Optimal Ramp Edge Detection Using Expansion Matching - IEEE Xplore

IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 18, NO. 11, NOVEMBER 1996

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Optimal Ramp Edge Detection Using Expansion Matching Zhiqian Wang, K. Raghunath Rao, and Jezekiel Ben-Arie Abstract-In practical images, ideal step edges are actually transformed into ramp edges, due to the general low pass filtering nature of imaging systems. This paper discusses the application of the recently developed Expansion Matching (EXM) method for optimal ramp edge detection. EXM optimizes a novel matching criterion called Discriminative Signal-to-Noise Ralio (DSNR)and has been shown to robustly recognize templates under conditions of noise, severe occlusion, and superposition. We show that our ramp edge detector performs better than the ramp detector obtained from Canny’s criteria in terms of DSNR and is relatively easier to derive for various noise levels and slopes. Index Terms-Canny edge detector, edge detection, optimal filters; ramp edges, step edges, Expansion Matching (EXM), Discriminative Signal-to-Noise Ratio (DSNR).

+ 1 INTRODUCTION DUE to camera frequency characteristics and other optical effects, ramp edges occur more frequently in real images than step edges [SI. In this paper, we develop a novel ramp edge detector based on the Expansion Matching (EXM) method for template recognition which optimizes the Discriminative Signal-to-Noise Ratio (DSNR) criterion 121, [31, 1101. DSNR emphasizes sharpness of the matching peak and penalizes any off-center responses. We compare our ramp edge detector with a recently developed ramp edge detector [8] which optimizes the three criteria originally formulated by Canny [4]: good signal-to-noise ratio, good localization, and maximum suppression of false responses. We find our ramp edge detector to perform better under a wide range of noise levels. A general survey of edge detection algorithms can be found in 161. Of the many works which have been performed in obtaining an optimal step edge detection operator, the most prominent is the work of Canny [4] who showed that the ideal operator that maximizes the conventional Signal-to-Noise Ratio (SNR) in detecting a particular edge, is correlation with the same edge model itself. However, this detection is not well localized and requires an additional localization criterion. A third criterion that suppresses multiple responses is also included, and numerical optimization results in the desired edge detector, which can be approximated by the Derivative of a Gaussian. While Canny [4] worked with finite extent filters, Deriche I51 used the same approach with infinite extent filters, with the objective of obtaining an efficient recursive implementation. By introducing a more appropriate term for the width of an IIR filter,

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Z . Wang and K.R. Rao O Y P with the Elrctricai and Computer Engineering Department, Illinois Irrstitutc~of Trchnology, Chicngo,I L 60616. E-mail: zwan~@~~ecs.~tic.cdu. ,/. Ben-Arie is with tlir Electrical Engineering and Coniputer Sciencr Depiirtment, University of Illinois at Chicago, Chicago, I L 60607. E-mnil: benarie@eecs,uic.edu.

Manuscript received J a n . 16, 1995; revised Feb. 5, 1996. Recommended far acceptance by K.Royer. For information on obtaining reprints of this article, please send e-mail to: [email protected]~, and reference I E E E C S Log Nuinher P96019.

Sarkar and Boyer [I] modified the low multiple response criterion and derived optimal IIR step edge detection filters based on those three criteria. While the above works jointly optimized SNR and localization and constrained multiple responses, Spacek 1131 combined all three criteria to form a performance measure for the edge detector and derived a more accomplished form of the filter equation. Another kind of optimal step edge detector can be derived [lo] from our newly developed Expansion Matching (EXM) method [2],131. EXM is a technique for robust recognition of templates in an image. The fundamental approach here is to match a given template with a given image by expanding the image signal in terms of nonorthogonal Basis Functions (BFs) which are all translated versions of the template. The expansion coefficients obtained at a particular location signify the presence of a template-similar signal (pattern) at that location. Since in practice, the shifted templates form a complete nonorthogonal basis 121, this entails a nonorthogonal expansion which can be quite complex if performed directly. However, since all the bases are shifted versions of the same function, this task can be significantly simplified by using frequency domain techniques [2], [31. EXM actually maximizes a matching quality criterion called Discriminative Signal-to-Noise Ratio (DSNR). In order to define DSNR, consider an M-point 1D signal s(x), in which it is desired to recognize a template Hx) by convolving the signal s(x) with a filter to yield a result c(x), i.e., c(x) = s(x) * We assume that the signal s(x) contains the template as a subimage at location I with additive noise, i.e., s ( x ) = w(x I ) + A(x) . The DSNR is defined by the ratio of the power of the desired peak vs. the total power of the undesired off-center response in the filter output:

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Thus, high DSNR implies a large matching peak at the template center I with small response elsewhere in the output c(x).The ideal response quested (with infinite DSNR) is a delta function, i.e.,c(x)= 8x - I). DSNR is more relevant to matching than the traditional SNR since it considers as “noise,” not only the filter’s response to the noise, but also the filter’s off-center response to the pattern. In contrast, the widely used correlation matching (also known as matched filtering) maximizes the SNR, which is defined as:

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where the filter output c(x) is split into two parts: cdx) = y(x - 2) * Bx) which is the filter’s response to the shifted template in the input signal, and cn(x) = A(x) * which is the filter’s response to the additive noise. The SNR approach overlooks the filter’s off-center response to the pattern, which can be quite substantial. It is found 121, [3]that this negligence results in filters which respond to their matched patterns with broad and ”unsharp” peaks which are quite hard to detect. In comparison, EXM yields sharp peaks since the ideal response quested is a delta function. EXM is found to be superior to the conventional correlation approach (matched filtering), especially in conditions of severe noise, occlusion, and superposition. We have shown 121 that maximization of the DSNR criterion of (1)yields the EXM filter in frequency domain as:

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IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 18, NO. 11, NOVEMBER 1996

(3)

where S,,(o)= Y(w)F(o),Y ( w ) is the Fourier transform of the template, S ~ l ( wis) the power spectrum of the additive noise, and the bar symbol represents complex conjugation. One can recognize this result as the Wiener MSE restoration filter with the template as the blurring function. In fact, we have shown in [21, [31 that nonorthogonal expansion with specific set of dense self-similar BFs is exactly implemented by MSE restoration. Recently, EXM has also been extended to the recognition of multiple-templates with a single filter [ 3 ] . By optimizing a weighted sum of DSNR over the set of template-similar patterns to be recognized, we have realized a single filter that elicits any userspecifiable response from each of these patterns. This paradigm has successfully been used for generic face recognition and corner extraction [15]. Also the EXM in conjunction with K-L transform can be used as generic feature extractor [16] and in general can be used to design distortion invariant filters. It is reasonable to consider edges as particular patterns existing in images. Thus, we can cast edge detection as a pattern matching problem, with the pattern to be detected as the edge model. Since EXM is an optimal DSNR template matching scheme, we can easily design optimal DSNR edge detectors using this paradigm. In a previous work, two of the authors have used the above paradigm to design an optimal-DSNR step edge detector [lo]. Experimental results comparing the EXM-based edge detector (called the Step Expansion Filter or SEF) to the widely used Canny Edge Detector (CED) show that the SEF outperforms the CED by about 5 dB DSNR even at extreme noise levels [lo]. Experimental results on 2D images [lo] show that when compared to the CED over a wide range of input noise levels, the SEF consistently yields better Pratt’s Figure of Merit (FOM) [9], fewer spurious edge elements, less blurring of detail at high noise levels, and less sensitivity of the design parameter with respect to variation of the input noise. The EXM filter also has other advantages. Other methods follow the approach of defining a specific set of criteria for the given edge model, and optimizing them-in most cases using numerical methods. In contrast, the EXM approach is analytical and generalized so that it easily yields the optimal DSNR detectors for any desired edge model and can also easily incorporate colored noise models. In this paper, we use the EXM approach to design an optimalDSNR edge detector for ramp edges. Although the basic principle used h’ere(EXM-based edge detection) is the same as the previous work in [IO], the edge model is more generalized in this paper. In fact, as the slope of the ramp edge model tends to infinity, we obtain the step edge model, and the result of this paper reverts to the special case considered in [lo]. Thus far, most optimal edge detection work concerns ideal step edges in white Gaussian noise. Unfortunately, practical image edges are never ideal steps. Even if the edges in scene were originally in the form of ideal steps, during the process of image capture and digitization the ideal steps turn out to be ramp edges. This is because any practical imaging system has a finite bandwidth and therefore will behave approximately as a low pass filter, blurring the edges [6], [71. Petrou and Kittler [81 extended Spacek’s work by applying it to ramp edges and derived optimal edge detectors for ramp edges. They showed that their edge detectors are superior to optimal edge detectors for ideal step edges when used in real image edge detection. This is the only work that we find on optimal ramp edge detection. As mentioned above, the EXM approach can be applied to any edge model since the DSNR criterion is consistent with the desired properties of edge detection. In this paper we derive an

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optimal ramp edge detector based on the DSNR criterion and compare our results with performance of the edge detector of Petrou and Kittler. Our optimal ramp edge detectors are much easier to obtain and perform better in terms of DSNR.

2 OPTIMALDSNR RAMPEDGEDETECTION In order to obtain the optimal DSNR ramp edge detector, it is first necessary to set up a model for the edge. In their work, Petrou and Kittler modeled the edge profile by the following function

1 - e? / 2 if x 2 0 esx / 2 otherwise

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It is quite reasonable to choose this function as the edge model in a practical image [8]. For a particular imaging system, all the step edges can be successfully modeled by (4) with the same value of s, which implicitly represents the low-pass effect or bandwidth limitation of the imaging system. Here, we modify the model slightly and use the bipolar (zero DC) form as our edge model. The bipolar model is more convenient for analysis than (4) and also is relevant to image edges even though images usually have only positive values. Thus, we obtain as our edge model: (5)

This ramp edge model has a Fourier transform given by -

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Using this edge model with (3), the Ramp Expansion Filter (REF) for this template can be analytically derived. We assume il (x)to be white noise with variance 0 : . The Ramp Expansion Filter (REF) in the frequency domain is then given by :

From (7), we can see that it is not easy to derive an analytical close form of the filter in the spatial domain. However, we can implement the filter directly in the frequency domain or obtain the filter in the spatial domain numerically in a straightforwardmanner. On obtaining the optimal Petrou-Kittler Ramp Edge Detector (PKRED) [8], Petrou and Kittler combined all three of Canny’s criteria into one performance measure, used the method of variational calculus to obtain a differential equation, and simplified the differential equation to yield the optimal filter. Under some assumptions on the form of the optimal edge detector, they obtained the close form of the solution to that differential equation which depends on four independent parameters. Numerical methods were used to maximize the performance measure with respect to these parameters. The PKRED obtained is a finite support edge detector, and thus does not lead itself to efficient IIR implementation, as does the REF. The REF impulse responses (solid line) are illustrated in Fig. 1 and Fig. 2 along with the PKRED impulse responses (dashed line). These two groups of filters are chosen such that they have the maximal DSNR performances when applied to ramp edge with s = 1 immersed in white Gaussian noise with 0: = 0.1 and 0: = 0.2, respectively. It can be seen that the REF edge detectors and the PKRED edge detectors are somewhat similar in profile. The main differences between these two kinds of edge detectors is that the REF is steeper around the center and smoother on the side-lobe. The response of the REF edge detector to a comparable PKRED

IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 18, NO. 11, NOVEMBER 1996

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can be expected to be sharper, due to the sharper slope at the origin and smoother side-lobe. Also we find that as the noise level increases, the width of REF grows slightly wider than the PKRED, so the REF could be more noise resistant. A noisy ramp edge is shown in Fig. 3. The enlarged profile of the noisy ramp edge around the center is shown in Fig. 4. The responses to the noisy ramp edge of both detectors in Fig. 2 are shown in Fig. 5. From Fig. 5, we can see that the REF response (solid line) is sharper and the PKRED response (dashed line) has less multiple responses around the edge. However the DSNR for REF (17.62dB) is slightly better than the DSNR for PKRED (17.43dB). 1

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Fig, 1 , The REF filter and the PKRED filter for a ramp edge with s = 1 , that yield best possible DSNR with additive Gaussian noise of variance 0:

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Fig. 2. The REF filter and the PKRED filter for a ramp edge with s = 1 , that yield best possible DSNR with additive Gaussian noise of variance a: = 0.2.

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We compare both the methods in terms of DSNR over a wide range of noise levels and different ramp characteristic slope s. Fig. 6 shows the DSNR evaluation for characteristic slope s = 1. These experiments reveal that the REF consistently performs slightly better than PKRED in terms of DSNR. We should mention that for each s and each noise level, the REF is redesigned (as is the PKRED)to obtain the maximum possible DSNR. Since the location of the ramp edge is known, the experimental DSNR is calculated on the output of the edge detector using (1)where I is the location of the step ramp edge. It is worthwhile to note here that maximizing the SNR alone (the matched filter approach) would yield an edge detector identical to the edge model assumed-in our case a ramp function. On the other hand, the PKRED simultaneously optimizes SNR and localization, while minimizing multiple responses. In our approach, maximizing the DSNR criterion enhances the peak desired response and improves the localization. Furthermore, responses are wppressed to an extent since the DSNR penalizes all offcenter responses as noise. Thus, the basic objectives of the two approaches are not altogether different. High DSNR implies that the peak is in the right location, which conforms to Canny's localization criterion. High SNR in Canny's formulation is akin to high DSNR since the DSNR also penalizes the noisy response of the filter output with respect to the peak desired output. Furthermore, high DSNR also means that the peak response is large in amplitude with respect to all off-center responses, not only those due to the additive noise. This is basically what Canny's multiple response criterion demands. It is therefore justifiable to compare the PKRED with the REF, since the objectives of both the approaches are baisically the same even though the actual criteria optimized are different. Furthermore, since we compare the methods in terms of DSNR, for a given amount of input noise we adjust the PKRED parameters to yield best-possible DSNR, and thus it is fair to compare the optimal DSNR REF with the corresponding PKRED. In addition, in the following section, we compare the actual 2D edge detection results of the two methods, which is a direct measure of their relative performance. 24

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sponses to get the gradient magnitude and the direction of the ramp edge. Petrou and Kittler extended the PKRED to two dimensions as follows: first integrate the PKRED in one dimension to derive a smoothing filter, then rotate the 1D smoothing filter to obtain a 2D filter, once the image has been convolved with the 2D smoothing filter, the edges can be detected by computing the derivative along two orthogonal directions [8]. However, it is not easy to obtain an analytical form of REF in spatial domain and thus the integration cannot be obtained exactly. Sarker and Boyer [11] accomplished the extension to 2D by applying their optimal step edge detector in one direction and a projection function in the orthogonal direction. They derived two orthogonal 2D edge detectors respectively for horizontal and vertical directions. The projection function is chosen as the integral of the edge detector. For the 2D REF, we use this extension method which is easier to implement. The projection function is mainly for noise suppression, so the requirement for the projection function is not very strict. As in [ll],the projection function we use is the numerical integral of the ramp edge detector. We compare the REF and the PKRED using a synthetic test image. The noiseless image is shown in Fig. 8, which is obtained by convolving the binary image shown in Fig. 7 with a ramp edge generating kernel [81. The ramp edge in Fig. 8 has a characteristic slope of s = 1 . Various levels of Gaussian noise are added to the blurred image. The SNR of the noisy image in Fig. 9 is 10 dB, and the image in Fig. 12 has SNR of 0 dE.

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3 TWO-DIMENSIONAL IMPLEMENTATION One possible way of extending the 1D results of the previous section to 2D is to design a set of 2D detectors, one for every rotated ramp edge. However, it is sufficient to design two edge detectors for ramp edges in orthogonal directions, and combine these re-

Fig. 9. The noisy synthetic image with SNR = 10 dB.

We adjust the parameters of the REF and PKRED for these two noisy images such that they obtain the best performance in terms of DSNR. The edge map detected by the REF from the noisy image (SNR = 10 dB) in Fig. 9 is shown in Fig. 10. Fig. 11 shows the edge map detected by the PKRED from the same noisy image. For the noisy image in Fig. 12 (SNR = 0 dN), the edge map for the REF is shown in Fig. 13 and the edge map for the PKRED in Fig. 14.

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Fig. 10. The edge detected by the REF from the noisy image in Fig. 9 (SNR = 10 dB).

Fig. 12. The noisy synthetic image with SNR = 0 dB

In our experiments, all the images are 128 x 128 in size. The two dimensional edge detectors are implemented using 512-point FFTs to avoid wrap-around errors. The REF is truncated such that the discontinuity at the truncation is less than 0.1% of the peak amplitude and the PKRED masks are all 13 x 13 pixels. We calculate the gradient magnitude at each pixel from the output of two orthogonal 2D edge detectors, and knowing the edges of the synthetic image, the DSNR performance for the outputs of both filters is computed from (11, and given in Table 1. In order to obtain a binarized edge map, a conservative threshold is applied to the gradient magnitude, and all remaining points are suppressed except those that are local maxima in a 3 x 3 neighborhood. Pratt's Figure of Merit (FOM) 191 is used to evaluate the quality of the edge detection and is given in Table 2. Such a simple postprocessing is intentionally used, since we are interested in comparing the intrinsic performance of the two edge detector methods. In general, improved edge maps can be expected by optimizing the postprocessing for each method.

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Fig. 13. The edge detected by the REF from the noisy image in Fig. 12 (SNR = 0 dB).

Fig. 11. The edge detected by the PKRED from the noisy image in Fig. 9 (SNR = 10 dB). Note the more rounded features compared to Fig. 10.

From the DSNR and FOM comparisons in Table 1 and Table 2, respectively, we can see that the REF performs slightly better than the PKRED. In the edge maps, the REF introduces less distortion and less break points, while the PKRED has a tendency to smooth out sharp features and also has larger gaps in the final edge traces (see the corners in Fig. 10, Fig. 11, Fig. 13, and Fig. 14). An exception is the concave triangular feature which seems to be more accurately captured in the PKRED output (Fig. 14) than in the REF output (Fig. 13), although slightly rounded and with missing gaps. In fact, the edge map of the REF in Fig. 13 is not rounded and has many edge elements along the actual boundary, but there are many spurious elements in the vicinity. We attribute this to the simple postprocessing we use, which is more suitable for smoother outputs like the PKRED, rather than the REF, which is, by design, optimally sharp. We also apply the REF and the PKRED to the widely used "Lena" test image. We do not perform tests over various levels of additive noise, since we are only subjectively comparing the relative performance of the two methods on a real image. We practically measure the characteristic slopes of edges in "Lena" and find that the characteristic slope distribution is around 1, so both filters are designed with characteristic slope of s = 1. The results are shown in Fig. 15 and Fig. 16. From the edge maps we can see that although both filters have almost the same performance, the REF maintains more accurate details, especially around the eyes and the nose.

4 CONCLUSION We find that the only previous work done on ramp edge detection is that of Petrou and Kittler 181. In their paper, they determine that the PKRED performs better than other optimal step edge detectors when applied to real images. The EXM method applied in this paper is a powerful method in pattern recognition. The DSNR criterion maximized by EXM emphasizes the sharpness of the matching peak and penalizes any off-center response. We regard edges in images as patterns and can derive an optimal DSNR edge detector for any edge model. The implementation of EXM is straight forward and easy. When EXM is used for ramp edge detection, it generates an optimal ramp edge detector which performs marginally better in terms of DSNR than the ramp edge detector that Petrou and Kittler derived from Canny's criteria. Experiments with 2D edge maps also show that the REF introduces less distortions and yields higher figures of merit as well.

Fig. 14. The edge detected by the PKRED from the noisy image in Fig. 12 (SNR = 0 dB).

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Fig. 15 The edge map detected by the REF from image “Lena” using characteristic slope s = 1. In comparison to Fig. 16, the details are more complete and accurate.

ACKNOWLEDGMENT This work was supported hy the Advanced Research Projects Agency under ARPA/ONR Grant No. N00014-93-1-1088.

REFEIRENCES J. Ben-Arie, “Multi-Dimensional Linear Lattice for Fourier and Gabor Transforms, Multiple-Scale Gaussian Filtering, and Edge Dmetection,” Neural Networks for Human and Machine Perception, H. Wechsler, ed., chapter 112, pp. 214-233. Academic Press, 1992. J. Ben-Arie and K.R. Rao, ”A Novel Approach for Template Matching by Non-Orthogonal Image Expansion,” IEEE Trans. Circuits arid Systems for Video TechnologLJ, vol. 3, no. 1, pp. 71-84, Fcb. 1993. K.R. Rao and J. Ben-Arie, ”Multiple Template Matching Using Expansion Matching,” IEEE Trans. Circuits and Systems for Video Tdznology, vol. 4, no. 5, pp. 490-503, Oct. 1994. J. Canny, ”A Computational Approach to Edge Detection,“ IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 8, no. 6, pp. 679697, Nov. 1986. R. Deriche, ”Using Canny’s Criteria to Derive a Recursively Implemented Optimal Edge Detector,” Int‘l J. Computer Vision, pp. 167187, 1987. D. Lee, “Edge Detection, Classification and Measurement,” Proc. IEEE CS Conf. Computer Vision and Pattern Recognition, p. 2, June 1989. V.S. Nalwa and T.O. Binford, “On Detecting Edges,” TEE€ Trans. Pattern Analysis aid Machiize Intelligence, vol. 8, no. 6, pp. 699-714, June 1986.

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Fig. 16. The edge map detected by the PKRED from image “Lena” using characteristic slope s = 1.

M. Petrou and J. Kittler, ”Optimal Edge Detectors for Ramp Edges,“ IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 13, no. 5, pp. 483-495, May 1991. [YI W.K. Pratt, Digital Image Processing. John Wiley and Sons, 1978. 1101 K.R. Rao and J. Ben-Arie, ”Optimal Edge Detection Using Expansion Matching and Restoration,” IEEE Trans. Pattern Analvsis and Machine Intellygence, vol. 13, no. 12, pp. 1,169-1,182, Dec. 1944. [ll] S. Sarkar and K.L. Boyer, ”On Optimal Infinite Impulse Response Edge Detection Filters,” I E E E Trans. Pattern Analysis and Machine Infelligenre, vol. 13, no 11, pp. l,154-1,17l, Nov. 1991. L121 S. Sarkar and K.L. Boyer, ”Optimal Infinite Impulse Response Zero Crossing Based Edge Detectors,” Computer Vision, Graphics, arid lrnage Processing: Image Understanding, vol. 54, pp. 224-243, Sept. 1991. 1131 L.A. Spacek, ”Edge Detection and Motion Detection,” Image Vision Computing, vol. 4, p. 43, 1986. [141 J. Ben-Arie and K.R. Rao, “Nonorthogonal Representation of Signals by Gaussians and Gabor Functions,” IEEE Trans. Circuits and Systems, vol. 42, no. 6, pp. 402-413, June 1995. [151 K.R. Rao and J. Ben-Arie, “Generic Face Recognition, Feature Extraction and Edge Detection Using Optimal DSR Expansion Matching,” Proc. I E E E Int’l Symp. Circuits and Systems, pp. 547550, Chicago, May 1993. 1161 D. Nandy, Z.Q. Wang, J. Ben-Arie, K.R. Rao, and N. Jojic, ”A Generalized Feature Extractor using Expansion Matching and the Karhunen-Loeuoe Transform,” Proc. ICASSP-96, Feb. 1996, to appear. 181