The multiscale medial properties of interfering image structures

Abstract

The multiscale medial axis (MMA) is a robust shape representation in analysing objects in greylevel images. It is computed as ridges in the medialness scale-space of an image. In this paper the multiscale medial properties of overlapping objects are investigated, the result of which is novel to the community in that the existing studies only concentrated on isolated objects. To improve the distinguishability of these objects, a sliding window algorithm is proposed to detect locally optimal scale ridges in scale space. When used in the medialness scale-space of an image, this efficient method can extract a complete set of MMAs for interfering objects, in contrast with the globally optimal scale ridge definition which has been widely used. This is demonstrated in the applications in medical images.

Introduction

The success of many computer vision algorithms depends on the ability to adequately represent the shapes of objects in a scene. One of the most famous shape representations is the medial axis transform (MAT) (Blum, 1967), which represents an object by the locus of centres of maximal disks inscribed within the object boundary, together with the radii of these disks. The MAT provides a direct encoding of global properties of object shape, such as overall orientation and end-to-end length, and of local properties, such as edge orientation and curvature (Blum and Nagel, 1978). Another attractive property of the MAT is that the branching structure of an object is reflected by the branching of the axes. This yields a natural correspondence between components of the object and the shape description.

The classical approach to MAT in greylevel images has been to apply some sort of edge detection and then derive the medial axes from the boundary contour. However, edge detection is especially sensitive to intensity variations and noise. In a discrete image, edges cannot be reliably extracted without some notion of spatial scale over which to measure discontinuity. The use of an unduly small scale emphasizes fine details including noise. The use of an unduly large scale distorts the form of detected objects and can lead to many details being lost. Often an image involves structures at several scales. Therefore, it is not trivial to choose an appropriate scale.

The solution to the difficulty of selecting an appropriate scale lies in the scale-space theory, which was pioneered by Marr and Hildreth (1980) and formulated by Witkin (1983), Koenderink (1984) and Lindeberg (1994). The scale-space of an image is generated by convolution of the image with progressively larger Gaussian kernels or their derivatives. Different combinations of the Gaussian-family derivatives are sensitive to different features in an image. When the scale and structure of a feature detector are best fit to a local region of the image, the strongest response will be generated at a scale proportional to the characteristic length of the feature of interest. The characteristic length corresponds to the wavelength of a periodic signal, the width of an elongated object, or the diffuseness of an edge (see Mallat and Hwang, 1992; Fritsch, 1993; Lindeberg, 1998). The significance of automatic scale selection is in the adaptive use of blurring scales across an image. Robustness is enhanced by using the largest degree of smoothing that does not sacrifice image structure.

The MAT, combined with the scale-space theory, leads to the concept of multiscale medial axis (MMA) jointly contributed by Fritsch (1993), Pizer et al. (1994), Morse et al. (1994), Lindeberg (1998) etc. The MMA for an object in a 2-D image is a set of curves in 3-D scale-space. For each MMA point, the spatial coordinate, x, indicates the middle position of the object; the scale parameter, σ, specifies the approximate width of the object at that position. The MMA curves are obtained by first computing a measure called “medialness” over scale-space and then detecting scale-space ridges in the medialness function. The medialness function, $\text{M(}\text{x}\text{,σ)}$, is defined as the degree to which a position x resembles an object middle when examined at a particular scale σ (see Fritsch, 1993; Morse et al., 1994; Xu and Pycock, 1999). An example of medialness functions is the normalized LoG operator: $\text{M(}\text{x}\text{,σ)=−σ}{}^2\text{∇}{}^2\text{G(}\text{x}\text{,σ)}$. At a given scale σ, the medialness function gives the strongest response for those objects with a specific width in proportion to σ. For variable width objects, the medialness values are relatively high along a track through the middle of the object and going up and down in scale proportional to the local object width. This track of high medialness, i.e. the ridges in medialness, is the MMA. Due to the robustness to image disturbances, the MMA provides a useful tool for many image analysis tasks including segmentation, recognition and registration, as shown by Fritsch (1993).

The accuracy of the MMA analysis to estimate the position and width of a single object has been demonstrated by Fritsch (1993), Koller et al. (1995) and Lindeberg (1998). However, the quantitative performance evaluation of the MMAs for multiple objects has not been focused, despite the reality that multiple objects existing together is very common in the existing applications (e.g. an organ embedded in or close to another) and even a single object often contains multiple object components (e.g. the fingers of a hand). Ideally the multiple objects are expected to have their own, accurate MMA representations and additional MMA branches for the group of objects, which constitutes a hierarchy of grouping relations. Unfortunately, the major problem with any multiscale description is that the location and existence of the scale-space representation for an object is influenced by nearby objects. To illustrate how object interference may affect the MMA extraction and representation, the scale-space behaviours of an overlapping object model with adjustable width, intensity contrast and position are investigated in this paper.

The optimal scale ridge definition proposed by Fritsch (1993) is a method to extract scale-space ridges and take scale’s distinguished role into account. In this method, the maximal response over scale at each position is located and projected onto the spatial space, forming an optimal scale response over spatial space. Then scale-space ridges are computed as the spatial ridges in the optimal scale response. A benefit of the optimal scale ridge definition is that the search space is readily reduced by one dimension (the scale dimension) and scale-space ridges can be detected in remaining spatial space. This is in contrast with the maximal convexity ridge definition used by Morse et al. (1994), which is an extension of the height ridges (Haralick, 1983; Eberly et al., 1994) from spatial space to scale-space and jointly considers spatial space and scale.

The original definition of optimal scale ridges aims to locates all local maxima over scale at each position, i.e. $\text{S={(}\text{x}\text{,σ)}\text{|}\text{M}{}_{\mathrm{\sigma }}\text{(}\text{x}\text{,σ)=0,M}{}_{\mathrm{\sigma \sigma }}\text{(}\text{x}\text{,σ)<0}}$. These maxima in scale-space may then be partitioned into connected subsets, S_k, k=1,2,… Each connected subset for a 2-D image is a surface patch in 3-D scale-space. The projection of its medialness response onto spatial space, , forms a 2-D subimage for the locally optimal scale response. The spatial ridges of the 2-D subimage constitute a part of the MMAs in the original image. However, for complicated images, determining the connectivity of the subsets of maxima for a surface has proven to be a difficult task by Fritsch (1993). Instead he proposed a simplified strategy, in which only the global maximum through scale at each position is considered and the ridge search is conducted over a single image of the globally optimal scale response, $\text{M}{}_{\mathrm{<mtext>opt</mtext>}}\text{(}\text{x}\text{)=}\text{max}{}_{\mathrm{\sigma }}\text{{M(}\text{x}\text{,σ)}}$. This method can capture a significant portion of the MMA, but inhibits a single position to belong to separate MMA branches at different scales, as in the case of overlapping objects. In addition, it aggravates the interference among the MMAs for objects in close distance. Despite these disadvantages, this simplified strategy for optimal scale ridges has been widely used to detect blood vessels in medical images and roads in remotely sensed images (see Koller et al., 1995; Feldmar et al., 1997; Krissian et al., 1998; Sato et al., 1998). In these cases it is appropriate that a point should not be the centre of several vessels or roads of different widths.

In this paper a sliding window algorithm is proposed for extracting a complete set of locally optimal scale ridges. It is based on but in contrast with Fritsch’s globally optimal scale ridges. The implementation avoids the non-trivial task of constructing sub-surfaces of connected maxima in the optimal scale ridge definition. It is useful and efficient in analysing the multiscale medial properties of a variety of complicated and interfering objects, e.g. overlapping or adjacent objects.

This paper is organized as follows: In 2 The MMA of single objects, 3 The MMA of overlapping objects, the MMAs of an isolated pulse and an overlapping pulse model are investigated. In Section 4, the importance of locally optimal scale ridges is emphasized and the sliding window algorithm is described. In Section 5, applications of the sliding window algorithm to medical images are demonstrated. This is followed by a discussion on the relationship between the sliding window algorithm and existing scale-space ridge definitions in Section 6.