Noise tolerance of moment invariants in pattern recognition
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:
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How sensitive is the rotation invariance to spatial discretization?
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How does this sensitivity depend on the area-to-edge ratio of the image components?
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How does this sensitivity depend on the order of the moments?
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Does different mappings of moments into feature space in classical pattern recognition give the same results?
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How does combined rotation and scaling compare with rotation alone?
Section snippets
Hu's moment invariants
In the spatially continuous limit we define the simple moments aswhere (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 byNote that these central moments can be expressed by the simple moments m. Scale invariant moments are
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 by
We assume here that at least one of the quantities μ1,1 and μ0,2−μ2,0 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.
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2000, Pattern Recognition LettersCitation Excerpt :If μ20o−μ02o and μ21o are close to zero, then the angle determination becomes uncertain due to noise. This situation was detected for the image of boldface `a' in the font `New Courier' (Balslev, 1998). If the above-mentioned annoying situation is accidental, then the introduction of weighted moments is useful, because the weighting is able to shift the set of moments away from the singularity.
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