using the derivative of the Gaussian function as the wavelet func- tion. Based on their analysis, a simple and computationally effi- cient algorithm is proposed here. ‘Corner’ is a relative term. Corners are largely dependent on the shape of the object under consideration. If there are sharp comers at the boundary, smooth curvature changes will not be recognised as corners. On the other hand, if the shape consists only of smooth curvature changes then these curvature changes will be recognised as corner points. To provide such robustness, this algo- rithm starts with the normalisation of the wavelet transform mod- ulus maxima. This normalisation is obtained at each level with the global maxima of that level; it also enables the algorithm to still be suitable when another wavelet is to be used for analysis. This normalisation is carried out in step 1 of the algorithm. (i) zyxwvutsrqponm Step zyxwvutsrqponm 1: The orientation profile is decomposed using a wavelet transform at scales 2I, zyxwvutsrqponm 22, zyxwvutsrqpon 23 and 24. The wavelet transform modu- lus maximum (WTMM) is computed at each of these levels. These WTMMs at each level are normalised with respect to the maxi- mum peak at that level. (ii) Step 2: The events (valid corners) are observed at the highest scale (24). Peaks higher than some threshold 2, are recognised as valid events. These events are successively tracked at lower scales (2,, 22, 29 to find their exact locations. (iii) Step 3 Those events which are increasing at decreasing scales and are greater than some threshold zyxwvutsr T~ zyxwvutsrqpon (< at scale zyxwvuts 24 are also taken as valid events. These events are also successively tracked at lower scales (z3, 22, 2’) to find their exact locations. (iv) Step 4: Find large (greater than some threshold T~) events in the vicinity of previously detected events in the preceeding steps from the lowest scale (i.e. 29. Results and discussion: The result of comer detection using the proposed scheme is shown in Figs. 1 and 2, where q = 0.4, 22 = 0.1 and T, = 0.65. These values of threshold are found after testing the algorithm with a large number of images. We used similar images to those of other groups such as Asada and Brady [8] (Fig. la), Lee et al. [4] (Fig. lb) and Rattarangsi and Chin [9] (Fig. 24. Here we observe that the performance of the proposed technique is better than, or at least the same as, that presented in the litera- ture. I zyxwvutsrqponmlk C Fig. 3 Corner detection results under noise (a) zyxwvutsrqpo 0, = 2.82 (c) 0, = 3.85 (b) 0, = 3.59 As far as computational efficiency is concerned, the technique reported by Asada and Brady [8] is time consuming because it requires the computation of decompositions using both first and second derivatives of the Gaussian function. These derivatives are not orthogonal and no fast computing algorithm for discrete dyadic wavelet transforms exists. The technique presented in [4] is computationally complex because it requires taking the derivative of the ratio of wavelet transform modulus maxima and then split- ting the orientation profile to detect corners. The proposed tech- nique does not require these complex steps. Hence, the proposed technique is computationally efficient and simple to implement. We also performed corner detection tests under additive white Gaussian noise. Noise was added at the co-ordinates of the tracked boundary. We performed corner detection tests with dif- ferent standard deviations (0,) of Gaussian noise. These results are shown in Fig. 3. Conclusions: Here a wavelet based scheme is presented for the detection of comers in 2D planar curves. We observed that the proposed method gives competitive results with respect to the other available techniques. The proposed technique has the advan- tage that it is computationally inexpensive when compared with the other techniques. It also performs well under noise. Moreover, the proposed algorithm is robust with respect to object geometry and the type of wavelet used for the decomposition. zyx 0 IEE 1999 Electronics Letters Online No. 19990121 DOI: 10.1049/el:19990I21 A. Quddus and M.M. Fahmy (Department of Electrical Engineering, King Fahd University of Petroleum and Minerals. Dhahran 31261, Saudi Arabia) E-mail: aquddus@kfupm.edu.sa 8 December 1998 References ATTNEAVE, F.: ‘Some informational aspects of visual perception’, Psychological Rev., 1954, 61, (3), pp. 183-193 KOENDERINK, J.: ‘The structure of images’ (Springer Verlag, New York, 1984) LEE, J.S., SUN, Y.N., CHEN, c.H., and TSAI, C.T.: ‘Wavelet based corner detection’, Pattern Recogn., 1993, 26, (6), pp. 853-865 LEE, J.s., SUN, Y.N.,and CHEN, c.H.: ‘Multiscale corner detection by wavelet transform’, IEEE Trans., 1995, IP-4, (l), pp. 100-104 MALLAT, s.G., and ZHONG, s.: ‘Characterization of signals from multiscale edges’, IEEE Trans., 1992, PAMI-14, (7), pp. 710-732 MALLAT, s.G., and HWANG, w.L.: ‘Singularity detection and processing with wavelets’, IEEE Trans., 1992, IT-38, (21, pp. 617- 643 LEE, J.s., SUN, Y.N.,and CHEN, c.H.: ‘Wavelet transform for corner detection’. Proc. IEEE Int. Conf. Systems Engineering, Kobe, Japan, 1992, pp. 596-599 ASADA, H., and BRADY, M.: ‘The curvature primal sketch’, IEEE Trans., 1986, PAM-8, (l), pp. 2-14 RATTARANGSI, A., and CHIN, R.T.: ‘Scale-based detection of corners on planar curves’, ZEEE Trans., 1992, PAMI-14, (4), pp. 430449 Fingerprint feature extraction using Gabor filters Chih-Jen Lee and Sheng-De Wang A Gabor filter-based method for directly extracting fingerprint features from grey-level images without pre-processing is introduced. The proposed method is more ef€icient and suitable than conventional methods for a small-scale fingerprint recognition system. Experimental results show that the recognition rate of the k-nearest neighbour classifier using the proposed Gabor filter-based features is 97.2%. Introduction: Most methods for fingerprint identification use minutiae as the fingerprint features. The main steps for minutiae extraction are smoothing, local ridge orientation estimation, ridge extraction, thinning, and minutia detection [l]. For a small-scale fingerprint recognition system, however, it would not be efficient ELECTRONICS LETTERS 18th February 1999 Vol. 35 No. 4 288