The normalized nth central moment is the nth central moment divided by σn; the normalized nth central moment of
x = E ((x − μ)n)/σn.
Theory of two-dimensional moment invariants for planar geometric figures was introduced by M K Hu in 1962 [34]. Hu established a fundamental theorem to relate moment invariants to algebraic invariants. He derived a complete system of moment invariants, invariant under translation, similitude and orthogonal transformations. The set of normalized central moments (second order moments) of images are shown to be invariant to such linear transformations. This property of moment invariance may be used to compare images of the drill-bit acquired during different stages of its life, for wear or
396 A. Jothilingam and B. Rajmohan breakage, with no specific requirement of identical conditions while acquiring these images. Various moments and their properties and their role in recognizing patterns are examined in detail in [6]. M Rizon et al [27] used geometric moment invariant feature vectors that are invariant under shifting, scaling and rotation to extract the global features for pattern recognition due to the discrimination power and robustness of moment invariants. Moment invariants are evaluated as a feature space for pattern recognition in terms of discrimination power and noise tolerance in [22,23,26,27,30,31].The need for using current technology against the conventional measurement methods for the economical influence of tool replacement costs and the use of different moments to describe tool wear images and to classify the tool condition in wear classes has been studied by J Barreiro et al[26] concluded that Hu and Legendre moment descriptors provide the best performance in the context of tool wear monitoring. In this work, the use of invariant moments has been extended to drill-bit wear monitoring. Apart from the discriminatory effect of the second order moments in classifying images of the drill-bit captured at its different usage points in the context of drill-bit monitoring, the interest in image moments stems from the following situation. Generally, vision based drill-bit wear monitoring schemes proposed in literature insist on or assume that the series of images of the drill bit at various stages of its usage, are captured under identical conditions and orientations. These algorithms use mean square error as a measure of image-distance for determining the distance between images of the drill-bit acquired at various stages of its life cycle. This distance measure is used to declare a drill bit as worn or otherwise, as well as to indicate the extent of wear. Changes in the conditions or slight changes in orientations during image capture may lead to image degradations and distortions (such as rotation, scaling, image blurring, etc.). Such distortions introduce distances between images that may not be attributed to drill- bit wear, leading to erroneous measurements.
In practice, it may not be possible to satisfy circumstances of uniform conditions & orientations during the image acquisition process. There is a distinct possibility that as time progresses, at different stages of measurement, the conditions of image acquisition such as lighting and background may differ or the orientation of acquired images may be different due to slight, miniscule shift in the camera position or the position of the drill bit. This leads to the introduction of linear transformations of translation, rotation and scaling between the acquired images, altering the position of the drill bit in the image, though by a few pixel distances. These are not perceptible to human interpretation, but may lead to the introduction of distortion that cannot be attributed to drill-bit wear. The set of second order moments, being invariant to such transformations, is a good candidate to act as a descriptor for measuring drill-bit wear. The property that second order moments are invariant to linear transformations is made use of to neutralize the effects of such distortions occurring between multiple image acquisitions.
3.1. Normalized Central Moments
Moments are descriptors obtained from point coordinates in a region and their gray level. They are applied in several areas of pattern recognition and object classification. Moments and functions of moments are invaluable tools in the literature for the measurement of the properties of a distribution [22, 23, 26, 27, 30, and 31]. In the field of image analysis, their use as image descriptors was pioneered by Hu [34] when he used the 2D geometric moments for characterizing the visual patterns in images. For different images, the respective sets of moments are unique and this makes them particularly useful for the task of pattern recognition. This is further added by the advantage of being able to construct moment invariants which are insensitive to rotation, scaling and translation. Hence, geometrical moments are effective descriptors for images under different perspectives. Geometric moment mpq of
image f(x, y ) of size M x N is defined as mpq =
Mx01 Ny01xp yq f(x, y)where p, q are non-negative integers and (p + q) is called the order of the moment. The corresponding central moment of order (p+q) is defined as
Drill-Bit Wear Monitoring Based on Invariant Normalized Central Image Moments 397 µpq = 1 1 0 0 M N x y
(x – x')p (y - y')q f(x, y)for p = 0, 1, 2,…. and q = 0, 1, 2,…, where x' = m10/m00 and y' = m01/m00
The normalized central moments, denoted ηpq, are defined as ηpq = µpq / µγ00 where γ =
2
p q
+ 1 for p+q = 2, 3….
Hu[34] derived seven invariant moments using nonlinear combinations of normalized central moments: Φ1 = η20+ η02 Φ2 = (η20 – η02)2 + 4 η211 Φ3 = (η30 - 3η12)2 + (3η21 – η03)2 Φ4 = (η30 + η12)2 + (η21 + η03)2 Φ5 = (η30 - 3η12) (η30 + η12)[ (η30 + η12)2 - 3(η21 – η03)2] + (3η21 – η03)( η21 + η03) [3(η30 + η12)2 - (η21 + η03)2] Φ6 = (η20 – η02) [(η30 + η12)2 - (η21 + η03)2] + 4η11 (η30 + η12)( η21 + η03) Φ7 = (3η21 – η03) (η30 + η12) [(η30 + η12)2 - 3(η21 + η03)2] + (3η12 – η30) (η21 + η03) [3(η30 + η12)2 - (η21 + η03)2]
The above mentioned set of seven moment invariants is insensitive to rotation, scale change, mirroring and translation.