f
W f
2
W f
4
Mallat and Zhong (1992).
Figure 6.2: Evolution of Coefficients Across Scales: Wavelet coefficients gen-erated by sharp peaks (middle right), which usually correspond to speckle noise, decrease in amplitude while resolution is decreasing, whereas the coefficients’ am-plitude is constant for step edges (middle left), or increasing for smooth edges (outer left and right).
6.3 Edge Validation Across Scales
Edges on multiple scales are thus detected by filtering the image with low-pass filters of different sizes followed by an application of the gradient operator. On each scale, the application of the low-pass filter reduces noise, but it also changes the location of edges due to a blurring effect. This change in location has to be taken into account when information about edges is combined across scales. For a finite number of scales s < s0, gradient extrema corresponding to the same edge may be found in a certain neighborhood at the adjacent scale. According to Mal-lat and Hwang’s (1992) theory the wavelet coefficients of the TIWT generated by speckle noise (i.e. sharp narrow peaks) decrease and finally disappear from higher to lower resolution scales whereas step edges or smoother edges generate constant or increasing wavelet coefficients3. Figure 6.2 sketches the evolution of wavelet coefficients for the most frequent edge types encountered in images. However, in 3D and in higher dimensional spaces the search space grows exponentially with the dimension of the space and the uncertainty of existing correspondences between spatially close events found on neighboring scales increases, such that the tracing across more than two scales of modulus maxima becomes a computationally inten-sive task. Transposing Mallat’s (1997) theory about the evolution of coefficients along scales to discrete scale space, and assuming additionally that the gradient di-rections of coefficients corresponding to the same edge are preserved across scales, a new edge validation method is developed. This method strongly enhances edges
3For an exact description of the coefficients’ evolution see Appendix D.2
66 Edge Detection
belonging to object boundaries via a confidence measureG. G assigns high values to edge points which can be found on two or more neighboring scales and small values to edge points which lack those correspondences. The measure G is based on the following assumptions about edges which result from object boundaries:
1. Edges are present in the primary edge images Es1 and Es2 of at least two neighboring scales, s2 = 2±1s1.
2. The corresponding edge locations t and q are spatially close.
3. The corresponding edge points have similar gradient directions4, i.e. γs1(t) γs2(q).
Since scale discretization into octave bands is used, the spatial drift of correspond-ing edges between two adjacent scales is large, such that the correspondences cannot be determined uniquely and only adjacent scales can be used for edge vali-dation. Therefore a measureC is established to determine the consistency degree between two nearby edges belonging to adjacent scales (see Rehse et al. (1996) for a 2D version):
C(t, q, s1, s2) = max[Ms1(t), Ms2(q)]Ms1(t)Es1(t) max
u∈Vt R(u, s1, s2), (6.6) where
R(q, s1, s2) = Ms2(q)Es2(q)
maxu∈B(Ms2(u)Es2(u))e−t−q →γTs
1 (t) →γs
2 (q), and (6.7) q = arg max
u∈Vt R(u, s1, s2), (6.8)
where Ms1(t), Ms2(q) are the absolute values of the gradient at scales s1, s2 and locations t, q (cf. Eq. 6.3); B is the whole image and Vt is the neighborhood considered at scales2 for every locationtat scales1. The size of the neighborhood Vt depends on the spatial extent of the low-pass and high-pass filters used for the wavelet transform and is derived in Appendix D.3.
In Eq. 6.6, Assumption 1 is implemented by the terms Es1 and Es2, which are zero, if no edge is present, Assumption 2 is implemented by the exponential term inR, and Assumption 3 is implemented by the scalar product betweenγs1(t) and γs2(q) in R.
Due to the good approximation of Gauss curves by Splines (see Figure B.1 in Appendix B.3 for a comparison), results from scale-space theory can be applied here as well. According to scale-space theory (Lindeberg, 1998) gradients corre-sponding to boundary edges are largest at scales which correspond to the extent of the blurring of an edge (cf. to the diffusion equation). The characterization of the
4The consideration of gradient directions as well as of their orientations for across-scales validation is a novelty. Until now gradient direction was used only for the detection of extrema or for further differential computations on the same scale (Elder and Zucker, 1998). For the 2D case a similar approach was done by Rehse et al. (1996), who considered gradient directions but not their orientation.
6.3 Edge Validation Across Scales 67
edge at its most representative scale is accomplished inC by a factor proportional to the absolute value of the maximum of the gradients at the two scales s1 and s2. Depending on the object sizes, their contrast levels, and the noise level, edges reflect the analyzed objects either at the higher, at the lower, or on both resolution scales. Since the consistency measure C is asymmetric w.r.t. the scales s1 and s2, there are three ways to define the measureG(t) to assign the confidence value for an edge, upon which the boundary edges are separated from noise in a later step:
Algorithm 6.1 (High to Low Edge Validation) evaluation of C from high to low resolution (i.e. preference of higher resolution edge locations):
G(t) =C(t, q, s1, s2), if s1 < s2 (6.9) Algorithm 6.2 (Low to High Edge Validation) evaluation of C from low to high resolution (i.e. preference of lower resolution edge locations):
G(t) =C(t, q, s1, s2), if s1 > s2 (6.10) Algorithm 6.3 (Symmetric Edge Validation) symmetric evaluation of C (for an adaptive choice of edge locations from both resolution levels):
G(t) = C(t, q, s1, s2), if Ms1(t)> Ms2(q), ∀t s.t. Es1(t) = 1, G(q) = C(t, q, s1, s2), if Ms1(t)< Ms2(q), ∀t s.t. Es1(t) = 1,
G(t) = C(t, q, s2, s1), if Ms2(t)> Ms1(q), ∀t s.t. Es1(t) = 0, (6.11) G(q) = C(t, q, s2, s1), if Ms2(t)< Ms1(q), ∀t s.t. Es1(t) = 0,
with s1 > s2.
For the “symmetric” method the location of the edge at the scale for which the absolute value of the gradient is largest is used for its placement in the boundary edge image and therefore Algorithm 6.3 automatically selects the scale at which a feature is present. Algorithm 6.3 is illustrated in Figure 6.3.
The choice of the method depends on the analyzed data. Algorithm 6.1 (Fig-ure 6.8.a) reliably detects small struct(Fig-ures, for example the terminal branches.
Since small neuronal structures cannot be distinguished from noise, it cannot be filtered optimally by this method. Algorithm 6.2 (Figure 6.8.b) finds wider struc-tures and therefore suppresses more noise, but small strucstruc-tures may also be lost.
Algorithm 6.3 (Figure 6.8.c) can fill in gaps left by Algorithm 6.2 and provides re-sults with less noise than those given by Algorithm 6.1, being thus a good compro-mise between the Algorithms 6.1 and 6.2. However, since Algorithm 6.3 completes boundaries detected at low resolution with points taken from the higher resolution at their respective locations, global edges are more fringed than those computed by Algorithms 6.1 and 6.2.
The computational complexity of these methods is O(n ·Vt3), where n is the number of edge points in the image and Vt is the lateral length (in pixels) of the considered neighborhood cube (Vt is illustrated in 1D in Figure 6.8)
68 Edge Detection
Boundary edge image Primary edge image
Primary edge image Scale s2
Scale s = 2s1 2 Vt>
Vt>
Figure 6.3: Edge Validation Across-Scales - Symmetric Algorithm: The figure shows two primary edge images at adjacent scaless2 ands1 (top and center, s1 > s2) and the corresponding boundary edge image of validated edges (bottom).
The location of edges is indicated by vertical bars, where the height indicates the absolute valueMsof the gradient. For every edge point at the low resolution image s1 we search for the edge point with the highest absolute valueMs2 of the gradient in a neighborhoodVt within the high resolution images2. Left: if Ms2 > Ms1 the edge is placed at the location of the edge point in scales2. Center: if Ms2 < Ms1 the edge is placed at the location of the edge point in scale s1. Right: If there is no edge point in the low resolution scale, but there is an edge point at the same position in the high resolution scale, then the correspondence in the lower resolution scale is looked up.
After the enhancement of boundary edges by the validation measure, noisy edges can be reliably eliminated by a global threshold operation. The behavior of the validation measure is discussed in the next section on several datasets.