Distance-based functions for image comparison

Abstract

The interest in digital image comparison is steadily growing in the computer vision community. The definition of a suitable comparison measure for non-binary images is relevant in many image processing applications. Visual tasks like segmentation and classification require the evaluation of equivalence classes. Measures of similarity are also used to evaluate lossy compression algorithms and to define pictorial indices in image content based retrieval methods. In this paper we develop a distance-based approach to image similarity evaluation and we present several image distances which are based on low level features. The sensitivity and effectiveness are tested on real data.

Introduction

Estimation of image similarity is an important problem of image analysis. Measures of similarity between two images are useful for the comparison of algorithms devoted to noise reduction, image matching, image coding and restoration. Visual tasks are often based on the evaluation of similarities between image-objects represented in an appropriate feature space.

Content-based query systems may process a query on the basis of a classification procedure, that assigns the unknown to the closest available prototype. The performance of the whole query system depends on the definition of suitable similarity measure based on Image Distance Functions (IDFs) (Danielson, 1980; Russ and Russ, 1989). Image segmentation and classification require the evaluation of equivalence classes. Measures of similarity are also used to evaluate compression algorithms (Wilson et al., 1997), to define pictorial indices in image retrieval methods and to compare image restoration methods (Zamperoni and Starovoitov, 1996).

Image comparison is often performed by computing a correlation function, the root of the mean square-error or measures of the signal-to-noise ratio. The last approach is applicable only if there is enough knowledge of the image content. In the case of binary images, the comparison problem is much simpler; in fact the image content is easily extracted by assuming the set of black pixels as objects, and the remaining pixels as the background. In (Klette and Zamperoni, 1987) several measures of correspondence between binary images were described and compared. The authors have shown that distance-based measures perform better comparison of binary images than measures based on set memberships. Baddeley (1992) presented a new error metric for binary images. He calculated distances from every point of a two-dimensional image space to the nearest object pixel for both images.

If we operate on gray scale or color images, there are two basic means of comparison: (1) to extract some objects of interest by thresholding, segmentation, edge and shape detection, and then to compare the objects; (2) to compare images as whole entities. The first method leads to high level image recognition, while the second leads to low level image analysis. In both cases the choice of the suitable feature space (type and number of feature parameters) is a critical and not fully solved problem.

Global features directly derived from gray levels (e.g. first and second order statistics, color) can give a coarse indication of image similarity. However, they may produce unstable indications, because quite different images may have similar histograms. On the other hand, structural features (e.g. edges, skeleton, medial axis, convex hull, object symmetry) are very sensitive to the level and kind of noise in the image; moreover their estimation depends on the operators applied. For example, Mokadem et al. (1996) and Ghorbel (1994) developed an invariant shape distance for comparing geometric objects extracted from images. The distance was based on Fourier coefficients. In (Moghaddam et al., 1996) a method, invariant to affine transformations, is also proposed.

Distance functions, combining global and structural information, seem to be more adequate to characterize low-level similarity of images. Several distance-based ideas were developed for a measure design for gray-scale image comparison. Huttenlocher et al. (1993) and Dubuisson and Jain (1994) have developed object similarity measures, which were based on the Hausdorff metric. The new distance-based measures (mainly non-metrics) were applied to edge matching of binarized gray level images. Distance-based measures were tested for image segmentation by Kara Falah and Bolon (1992). Di Gesù (1994) proposed a new version of the Mahalanobis distance in order to cluster homogeneous pixels for image segmentation. Pal and Majumder (1986) considered similarity measure based on fuzzy metrics defined in a feature space. Tegolo (1994) partitioned two images D₁ and D₂ in n×m sub-images, and calculated the generalized invariant moments for each sub-image. The IDF$\text{(D}{}_1\text{,}\text{D}{}_2\text{)}$ was defined as the Euclidean distance between vectors containing the list of invariant moments related to the images. It was applied in a content-based query system.

Wilson et al. (1997) extended the idea of Baddeley's error metric. However, computation of the new measure, which is used on gray scale images, is time consuming. The same authors tested also the Sobolev norm for image comparison purposes. It is calculated in the frequency domain by the Fourier transform of compared images. However, the Sobolev norm was not normalized and the results of its application were similar to those obtained by the simpler well-known root-mean-squared error.

In this paper we present new versions of IDFs based on local distances combining intensity and structural image features in different ways. The functions may also be named hybrid IDFs. They are implemented for the direct comparison of non-binary images without calculating visible features like edges or shapes.

Our experiments demonstrate the benefits of the new IDFs. The results indicate that better sensitivity in comparison of similar digital images can be reached by using the distance-based measures.

The paper is organized as follows. Section 2describes the IDFs introduced. Section 3illustrates part of our experimental results performed for real images. Final remarks are given in Section 4.