Image Processing Issues in Digital Strain Mapping - Semantic Scholar

Image Processing Issues in Digital Strain Mapping W. F. Clocksina , J. Quinta da Fonsecab , P.J. Withersb and P.H.S. Torrc a

b

School of Computing and Mathematical Sciences, Oxford Brookes University, Wheatley, Oxford OX33 1HX, UK Manchester Materials Science Centre, Grosvenor Street, Manchester M1 7HS, UK c Microsoft Research, 7 JJ Thomson Avenue, Cambridge CB3 0FB, UK ABSTRACT

We have developed high density image processing techniques for finding the surface strain of an untreated sample of material from a sequence of images taken during the application of force from a test rig. Not all motion detection algorithms have suitable functional characteristics for this task, as image sequences are characterised by both short- and long-range displacements, non-rigid deformations, as well as a low signal-to-noise ratio and methodological artifacts. We show how a probability-based motion detection algorithm can be used as a high confidence estimator of the strain tensor characterising the deformation of the material. An important issue discussed is how to minimise the number of image brightness differences that need to be calculated. We give results from two studies of materials under axial tension: a sample of aluminium alloy exhibiting a propagating plastic deformation, and a preparation of deer antler bone, a natural composite material. Keywords: strain mapping, optical flow, image correlation, probabilistic image processing

1. INTRODUCTION The detection of motion from a sequence of digital images is used widely in the field of computer vision, and many algorithms have been developed for this purpose.1 However, there has been relatively little work in which such techniques are applied within experimental mechanics to aid the understanding of the deformation of materials via strain mapping. Image processing approaches are of potential value for measuring local displacements and strains over a range of scales, without obscuring the surface or requiring contact with it. They may thus be competitive with more mature technologies such as strain gauges, moir´e interferometry and speckle metrology. Such techniques suffer from characteristic limitations. For example, strain gauges only inform about discrete local areas, and the upper strain limit is relatively low. Moir´e interferometry2 gives a more complete displacement and strain map, but the application of the required grating is technically demanding, limiting, time consuming, and can obscure underlying features. Speckle displacement analysis with digital image correlation has been used for finding displacements in materials under stress including metals, 3, 4 wood and rubber,5 and stir friction welds.6 However, this technique often requires artificial contrast enhancement, such as by painting the surface with black-and-white speckles,3, 4 dusting with a reflective powder,7 or sputtering a grid of fine gold dots,8 all of which improve the texture but can obscure important metallographic detail. The method employed in this paper is based on the analysis of photomicrographic sequences of free surfaces, which can be polished and etched, so that optical flow patterns (displacement maps) can be compared with surface microstructure. The most common algorithms used for strain mapping are based on correlation 9 and Fourier methods3 ; a more recent method is based on maximising the posterior probability of displacement. 10 Because long-range displacements (of more than say 8 pixels) are common in strain experiments, correlation-based methods have been favoured over the local gradient-based methods.11 Yet, the non-rigid deformations inherent in strained materials are not ideal for correlationbased approaches. In previous work, Chivers12 carried out a comparative study of the performance of various algorithms Correspondence to W.F.C.: [email protected]. Copyright 2002 Society of Photo-optical Instrumentation Engineers. This paper will be published in Proceedings of the SPIE Volume 4790 and is made available as an electronic preprint with permission of SPIE. One print or electronic copy may be made for personal use only. Systematic or multiple reproduction, distribution to multiple locations via electronic or other means, duplication of any material in this paper for a fee or for commercial purposes, or modification of the content of the paper are prohibited.

(correlation, optical flow, probability) applied to strain measurement problem on metal alloys, and found that the probability method13 gave the best performance on a range of trial problems. This method computes, at each point where a displacement vector is required, a multi-modal probability distribution function to estimate the maximum likelihood of a displacement. Both short-and long-range displacements can be found, as well as the displacements that form non-rigid deformations. The method is particularly accurate at motion contrast borders, an essential property for strain measurements. We have used this method for assessing deformations in a variety of materials including steel, aluminium welds, and carbon-fibre composites.10, 14 In this paper we summarise a number of issues that have arisen from strain mapping studies. We also demonstrate our methods on two types of material: an aluminium alloy and a sample of deer antler bone. The aluminium sample exhibits a propagating deformation, difficult to characterise using previous techniques. The deer antler bone is a porous composite material. The existence of many pores causes numerous large localised motion contrasts, which pose problems of accuracy and convergence for previous methods. In order to estimate velocities in the image, one must first decide which property of the image to track over time. One common approach is based on the assumption that light reflected from a part of the object surface remains constant through time, in which case one can track points of constant image intensity. Where I(x, y, t) is the image intensity function, the conservation of intensity assumption11 is given by the equation I(x, y, t) = I(x + u, y + v, t + δt) where (u, v) is the displacement of the local image region after time δt. Letting u = temporal derivative of Eq. 1, the gradient constraint equation is defined:

(1) dx dt

and v =

∂I ∂I ∂I u+ v+ = 0. ∂x ∂y ∂t

dy dt

and taking the

(2)

The solutions to Eq. 2 in velocity space lie along a line, which represents all the 2D velocities that are consistent with the derivative measurements in Eq. 2. This fundamental ambiguity is resolved in practice by choosing the solution (u, v)⊥ , the motion component in the direction of the local gradient of the image intensity function. Thus, measuring spatiotemporal derivatives allows the estimation of normal image velocity. For simplicity when a pair of images I 1 and I2 are considered, we shall henceforth abbreviate I1 (x, y) = I(x, y, t) and I2 (x + u, y + v) = I(x + u, y + v, t + δt). The conservation intensity assumption together with the gradient constraint equation lie at the heart of all methods for computing displacement maps from a sequence of images.

2. CORRELATION METHODS The cross correlation between two images is a standard approach to feature detection, and has been applied widely as a tool for measuring surface deformations. However, its motivation is frequently unexamined. Cross-correlation derives from the distance between two images. Given two images I1 and I2 , a N × N region of interest is defined for each image. With n = bN/2c, the squared Euclidean distance between the two regions of pixels is d2 (u, v) =

n n X X

(I1 (x, y) − I2 (x + u, y + v))2 .

(3)

x=−n y=−n

In a template matching context, I1 is the template, and the above operation has the effect of pairing each pixel in the template with a corresponding pixel in the image centred at u, v. In the expansion of d 2 , d2 (u, v) =

n n X X

(I12 (x, y) − 2I1 (x, y)I2 (x + u, y + v) + I22 (x + u, y + v)),

(4)

x=−n y=−n

P 2 P 2 the template term I1 (x, y) is constant. If the image energy term I2 (x + u, y + v) is approximately constant (an assumption challenged below), then the remaining cross-correlation term c(u, v) =

n n X X

x=−n y=−n

I1 (x, y)I2 (x + u, y + v)

(5)

is a measure of the similarity between the regions of interest. In spite of the popularity of this approach, there are several disadvantages to using Eq. 5 for for finding displacements: PP 2 • If the image energy I2 (x + u, y + v) varies with position, matching using Eq. 5 can fail. For example, Eq. 5 implicitly gives higher values for correlation in brighter parts of an image. • The range of c(u, v) depends on the size of the feature. • Eq. 5 is not invariant to changes in image amplitude such as those caused by changes in illumination across the image sequence. • The cross-correlation is insensitive to rotation and affine transformation of the feature. For example, correlation may be lost in the presence of non-rigid deformations of the image sequence. To overcome the first three disadvantages, it is customary to normalise c(u, v). One way is to divide by the sum of grey levels in one or both regions of interest. Where henceforth the sums are taken as above, P P I1 (x, y)I2 (x + u, y + v) x 0 Py P (6) c (u, v) = x y I1 (x + u, y + v)

or alternatively,15

P P

x c (u, v) = qP P 0

x

y I1 (x, y)I2 (x

2 y I1 (x, y)

P P x

+ u, y + v)

2 y I2 (x

.

(7)

+ u, y + v)

Another approach16, 17 is to use a centralised correlation coefficient. Where µ ¯ 1 is the mean pixel value in the template and µ ¯2 is the mean pixel value in the image region of interest, P P ¯1 )(I2 (x + u, y + v) − µ ¯2 ) x y (I1 (x, y) − µ γ(u, v) = qP P (8) P P 2 (x + u, y + v) − µ 2 (I (x, y) − µ ¯ ) (I ¯ ) 1 1 2 2 x y x y

Thus γ(x, y) is normalised with respect to both the image and the template, and lies in the range [−1, 1].

One way to ameliorate the problem of insensitivity to non-rigid distortions is to use a parameter vector that relates pixel coordinates in the first image with coordinates in the second image. 15, 18 For example, let ξ(x, y, θ) = θ1 + θ3 x + θ4 y,

(9)

η(x, y, θ) = θ2 + θ5 x + θ6 y.

(10)

The parameters (θ1 , θ2 ) describe the (u, v) displacement components between the images, and the remaining parameters allow for an affine transform. Using the parameter vector θ, Eq. (7) may be augmented to give P P x y I1 (x, y)I2 (ξ(x, y, θ), η(x, y, θ)) 0 . (11) c (u, v) = qP P P P 2 2 x y I1 (ξ(x, y, θ), η(x, y, θ)) x y I2 (ξ(x, y, θ), η(x, y, θ))

This formulation has been proposed for strain mapping.15 It is necessary to solve for the parameter vector by iterative optimisation of Eq. 11, and the implications of this are discussed below. In what follows we shall use c(u, v) to stand for any correlation function.

3. PROBABILITY METHODS In Eq. 7 the values of c0 lie between 0 and 1, and could therefore be interpreted as a ‘probability’ of correlation. However, we would prefer a more principled approach based on probabilistic models. In this section we formulate a Bayesian framework for robustly estimating the displacement parameters θ within a support region R; this work extends and improves the scheme previously used.10, 13 The key issue is the definition of the conditional likelihood function. Within the framework it is possible to handle several different types of displacement model. Two models are considered here: (a) Constant displacement of the region, thus θ has two elements (θ1 , θ2 ) = (u, v), and (b) Affine deformation of the region, in which case the six elements of θ are given as in Eq. 9 and 10. In estimating image velocity, the conditional likelihood function expresses our belief that velocities are consistent with measurements taken from the image. Uncertainty arises for a number of reasons, first among which include the fundamental ambiguity expressed in the multiple solutions of Eq. 2. This ambiguity can give rise to the so-called ‘aperture’ effect and other illusions. Also, the partial derivatives must be approximated by discrete differences, which accentuates noise. In order to optimally estimate θ, a maximum a posteriori (MAP) estimate is made such that P (D|θ, X)P (θ|X) θˆ = argmaxθ P (D|X)

(12)

with θˆ corresponding to the estimate of the true θ. The two quantities D and X are defined as follows: D is the data, in this case the images; X is the prior information upon which all the probabilities are conditioned, in this case the shape of the region R in the first image, the search region in the second image S, and the assumption of the error distribution. The shape of support region R is a disc of radius r. Although practically all correlation methods assume a rectangular set of support, we use a disc to remove any possible bias owing to nonuniform phase angles. All the pixels that are completely enclosed by a disc of the support radius are included to form R. As the radius r increases, the stability of the result may be improved owing to the accumulation of more evidence, however accuracy may decrease, as the motion model θ may no longer be valid for the whole region. Furthermore the computation time required is governed by factor of approximately πr 2 . The radius s of the search region S determines the range of permissable θ, and should be large enough to encompass the correct solution. However the computation time required is governed by factor of approximately πs2 and if the radius is too large, the algorithm is more susceptable to aliasing effects. Next we formulate the distributions in Eq. 12. There are two terms to deal with: P (D|θ, X) is the conditional likelihood and P (θ|X) is the prior probability. Note that P (D|X) is a constant and hence discounted from hereon for the purposes of optimising θˆ . The likelihood term is based on the conservation of intensity assumption 11 that for the true displacement (u, v) then I1 (x, y) = I2 (x + u, y + v) plus some additive noise , such that  follows the distribution f (), i.e. the probability of observing the value  = d is f (d). Thus the likelihood of observing a particular displacement (ui , vi ) for a pixel (xi , yi ) within R is f (di ) such that di = I2 (xi + ui , yi + vi ) − I1 (xi , yi )

(13)

assuming that all the di are conditionally independent of one another within a region (note this is not the same as assuming that the displacements are independent). The likelihood of a region moving under displacement hypothesis θ can be determined by the product of all di arising from the hypothesised displacements within the region: Y P (D|θ, X) = f (di ). (14) i:(xi ,yi )∈R The prior term P (θ|X) is zero for values of θˆ which lead to disparities outside the search region S of radius s. Within S it is possible to assume a uniform prior, however, it is useful to use the prior to bias θˆ towards low displacement solutions. We discuss the prior further below. Taking logs, the MAP estimator is given as θˆ = argmaxθ log(P (θ|X)) +

X

i:(xi ,yi )∈R

log f (di ).

(15)

This is equivalent to the formulation given in previous work10, 13 with values of the likelihood chosen to be of the form −d2 f (di ) = exp(exp( σ2i )) for displacements that lie within S. This is also equivalent to an M-estimator using the Welsch weight function.19, 20 For pixels whose correspondence lies outside S the probability is suitably normalised as previously described.10, 13 Combining terms, we obtain the MAP estimator θˆ = argmaxθ log(P (θ|X)) +

X

i:(xi ,yi )∈R

exp



−d2i σ2



.

(16)

Note that this distribution is not a Gaussian. The exponential likelihood has particular advantage as it strongly limits the effects of outliers, and normalises the log likelihood to lie between 0 and 1. The choice of σ controls the effect of large errors di (potential outliers) on the resultant estimate of θˆ . The formulation of the MAP estimator given in Eq. 16 lends itself to a simple robust estimation scheme based on a previously described voting algorithm.13 In the case of the pure displacement model (θ1 , θ2 ) = (u, v) it is possible for each θ to evaluate Eq. 16 for all the pixels in R and select the optimal θ that maximises Eq. 16. This is a robust estimator as the double exponential function eliminates the effects of outliers. For efficiency, look-up tables are used to enumerate the pixels within R and S and to evaluate the exponentials. For the affine case, with θ = (θ1 , ..., θ6 ), the estimate (θ˜1 , θ˜2 ) = (˜ u, v˜) of the constant displacement model for the region is obtained, then an optimisation is conducted with start point (θ˜ 1 , θ˜ 2 , 0, 0, 0, 0) using a direct method. The optimisation method, discussed in the next section, does not use gradients, and typically requires 9 iterations and 73 evaluations of Eq. 16, and is justified because (˜ u, v˜) is guaranteed to be within 1 pixel of the true optimum, and the affine terms θ 3 , ..., θ6 are all much less than 1. Here the affine terms are discarded, though in principle they may convey useful information about local strain, and future work could investigate this. The resulting solution can show accuracy to 0.01 pixel within an arbitrary range, typically s = 20 pixels in our stressed materials data. A support radius of r = 9 pixels suffices for good quality and synthetic data, though a support radius of r = 16 pixels has been used for the examples given in Section 6. If noisy results are obtained, it is wise to increase the support radius. For example, a support radius of 20 contains 335 pixels, and gives good results on images that are otherwise unprocessable. In following sections, it will be useful to refer to the probability distribution of displacements over the search region S. Let p(u, v) = P (θˆ 1 , θˆ 2 ) for all (u, v) ∈ S. In later discussion, function p(u, v) will serve the same purpose as the correlation surface c(u, v) and will be used interchangeably. We now turn to the definition of a suitable prior. Priors can be defined for several uses. A prior defined over the support radius R will permit elements of the support region to be weighted differently. For example, the weight could decrease as a function of distance from the centre of the support region. Here we use equal weighting. It is more useful to consider priors over the search region S. It is possible to assume a uniform prior P (θ|X) = k for constant k, however, it is useful to use the prior to bias the answer toward (u, v) = (0, 0) (recall (u, v) ≡ (θ 1 , θ2 ). This is because areas of very low image contrast give relatively flat correlation surfaces with no significant peaks. In this situation, solutions for (u, v) are uninformative, and the prior should dominate the result to assign a displacement value of (0, 0). This is justified by interpreting the prior as the probability of displacement occurring at random within the search radius. For search radius s, the greatest probability is for displacements of lengths approaching 0, and the lowest probability is for displacements of length 2s. A suitable prior with these properties is  p u2 + v 2 (17) P (θ|X) = g λ ) is defined within the range −s ≤ λ ≤ s. Future work will concentrate on a prior that favours where g(λ) = cos( πs spatial coherence.

4. INTERPOLATION AND OPTIMISATION Textbook formulations of correlation show the correlation image (or surface) that results when the template is applied to every pixel location in the source image. To find the location of highest correlation, it is necessary to find the correlation

surface, and then search it for the maximum peak. However, this way of formulating the problem is time consuming. Not only is the correlation result computed in locations far from the peak, but the entire correlation surface must be searched for the peak. Furthermore, if subpixel accuracy is required, the image and template need to be supersampled, and consequently the area of the correlation surface may be increased by orders of magnitude. Because the correlation surface does not contain phase information from the original images, subpixel accuracy cannot be recovered reliably by interpolating the correlation surface itself. Normally, images are represented as arrays of pixels indexed by integers. Pixels can be directly accessed only at integer locations. However, when estimating displacements of non-integral length, it is necessary to sample the image at fractional locations. To accomplish this, some form of interpolation is required. Interpolation is also required if the search for the correlation peak is driven by the optimisation method. Linear interpolation has been proposed in a strain mapping context,21, 22 but this has been shown15 to result in errors of up to 20% of the actual strain level. Schrier et al15 have studied interpolation errors for a variety of interpolators including spline and polynomial methods, and have found good results for the standard cubic spline interpolator, with best results for the more expensive quintic B-spline interpolator. However they did not consider the Catmull-Rom spline, one of a family of C1-continuous interpolating cubics. Dodgson23 has shown how it has been derived differently in five different ways: as the quadratic B-spline blend of three linear functions; as a linear blend of two quadratics; as the C1-continuous cubic that matches a Taylor series expansion to the highest order; as the C1-continuous interpolating cubic with the best frequency response; and as a Hermite form with approximated tangents. The fact that the same cubic has been put forward independently as somehow the ‘best’ cubic by two separate methods, and is also the end result of three other derivations would seem to point to the Catmull-Rom spline as the best all-round cubic interpolant. Empirical results back this up: On strain mapping data we have found the Catmull-Rom spline to cause about half the interpolation error as the standard cubic spline used by Schrier et al,15 and is therefore comparable to the quintic B-spline but at the lower computational cost of the cubic. The optimisation methods referenced above begin with an initial estimate of the peak location, and use an iterative optimisation routine to find the local maximum of the correlation surface. At each iteration, a set of noninteger pixel coordinates will be generated by the optimisation algorithm, and image values at these coordinates need to be estimated by interpolation. Gradient-based optical flow methods11 also implicitly search for an optimum. The main assumption of local optimisation is that the initial estimate is in the vicinity of the true optimum. Because most optical flow methods are interested in short-range displacements (up to 3 pixels typically), an initial estimate of (0,0) makes sense. However, in the strain mapping context, when long range displacements are also encountered, optimisation can easily get lost if an initial estimate of (0,0) is used, and this was probably the reason for the poor performance of standard optical flow methods applied to strain mapping in a previous comparative study.12 The choice of optimisation method may be important also. In the strain mapping context, derivatives are not directly available, and it may be unwise to estimate them using differences of noisy image data. One study24 optimises the correlation coefficient by Newton’s iteration, which requires second-order derivatives. A successor study15 used the more sophisticated Levenberg-Marquart algorithm, which can give better results but which is also a curious choice given that it too requires derivatives, and estimating these from noisy data could be an additional source of systematic error. This is why we use a direct method similar to a Hooke-Jeeves method, which does not require gradients.

5. CONFIDENCE FACTOR It is useful to be able to associate a confidence factor with a displacement probability p(u, v) (or, for the correlation methods, a correlation value c(u, v)). The correlation value itself provides a measure of the strength of correlation, but intuitively the dispersion of the correlation peak is an indicator of the confidence of correlation: sharp peaks indicate a high confidence correlation, whereas broad peaks indicate a lower confidence correlation. In this application, where each displacement should have a unique endpoint, multiple peaks indicate an ambiguity of displacement. Because correlation surfaces are multimodal, the variance is not actually a good estimator of dispersion. However, a dimensionless confidence factor 0 ≤ cf ≤ 1 for the displacement value (u, v) can be found as follows. Suppose a set of correlation values c(u, v) are obtained over the search region S. A nonlinear function G can be applied to each correlation value to count significant displacements (those with correlation values greater than half the maximum value over the search region): P (u,v)∈S G(c(u, v)) cf = 1.0 − (18) NM

where M is the maximum of the c(u, v) values for all (u, v) ∈ S, and N = |S|. The hard limiter 1, x > 0.5M L(x) = 0, x ≤ 0.5M

(19)

can be used to make explicit counts, but it has a smooth approximation G(x) =

M 1+

e−k(x−0.5)

(20)

for which a parameter value of k = 50 is found to be satisfactory. A probabilistic perspective can give a more principled approach. Because the displacement probability (or, for the correlation methods, normalised correlation surface c(u, v)) over the search region S can be interpreted as a bivariate distribution given as a frequency table, the entropy of the distribution may be used as a confidence factor. Neglecting dependence of variables, entropy H(x, y) can be calculated XX H(x, y) = − c(u, v) ln c(u, v) (21) u

v

with higher values of H(x, y) meaning that c is more randomly distributed; the maximal value H(x, y) = |S| indicates a flat distribution. When H(x, y) = 0, the most confident correlation has been achieved: one in which one displacement in c is at the the maximum value and the rest of the correlation values are 0. However, in this application it is useful to consider a measure of association between the variables u and v. The symmetric uncertainty coefficient can be defined as " # H(y) + H(x) − H(x, y) U (x, y) = 2 . (22) H(x) + H(y) P P where H(x) = − u c(u, ·) ln c(u, ·) and H(y) = − v c(·, v) ln c(·, v). The two-dimensional entropy together with the symmetric uncertainty coefficient can reveal both the dispersion of the distribution and the dependence between the two variables.

6. STRAIN MAPPING The first step in mapping the strain of a material undergoing deformation is to estimate a strain tensor at specified points. The situation is modelled as the deformation of an extendible plane sheet. Let the point P , whose coordinates, when referred to axes fixed in space are (x, y), move to P 0 , with coordinates (x + u, y + v). A quantity known25 as the engineering strain tensor at point P 0 is defined as   ∂u ∂u    ∂x ∂y  e e12  = eij = 11 (23)  ∂v ∂v  . e21 e22 ∂x ∂y A general deformation is equivalent to a strain and a rotation, so the engineering strain tensor may be expressed in terms of the strain and rotation tensors as follows. Any second-order tensor may be expressed as the sum of a symmetrical and an antisymmetrical tensor. We may write eij = ij + $ij , where ij = 12 (eij + eji ) and $ij = 12 (eij − eji ). One useful measure, the effective plastic strain, may then be defined r s 2X 2 2 eq = (ij ) = (11 + 22 + 2212 ). (24) 3 ij 3 The strain tensor can be estimated from data in a number of ways. In our first study, 10 we fitted a single strain tensor using a large number of displacement vectors uniformly sampled from a pair of images. With each displacement vector is associated a confidence value. The aim was to find one strain tensor to explain the homogeneous deformation

50 µm

50 µm

Figure 1. Typical microstructure of fallow deer antler bone. The dark areas are Haversian canals around which osteons grow. Other displacement mapping techniques might obscure this structure to the detriment of analysis. The direction of bone growth is horizontal on the left image and vertical on the right image.

of the material. However, most interesting practical situations are inhomogeneous. Therefore we later 14 analysed such situations in the following way: (a) obtain the observed displacement map from a pair of images; (b) find the strain tensor that describes the displacements with least error; (c) use the strain tensor to synthesise a new displacement map; and (d) compare the synthetic map with the observed map by finding the map of residuals – the vector differences between each observation vector and its fitted value. However, both studies 10, 14 fit the strain tensor using a least-squares method, which is vulnerable to outliers. Following Zhang26 an alternative is to use a robust M-estimate. With strain tensors thus obtained we would expect to see a better fit together with higher residual values associated with outliers. However, even this approach has the more general drawback of using a single strain tensor to explain a situation of several heterogeneous deformations. Therefore, another approach we are currently exploring is to segment the surface into regions of approximately similar strain using a RANSAC method.27

7. APPLICATION CASE STUDIES 7.1. Antler: One of Nature’s Composite Materials Antler is a natural composite with remarkable toughness and strength, properties that are put to good use by deer during mating rituals. Antler has a porous core surrounded by cancellous, dense bone in a configuration reminiscent of the honeycomb structures used in aerospace components. This structure is ideal for resisting the dominant bending loads that it must survive, whilst keeping weight to a minimum. Most of the flexural strength of antler derives from the outer bone layer, which is characterised here. The microstructure of antler bone can be seen in Fig. 1. Oesteons are large (100 µm) structures made up of layers surrounding a central vessel that carries nutrients during bone growth. The layers surrounding these Haversian canals are themselves composites, formed by wound micro-fibrils of collagen and hydroxyapatite. Bone can be therefore be thought of as a long fibre composite, with long osteons aligned in the direction of growth and will therefore be highly isotropic. As a result the properties of the composite need to be evaluated in more than one direction. Traditionally this would involve using strain gauges to measure the strains on different directions. However, the attachment of strain gauges to natural materials is always difficult. Furthermore is it is likely that the strain gauge adhesive will interact with the material, affecting the value of strains measured. Two antler bone samples were machined from European fallow deer (Dama dama) antler, one parallel and one perpendicular to the direction of growth. A Deben 2000N microtester, controlled via a computer, was used to strain the samples to failure while viewed under an optical microscope. Images (1300×1000 pixels) were acquired with a cooled 12-bit CCD Axiocam from Zeiss connected to a computer via a frame grabber. The stage and camera were controlled by Axiovision software, also by Zeiss. This allowed real-time focus adjustment and real-time image acquisition. Loss of focus introduces errors in the displacement determination and must be kept to a minimum. Automatic focusing could be

(b)

(a)

(c)

Figure 2. Magnified region of antler image (a) before and (b) after cracking. (c): A displacement map calculated from a pair of intermediate frames. Note the outlier near the lower left corner.

(a)

(b)

Figure 3. Stress-strain behaviour measured for the antler: (a) loaded in the direction of growth; (b) loaded perpendicular to growth direction.

used but this has been found to be inaccurate due to natural contrast changes in the images due to deformation alone. A motorised stage also allows translation of the sample so that the same area can be imaged throughout the test. Successive frames were analysed using the probabilistic algorithm of Section 3 to build a picture of the local and field averaged strains throughout both tensile tests. Figure 2 shows the first and last frames and one of the interim displacement maps, calculated from the affine (uniform) strain component. In addition to the uniform component, strain localisation was observed, ultimately leading to microcracking even far from the final fracture location. This can be clearly seen by comparing the magnified images in Figure 2. This damage mechanism is responsible for the high strain to failure values measured using optical correlation of 3% in the longitudinal direction and 1.8% in the transverse direction and is probably the main toughening mechanism in antler. This microcracking is only visible when the bone is loaded; upon unloading, the cracks close. Therefore, a post-mortem investigation would not detect them. Because the algorithm used here does not obscure the surface, events such as microcracking can be observed and directly linked with the general stress strain curve, the local microstructure, and the general stress behaviour. Ek (GPa) 11

E⊥ (GPa) 6

νk 0.30

ν⊥ 0.21

σkmax (MPa) 145

max σ⊥ (MPa) 58

Table 1. Mechanical properties of fallow deer antler bone. E is Young’s modulus σ11 /e11 , ν is Poisson’s ratio −e22 /e11 , and σ max is tensile strength (stress value at failure). Subscript k denotes the specimen loaded parallel to the osteon growth direction, and ⊥ the specimen loaded perpendicular to the growth direction.


 2 mm

Figure 4. Part of an image of aluminium sheet specimen, showing abrasion. Here the sample is rotated 90 degrees for compact layout. The stress direction is indicated by the arrows.

0.1

0.05

0 0

0.15 Field Averaged Strain

0.15 Field Averaged Strain

Field Averaged Strain

0.15

0.1

0.05

0.02 0.04 Total strain

(a)

0.06

0 0

0.1

0.05

0.02 0.04 Total strain

(b)

0.06

0 0

0.02 0.04 Total strain

0.06

(c)

Figure 5. The dark line plots the field average strain for the (a) top, (b) middle and (c) bottom of the aluminium sheet specimen tested, compared with the total sample averaged strain (dashed line).

The field averaged strain response was evaluated from subsequent pairs of images obtained during loading as is plotted together with the corresponding stress in Figure 3. The stress was calculated from logged load data obtained from the load cell of the Deben microtester. The mechanical properties extracted from the resultant stress strain curve are summarised in Table 1. Note the extensive ‘plastic’ strain measured both parallel and perpendicular to growth direction. This data agrees well with previous published work.28 The magnified regions show that the cracks form between the lamella that make up the osteons, which are only weakly bonded, and only rarely cross them. Because of a lack of ordering of these osteons, the cracks tend to remain isolated and a very high density of non-percolating cracks can coexist. In effect, the traditional elastic plastic behaviour observed that indicates good energy absorption does not arise from plasticity at all. Instead, the ‘plastic’ strain is accommodated primarily by the occurrence of microcracking. The uncracked regions deform only elastically. Osteons in antler bone are much less organised than those in common bone and therefore a higher density of microcracks can build up without failure. This is part of the reason why antler bone is tougher than common bone.

7.2. Aluminium Alloy Strain mapping can be a important tool in metal forming. Aluminium alloys are widely used in the aerospace and beverage can industries, and are beginning to extensively replace steel in car bodies. Some of these light alloys, however, exhibit unpredictable heterogeneous deformation, which, during forming, affects the surface finish and imposes restrictions on the forming process. This phenomenon is not fully understood, and the example presented here is part of a project that aims to improve our understanding. Aluminium sheet 0.5mm thick was tested in tension at an elongation rate of 5 mm min −1 . Images were acquired at 2 second intervals as the specimen was loaded to failure. The probability algorithm used here needs high local contrast for best results. To ensure this, the sample was abraded with 500 grit silicon carbide paper. The marks introduced can be seen clearly in Figure 4, which shows an image used in the analysis. The sample imaged is 200mm long and 50mm wide. To investigate the possibility of heterogeneous deformation, the images were divided into three 60mm long fields and analysed separately using the approach described earlier. The strains calculated for the different fields are plotted in Figure 5. The results clearly demonstrate that localised plastic deformation initiates in the top third, where the strain reaches 2.5%, which is approximately 50% more than the overall strain. At 4% overall strain, the strain in the top third is the lowest and increases dramatically in the bottom third where it reaches 10%. This propagation of strain is typical of

the deformation in these aluminium sheets and is symptomatic of local variations in yield stress that are responsible for defects that arise during forming. The large strains measured here could not be measured practically with interferometry techniques. On the other hand, optical displacement revealed by image processing yields full field information with only a simple experimental setup. Future work will involve magnification studies on samples etched to reveal the underlying grain structure. In this way the algorithm will be used to relate the microstructure to the development of heterogeneous straining.

8. CONCLUSIONS The probabilistic method for computing optical displacement maps has given good results for the analysis of image sequences for strain mapping of materials. In the particular cases shown here, we used materials exhibiting high porosity and propagation of distortion, situations that pose challenges to previous methods. In summary, viewing the problem probabilistically has three advantages29: (1) it produces useful extensions including the incorporation of prior bias, (2) it provides principled confidence information, and (3) it provides a framework for combining displacement information with probabilistic information from other sources. Future work should include further evaluation of priors and confidence values, and putting the strain tensor calculations into a robust estimation framework.

ACKNOWLEDGMENTS This work was supported in part by EPSRC grant GR/M29177. We thank Dr Paul Mummery and the National Parks of England and Wales for supplying the deer antler sample.

REFERENCES 1. S. S. Beauchemin and J. L. Barron, “Computation of optical flow,” ACM Computing Surveys 27(3), pp. 433–467, 1995. 2. C. A. Walker, “A historical review of Moir´e interferometry,” Experimental Mechanics 34, pp. 281–229, 1994. 3. A. Bastawros and R. McManuis, “Use of digital image analysis software to measure non-uniform deformation in cellular aluminum alloys,” Experimental Techniques 22, pp. 35–37, 1998. 4. M. A. Sutton, W. J. Wolters, W. H. Peters, W. F. Ranson, and S. R. McNeill, “Determination of displacements using an improved digital correlation method,” Image and Vision Computing 1(3), pp. 133–139, 1983. 5. S. Samarasinghe and G. D. Kulasiri, “Displacement fields of wood in tension based on image processing: Part 1. tension parallel- and perpendicular-to-grain and comparisons with isotropic behaviour,” Silva Fennica 34(3), pp. 251– 259, 2000. 6. A. P. Reynolds and F. Duvall, “Digital image correlation for determination of weld and base metal constitutive behavior,” Welding Research Supplement , pp. 355–360, October 1999. 7. D. Zhang, X. Zhang, and G. Chen, “Compression strain measurement by digital speckle correlation,” Experimental Mechanics 39(1), pp. 62–65, 1999. 8. L. Allais, M. Bornert, T. Bretheau, and D. Caldemaison, “Experimental characterization of the local strain field in a heterogeneous elastoplastic material,” Acta Metallurgica et Materialia 42(11), pp. 3865–3880, 1994. 9. W. H. Peters and W. F. Ranson, “Digital imaging techniques in experimental stress analysis,” Optical Engineering 21(3), pp. 427–431, 1982. 10. K. F. Chivers and W. F. Clocksin, “Inspection of surface strain in materials using optical flow,” in Eleventh British Machine Vision Conference, pp. 392–401, (Bristol), September 2000. 11. B. K. P. Horn and B. G. Schunck, “Determining optical flow,” Artificial Intelligence 17, pp. 185–203, 1981. 12. K. F. Chivers, “Surface strain mapping of heterogeneous materials,” CPS dissertation, Department of Materials Science and Metallurgy, University of Cambridge, 1997. 13. W. F. Clocksin, “A new method for estimating optical flow,” Tech. Rep. 436, Computer Laboratory, University of Cambridge, 1997.

14. W. F. Clocksin, K. F. Chivers, P. H. S. Torr, J. Quinta da Fonseca, and P. J. Withers, “Inspection of surface strain in materials using dense displacement fields,” in 4th International Conference on New Challenges in Mesomechanics, (Aalborg), August 2002. 15. H. W. Schrier, J. R. Braasch, and M. A. Sutton, “Systematic errors in digital image correlation caused by intensity interpolation,” Optical Engineering 39(11), pp. 2915–2921, 2000. 16. J. H. Sun, D. A. Yates, and D. E. Winterbone, “Measurement of the flow field in a diesel engine combustion chamber after combustion by cross-correlation of high-speed photographs,” Experiments in Fluids 20, pp. 335–345, 1996. 17. Q. X. Wu, “A correlation-relaxation-labeling framework for computing optical flow – template matching from a new perspective,” IEEE Transactions on Pattern Analysis and Machine Intelligence 17(8), pp. 843–853, 1995. 18. M. J. Black and P. Anandan, “The robust estimation of multiple motions: parametric and piecewise-smooth flow fields,” Computer Vision and Image Understanding 63(1), pp. 75–104, 1996. 19. P. W. Holland and R. E. Welsch, “Robust regression using iteratively reweighted least squares,” Commun. Statist. Theor. Meth A 6, pp. 813–827, 1977. 20. P. J. Huber, Robust Statistics, John Wiley and Sons, New York, 1981. 21. S. Fu and T. Pridmore, “Image flow field detection,” in SPIE International Conference on Signal Processing, (Beijing), 1996. 22. S. Fu, “Strain distribution,” in Topical Discussion Meeting on the Application of Fractals in Engineering Analysis, (Wolverhampton), September 1999. 23. N. Dodgson, “Image resampling,” Tech. Rep. 261, Computer Laboratory, University of Cambridge, 1992. 24. H. A. Bruck, S. R. McNeill, M. A. Sutton, and W. H. Peters, “Digital image correlation using newton-raphson method of partial differential correction,” Experimental Mechanics 29, pp. 261–267, 1989. 25. J. F. Nye, Physical Properties of Crystals: Their representation by tensors and matrices, Oxford University Press, 1957. 26. Z. Zhang, “Parameter estimation techniques: A tutorial with application to conic fitting,” Image and Vision Computing 15(1), pp. 59–76, 1997. 27. P. H. S. Torr and D. W. Murray, “The development and comparison of robust methods for estimating the fundamental matrix,” International Journal of Computer Vision 24(3), pp. 271,–300, 1997. 28. S. A. Maskil, “Micromechanisms of failure in natural composite materials,” PhD thesis, University of Leeds, 1999. 29. E. P. Simoncelli, E. H. Adelson, and D. J. Heeger, “Probability distributions of optical flow,” in IEEE Conference on Computer Vision and Pattern Recognition, pp. 310–315, (Hawaii), June 1991.