L. Basano, B. Caprile, E. De Micheli, A. Geminiani, and P. Ottonello, "Edge-detection schemes highly suitable for hardware implementation," J. Opt. Soc. Am. A 5, 1170-1175 (1988)
Two schemes for edge detection of real images based on gradient maxima are presented. Images are filtered with narrow filters to increase localization. Experimental results and theoretical considerations suggest that the exact shape of the filter is not critical for good performance of the algorithm. Therefore a filter can be chosen to allow for a highly efficient hardware implementation, for example, a binary filter or a 4-bit finite-impulse-response filter. Because the digitized values of a binary filter are powers of 2, the hardware implementation does not require time-consuming computations, such as multiplication and time shift, but just appropriate addressings. The performance of this scheme, or a similar scheme using 4-bit filters, is as satisfactory as that of more sophisticated schemes. Therefore these low-cost schemes are likely to be more suitable for hardware implementation.
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Comparison between Edges Extracted as Maxima of Gradient from the Image of Fig. 1 Using a 12-Bit Gaussian Filter (σ = 1), a Deriche Filter (α = 2, ω = 0.01), a Binary Filter, and a 4-Bit Gaussian Filter (σ = 1) When No Threshold Was Seta
Gauss 12
Deriche
Binary
Gauss 4
Number of edges
51694
55679
56108
57691
Maximum slope
188
187
187
188
Mean slope
19.73
18.44
18.40
18.87
Rms
30.43
29.07
29.08
30.26
Coincident edges
—
43742 (79%)
44249 (79%)
43901 (76%)
Mean slope
—
22.01
21.78
22.03
Rms
—
32.16
32.02
32.09
Not-coincident edges (1)
—
7972 (15%)
7445 (14%)
7793 (15%)
Mean slope
—
7.25
7.55
6.79
Rms
—
12.23
12.67
12.03
Not-coincident edges (2)
—
11957 (21%)
11859 (21%)
13790 (24%)
Mean slope
—
6.51
6.71
6.27
Rms
—
10.81
11.19
11.63
Md
—
1.66 × 10−3
1.32 × 10−3
4.24 × 10−3
Rms
—
5.69 × 10−3
5.62 × 10−3
9.73 × 10−3
Coincident edges refer to the edges extracted with the 12-bit Gaussian filter that are present with the same spatial coordinates in the edge map obtained with the other filters. Not-concident edges (1) is the number of edges extracted with the 12-bit Gaussian filter but not with the other filters. Not-coincident edges (2) is the number of the edges not extracted with the 12-bit Gaussian filter but extracted with the other filters. For each quantity the mean value of gradient M and the related rms value are shown. The mean difference between two edge maps Md is defined as
where Mi1 are the gradient values of edges extracted with the 12-bit Gaussian filter, Mik refer to the gradient values of edges extracted with an other filter (k = 2, 3, 4), and i runs on the set of the coincident edges. The related root mean value is also shown.
Table 2
Comparison between Edges Extracted As Maxima of Gradient from the Image of Fig. 1 Using a 12-Bit Gaussian Filter (σ = 1), a Deriche Filter (α = 2, ω = 0.01), a Binary Filter, and a 4-Bit Gaussian Filter (σ = 1) by Setting An Absolute Threshold, Tmin = Tmax = 10 To Show the Dependence of the Edges on the Related Gradient Valuesa
Gauss 12
Deriche
Binary
Gauss 4
Number of edges
19518
19749
19837
20407
Maximum slope
188
187
187
188
Mean slope
45.51
44.43
44.45
45.65
Rms
37.13
36.44
36.54
38.36
Coincident edges
—
17880 (91%)
17886 (90%)
18240 (89%)
Mean slope
—
47.64
47.62
47.02
Rms
—
37.56
37.56
37.47
Not-coincident edges (1)
—
1638 (8%)
1632 (8%)
1278 (7%)
Mean slope
—
22.34
22.49
23.98
Rms
—
20.84
20.98
22.70
Not-coincident edges (2)
—
1869 (9%)
1951 (10%)
2167 (11%)
Mean slope
—
22.40
22.65
22.45
Rms
—
20.81
21.03
23.31
Md
—
4.81 × 10−3
4.19 × 10−3
7.35 × 10−3
Rms
—
6.95 × 10−3
7.08 × 10−3
1.39 × 10−2
See footnote a in Table 1 for a discussion of the values.
Tables (2)
Table 1
Comparison between Edges Extracted as Maxima of Gradient from the Image of Fig. 1 Using a 12-Bit Gaussian Filter (σ = 1), a Deriche Filter (α = 2, ω = 0.01), a Binary Filter, and a 4-Bit Gaussian Filter (σ = 1) When No Threshold Was Seta
Gauss 12
Deriche
Binary
Gauss 4
Number of edges
51694
55679
56108
57691
Maximum slope
188
187
187
188
Mean slope
19.73
18.44
18.40
18.87
Rms
30.43
29.07
29.08
30.26
Coincident edges
—
43742 (79%)
44249 (79%)
43901 (76%)
Mean slope
—
22.01
21.78
22.03
Rms
—
32.16
32.02
32.09
Not-coincident edges (1)
—
7972 (15%)
7445 (14%)
7793 (15%)
Mean slope
—
7.25
7.55
6.79
Rms
—
12.23
12.67
12.03
Not-coincident edges (2)
—
11957 (21%)
11859 (21%)
13790 (24%)
Mean slope
—
6.51
6.71
6.27
Rms
—
10.81
11.19
11.63
Md
—
1.66 × 10−3
1.32 × 10−3
4.24 × 10−3
Rms
—
5.69 × 10−3
5.62 × 10−3
9.73 × 10−3
Coincident edges refer to the edges extracted with the 12-bit Gaussian filter that are present with the same spatial coordinates in the edge map obtained with the other filters. Not-concident edges (1) is the number of edges extracted with the 12-bit Gaussian filter but not with the other filters. Not-coincident edges (2) is the number of the edges not extracted with the 12-bit Gaussian filter but extracted with the other filters. For each quantity the mean value of gradient M and the related rms value are shown. The mean difference between two edge maps Md is defined as
where Mi1 are the gradient values of edges extracted with the 12-bit Gaussian filter, Mik refer to the gradient values of edges extracted with an other filter (k = 2, 3, 4), and i runs on the set of the coincident edges. The related root mean value is also shown.
Table 2
Comparison between Edges Extracted As Maxima of Gradient from the Image of Fig. 1 Using a 12-Bit Gaussian Filter (σ = 1), a Deriche Filter (α = 2, ω = 0.01), a Binary Filter, and a 4-Bit Gaussian Filter (σ = 1) by Setting An Absolute Threshold, Tmin = Tmax = 10 To Show the Dependence of the Edges on the Related Gradient Valuesa
Gauss 12
Deriche
Binary
Gauss 4
Number of edges
19518
19749
19837
20407
Maximum slope
188
187
187
188
Mean slope
45.51
44.43
44.45
45.65
Rms
37.13
36.44
36.54
38.36
Coincident edges
—
17880 (91%)
17886 (90%)
18240 (89%)
Mean slope
—
47.64
47.62
47.02
Rms
—
37.56
37.56
37.47
Not-coincident edges (1)
—
1638 (8%)
1632 (8%)
1278 (7%)
Mean slope
—
22.34
22.49
23.98
Rms
—
20.84
20.98
22.70
Not-coincident edges (2)
—
1869 (9%)
1951 (10%)
2167 (11%)
Mean slope
—
22.40
22.65
22.45
Rms
—
20.81
21.03
23.31
Md
—
4.81 × 10−3
4.19 × 10−3
7.35 × 10−3
Rms
—
6.95 × 10−3
7.08 × 10−3
1.39 × 10−2
See footnote a in Table 1 for a discussion of the values.
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