Elsevier

Pattern Recognition Letters

Volume 19, Issue 13, November 1998, Pages 1183-1189
Pattern Recognition Letters

Noise tolerance of moment invariants in pattern recognition

https://doi.org/10.1016/S0167-8655(98)00106-8Get rights and content

Abstract

The noise tolerance of moment invariants in rotation and scale invariant 2-D pattern recognition is studied. The noise considered is that of discretization distortion using finite spatial resolution. The noise tolerance was explored for different resolutions, with and without scaling, and for different choices of moment orders.

Introduction

In pattern recognition it is often essential to transform the primary image data to features with the desired invariance. In the present paper we deal with features invariant to the simplest spatial transformations: translation, isotropic scaling and rotation. As discussed in several works (Gonzales and Wintz, 1983Gonzales and Woods, 1993), the moments of image components are useful in pattern recognition, since it is easy to derive transform invariant quantities from moments. Invariants on other transformations than Cartesian moments are discussed by Brandt and Lin (1996).

In the present paper we shall report on a simulation study of the noise tolerance of moment invariants in pattern recognition situations. Here noise is defined as the deviations between the bitmap treated in the computer and an image produced by an ideal scanner or camera with infinite spatial resolution. The most important noise contributions are (1) noise from detector and electronics in the scanning equipment (flat bed scanner or video camera/frame grabber), and (2) noise from pixel averaging in the spatial discretization.

The detector noise is present everywhere in the bitmap, whereas the discretization noise distorts mainly the edges of image components2. In the present work we neglect the noise in the scanning equipment for two reasons. First, in most cases the influence of the detector noise outside the edges can be eliminated by suitable preprocessing of the digital image, and so the character of the remaining noise contribution from the scanner is the same as that of spatial discretization. Secondly, as the CPU and memory related costs of high spatial resolution are considerable, there is always a need for reducing the spatial resolution so much, that the noise from discretization becomes dominant. The noise to be considered is illustrated in Fig. 1. Here the letter `a' is shown as a binary bitmap with high and low spatial resolution. The poorly resolved representations are shown for five different rotation angles. The aim of the present paper is to study how features, which are rotation invariant for high resolution, become less invariant for low resolution.

In the present work we try to answer the following questions:

  • How sensitive is the rotation invariance to spatial discretization?

  • How does this sensitivity depend on the area-to-edge ratio of the image components?

  • How does this sensitivity depend on the order of the moments?

  • Does different mappings of moments into feature space in classical pattern recognition give the same results?

  • How does combined rotation and scaling compare with rotation alone?

In the present paper we shall study two classes of scale-rotation invariants. The first class are the Hu type moment invariants (Hu, 1962). Later extensions of these invariants are reported by Li (1991). A recent review of algebraic and Fourier–Melin based derivation of Hu type moment invariants is given by Wood (1996). We include here the lowest 12 Hu type invariants which involves up to fourth order cartesian moments. The second class of moment invariants are based on finding the principal axes of the image components. These moment invariants, here denoted as PA invariants, will be explained in detail in Section 3. To the author's knowledge, they have not been reported previously.

Section snippets

Hu's moment invariants

In the spatially continuous limit we define the simple moments asmp,q=∫f(x,y)xpyqdxdy,where (x,y) are the coordinates in the image, p and q are nonnegative integers, and f(x,y) describes the image irradiance. Background areas have (x,y)=0. Translational invariant moments are given bymp,q=∫f(x,y)x−m1,0m0,0py−m0,1m0,0qdxdy.Note that these central moments m can be expressed by the simple moments m. Scale invariant moments areμp,q=mp,qm0,0(p+q)/2+1.

Rotation invariant quantities can be derived by

Principal axes moment invariants

We describe in this section an alternative class of moment invariants. The second order moments define principal axes in the same way as moments of inertia in mechanics. The orientation of the principal axes of an image component is described by an angle φ given bytan2φ=1,1μ0,2−μ2,0.

We assume here that at least one of the quantities μ1,1 and μ0,2μ2,0 are different from zero. We shall return to this requirement later. We define the rotation invariant moments by transforming the moments μ

Analysis of the noise tolerance

We report here a statistical study of tolerance to discretization distortion of the above two classes of moment invariants in case of binary image components (f(x,y) in Eq. (1)is 0 or 1) with the moments m of Eq. (1)calculated as simple sums over pixels of the image component. The transformations given in Table 1 leading to Ψ and Ψ will be applied to the discrete representation. The image components studied are those of 10 selected letters as they are represented in the font `Courier New'

Statistical results

We have calculated the Hu type invariants Ψ and the PA type invariants Ψ of the 10 different lower case letters (see Fig. 2). The reference letters used are `Courier New' with nominal size 200 generated by the Microsoft program Paintbrush. The tall letters `h,k,l' were about 200 pixels high. These letters were scaled down by factors between 1 and 11. Thus the typical height of the letters `h,k,l' were between 200 and 18 pixels. In the following the `letter size' is the height in pixels of the

Conclusion

In the present study of noise tolerance of scale-rotation invariants, we have tested a classification among 10 image components of comparable size. We find that invariants using moments up to fourth order are reasonably efficient for image components with average diameters of about 29 pixels. We found that the noise tolerance was poorer for objects subject to combined scaling and rotation compared to objects subject to rotation alone.

We expected the noise tolerance to be decreasing

Acknowledgements

Fruitful discussion with Henrik Gordon Petersen, Henrik I. Christensen and Peter Hauge is gratefully acknowledged.

References (6)

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