 1993.

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