Parallel Segmentation of Multi-Channel Images Using Multi-Dimensional Mathematical Morphology

Multichannel images are characteristic of certain applications, such as medical imaging or remotely sensed data analysis. Mathematical morphology-based segmentation of multichannel imagery has not been fully accomplished yet, mainly due to the lack of vector-based strategies to extend classic morphological operations to multidimensional imagery. For instance, the most important morphological approach for image segmentation is the watershed transformation, a hybrid of seeded region growing and edge detection. In this chapter, we describe a vector-preserving framework to extend morphological operations to multichannel images, and further propose a fully automatic multichannel watershed segmentation algorithm that naturally combines spatial and spectral/temporal information. Due to the large data volumes often associated with multichannel imaging, this chapter also develops a parallel implementation strategy to speed up performance. The proposed parallel algorithm is evaluated using magnetic resonance images and remotely sensed hyperspectral scenes collected by the NASA Jet Propulsion Laboratory Airborne Visible Infra-Red Imaging Spectrometer (AVIRIS). Parallel Segmentation of Multi-Channel Images 271 Copyright © 2006, Idea Group Inc. Copying or distributing in print or electronic forms without written permission of Idea Group Inc. is prohibited. INTRODUCTION The segmentation of an image can be defined as its partition into different regions, each having certain properties (Zhang, 1996). In mathematical terms, a segmentation of an image f is a partition of its definition domain D f into m disjoint, non-empty sets s 1 ,s 2 ,. ,s m called segments, so that f i m i D S = =1 and j i S S j i ≠ ∀ ∅ = , . Segmentation of intensity images in the spatial domain usually involves four main approaches (Haralick & Shapiro, 1985). Thresholding techniques assume that all pixels whose value lies within a certain range belong to the same class. Boundary-based methods assume that the pixel values change rapidly at the boundary between two regions. Region-based segmentation algorithms postulate that neighboring pixels within the same region have similar intensity values, of which the split-and-merge technique is probably the most well known. Hybrid methods combine one or more of the above-mentioned criteria. This class includes variable-order surface fitting and active contour methods. One of the most successful hybrid segmentation approaches is the morphological watershed transformation (Beucher, 1994), which consists of a combination of seeded region growing (Adams & Bischof, 1994; Mehnert & Jackway, 1997) and edge detection. It relies on a marker-controlled approach (Fan, et al., 2001) that considers the image data as imaginary topographic relief; the brighter the intensity, the higher the corresponding elevation. Let us assume that a drop of water falls on such a topographic surface. The drop will flow down along the steepest slope path until it reaches a minimum. The set of points of the surface whose steepest slope path reachs a given minimum constitutes the catchment basin associated with that minimum, while the watersheds are the zones dividing adjacent catchment basins. Another way of visualizing the watershed concept is by analogy to immersion (Vincent & Soille, 1991). Starting from every minimum, the surface is progressively flooded until water coming from two different minima meet. At this point, a watershed line is erected. The watershed transformation can successfully partition the image into meaningful regions, provided that minima corresponding to relevant image objects, along with object boundaries, are available (Shafarenko, et al., 1997). Despite its encouraging results in many applications, morphological techniques have not been fully exploited in applications that involve multichannel imagery, where a vector of values rather than a single value is associated with each pixel location. Many types of multichannel images exist depending on the type of information collected for each pixel. For instance, color images are multichannel images with three channels, one for each primary color in the RGB space. Images optically acquired in more than one spectral or wavelength interval are called multispectral. These images are characteristic in satellite imaging and aerial reconnaissance applications. The number of spectral channels can be extremely high, as in the case of hyperspectral images produced by imaging spectrometers (Chang, 2003). Finally, all image types above can be extended to the class of multitemporal images or image sequences, which consist of series of images defined over the same definition domain, but collected at more than a single time. Examples include magnetic resonance (MR) images in medical applications and video sequences. Segmentation of multichannel imagery has usually been accomplished in the spectral/temporal domain of the data only. Techniques include well known data clustering algorithms such as ISODATA (Richards & Jia, 1999). Other techniques, such as 272 Plaza, Plaza, Valencia, & Martinez Copyright © 2006, Idea Group Inc. Copying or distributing in print or electronic forms without written permission of Idea Group Inc. is prohibited. Soille’s watershed-based multichannel segmentation (Soille, 1996), are based on an initial spectral clustering followed by a post-classification using spatial information. This approach separates spatial information from spectral information, and thus the two types of information are not treated simultaneously. In this chapter, we develop a novel watershed-based segmentation technique that naturally combines spatial and spectral/temporal information in simultaneous fashion. While such an integrated approach holds great promise in several applications, it also creates new processing challenges (Tilton, 1999). In particular, the price paid for the wealth of spatial and spectral/temporal information is an enormous amount of data to be processed. For that purpose, we develop a parallel implementation of the proposed segmentation algorithm that allows processing of high-dimensional images quickly enough for practical use. The chapter is structured as follows. The following section provides a mathematical formulation for multidimensional morphological operations, and relates the proposed framework to other existing approaches in the literature. A multidimensional watershed-based segmentation approach is described next, along with its parallel implementation. A quantitative segmentation evaluation comparison with regard to standard segmentation techniques is then provided, using both MR brain images and remotely sensed hyperspectral data collected by the 224-channel NASA Jet Propulsion Laboratory Airborne Visible Infra-Red Imaging Spectrometer (AVIRIS) system. The chapter concludes with some remarks. MULTIDIMENSIONAL MATHEMATICAL MORPHOLOGY Our attention in this section focuses primarily on the development of a mechanism to extend basic morphological operations to multichannel imagery. In the following, we provide a mathematical formulation for classic and extended morphological operations. Classic Morphological Operations Following a usual notation (Soille, 2003), let us consider a grayscale image f, defined on a space E. Typically, E is the 2-D continuous space R or the 2-D discrete space Z. The classic erosion of f by B ⊂ Z is given by the following expression: ( )( ) ( ) ( ) ( ) { } t y s, x min , 2 , + + = Θ ∈ f y x B f B Z t s , ( ) 2 , Z y x ∈ (1) where Z(B) denotes the set of discrete spatial coordinates associated to pixels lying within the neighborhood defined by a “flat” SE, designed by B. The term “flat” indicates that the SE is defined in the x-y plane. Similarly, the classic dilation of f by B is given by: ( )( ) ( ) ( ) ( ) { } t y s, x max , 2 , − − = ⊕ ∈ f y x B f B Z t s ( ) 2 , Z y x ∈ (2) Parallel Segmentation of Multi-Channel Images 273 Copyright © 2006, Idea Group Inc. Copying or distributing in print or electronic forms without written permission of Idea Group Inc. is prohibited. Vector-Based Mathematical Morphology Let us denote a multichannel image with n channels as f, where the values of each pixel f(x,y) of the definition domain D f are given by an n-dimensional vector f(x,y) =(f 1 (x,y), f 1 (x,y),. ,f n (x,y)). Extension of monochannel erosion and dilation above the images defined on the n-dimensional space is not straightforward. A simple approach consists in applying monochannel techniques to each image channel separately, an approach usually referred to as “marginal” morphology in the literature. This approach is unacceptable in many applications because, when morphological techniques are applied independently to each image channel, there is a possibility for loss or corruption of information of the image due to the probable fact that new pixel vectors —not present in the original image— may be created as a result of processing the channels separately. In addition, no correlation between spectral/temporal components is taken into account. An alternative way to approach the problem of multidimensional morphology is to treat the data at each pixel as a vector. Unfortunately, there is no unambiguous means of defining the minimum and maximum values between two vectors of more than one dimension, and thus it is important to define an appropriate arrangement of vectors in the selected vector space. Several vector-ordering schemes have been discussed in the literature (Plaza, et al., 2004). Four classes of ordering will be shortly outlined here for illustrative purposes. Let us now consider an n -dimensional image f and let f(x,y) and f(x',y') denote two pixel vectors at spatial locations (x,y) and (x',y') respectively, with f(x,y) =(f 1 (x,y), . ,f n (x,y)) and f(x',y') = f 1 (x',y'),. ,f n (x',y')). In marginal ordering (M-ordering), each pair of observations f i (x,y) and f i (x',y') would be ordered independently along each of the