Estimating 3D Fibre Orientation in Volume Images
Estimating 3D Fibre Orientation in Volume Images Maria Axelsson Centre for Image Analysis, Swedish University of Agricultural Sciences Box 337, SE-751 05, Uppsala, Sweden [email protected]
Abstract Fibre orientation is an important structural property of fibre-based materials. For example, in paper the orientation of the fibres influences the dimensional strength of the sheet and the tendency of the sheet to curl and twist at moisture changes. Here, we present a threedimensional image analysis method for estimating the fibre orientation and the orientation anisotropy. The proposed method can be applied directly to greyscale volume images and is based on local orientation estimates using quadrature filters and structure tensors. From the tensor field the fibre orientation can be estimated together with a corresponding certainty measurement. Good results are obtained for both synthetic fibre data sets and fibre based materials imaged using X-ray microtomography.
1. Introduction Fibre-based materials are used in many applications. For example, fibre reinforced composites are used as strong lightweight materials and paper is used in many areas of our everyday lives. In structure characterisation of fibre based materials it is important to estimate the fibre orientation, since it determines the mechanical properties of the material. Here, the focus is on wood fibre based materials, such as paper, but the proposed method is directly applicable to other fibre based materials, e.g., glass fibre reinforced composites, fabrics, and press felts. The fibre orientation in paper influences, e.g., the dimensional strength of the sheet and the tendency of the paper to curl and twist at moisture changes. Two-dimensional (2D) methods have been proposed to measure the fibre orientation, e.g., in [1] a sheet splitting and gradient image analysis method is presented for paper. Stereological methods are common, but not suited for natural fibres due to large shape variations. Also, the three-dimensional (3D) fibre orientation can
978-1-4244-2175-6/08/$25.00 ©2008 IEEE
not be estimated using 2D methods. Volume images of materials have become available through advances in imaging techniques and high resolution images can be acquired using X-ray microtomography [5]. In such data sets it is possible to estimate the full 3D fibre orientation. However, there are few previously proposed 3D methods. In [4] a method using anisotropic Gaussian filters is presented. A fixed number of filters with varying orientation is used and the orientation of the filter with the largest filter output is selected as the fibre orientation in each voxel. This approach becomes computationally heavy when good angular resolution is needed. Also, there is no certainty measurement for the estimated orientations. In this paper, the fibre orientation is assumed to correlate directly with the local material orientation. Based on this assumption, the fibre orientation in a local neighbourhood is modeled as the orientation with the least signal variation. We propose a method for estimating the 3D fibre orientation. The method is based on a phase invariant orientation estimate using quadrature filters [2]. It can be directly applied to greyscale volume images and is not dependent on segmentations of individual fibres or segmentations of the image into fibre material and void space. Estimates of the fibre orientation are obtained for each voxel and can be averaged for parts of, or the whole, volume image. A certainty measurement is available for each estimated orientation. The fibre orientation distribution and anisotropy in the material can be calculated directly from the orientation estimates. The method is described in Section 2, experiments and results are presented in Section 3, and a discussion and concluding remarks are found in Section 4.
2. Method The local 3D orientation in a neighbourhood of a voxel can be estimated using a set of quadrature filters, where each filter is sensitive to signal variation in a cer-
tain orientation. The framework for local orientation estimation using quadrature filters is thoroughly described in [2]. In 3D, six quadrature filters are needed to get a symmetric distribution of the filter orientations. The filters are phase invariant and the real part of each filter corresponds to a line detector and the complex part corresponds to an edge detector. The shape of the filters is preferably optimised to have as small error as possible compared to the ideal filters both in the spatial and the Fourier domain [2]. The centre frequency of the radial function of the filters should be chosen to correspond to the size of the structures to be detected. The filters should be sensitive to the dimensions of the fibre walls in the case of tubular fibres and the cross section of the fibres in the case of solid fibres. The input volume image is convolved with the quadrature filters and the filter outputs are used to construct a structure tensor in each voxel. The tensor field is smoothed by convolving each component image with a small Gaussian kernel to obtain better estimates and reduce local errors in the orientation estimates. The fibre orientation is assumed to vary slower than the small scale variations and noise that are suppressed by the smoothing. The eigenvectors and the eigenvalues of the tensor contain information about the local orientations and local structure anisotropy. The eigenvectors of each tensor are calculated and sorted according to the size of the eigenvalues in descending order. The sorted eigenvalues are denoted λ1 , λ2 , and λ3 and the corresponding eigenvectors e1 , e2 , and e3 . We calculate certainty measurements that describe the relative anisotropy of the orientation tensor as, c1 =
λ 1 − λ2 , λ1
c2 =
λ2 − λ3 , λ1
c3 =
λ3 λ1
The fibre orientation is estimated from the smoothed tensor field using the eigenvector e3 of each tensor. The c2 measurement is used as certainty for the estimated orientation. c2 is large in neighbourhoods where the signal varies much more in the e1 and e2 orientations than in the e3 orientation. In the input volume, this corresponds to the case where e3 is parallel to a fibre and e1 and e2 are parallel to the plane of the fibre cross section. If c2 is small, it means that the variation is either found in only one main orientation, indicated by a large c1 value, or in all three orientations, indicated by a large c3 value. c3 is also large if the signal is approximately constant in all orientations. The fibre orientation is estimated in each voxel neighbourhood and estimates for any subvolume can be obtained by averaging the estimates. In the averaging,
the certainty value c2 and the image intensities should be taken into account as weights for each measurement. The orientation distribution and the anisotropy of the distribution are interesting measurements derived from the orientation estimates. These measurements indicate potential differences in material properties in different orientations. When measuring the orientation distribution, each local orientation estimate should be weighted by the image intensity in order to measure the orientation distribution using only voxels that contain fibres. The estimate should also be weighted by the certainty c2 in order to use only the reliable estimates. The anisotropy can be calculated from the sum of outer products of the e3 vectors in the fibre voxels, A, by taking the ratios between the eigenvalues of the matrix. Note that the tensor anisotropy is different from the fibre orientation anisotropy. The tensor anisotropy holds information about the local neighbourhood whereas the fibre orientation anisotropy holds information about the fibres throughout the image.
3. Experiments and results Synthetic test volumes were used to evaluate the method and to find suitable parameters for different fibre dimensions. Volume images with straight synthetic fibres were generated with known dimensions and orientations of the individual fibres and known fibre orientation distribution. Tubular fibres were generated to simulate wood fibres. Here, some results that illustrate the performance of the method are presented for six volumes, V1 – V6 , with different orientation anisotropies. The size of the test volumes was 300 × 300 × 300 voxels, the fibre diameter 5 voxels, and the fibre length 50 voxels. The spatial filter size was 7 × 7 × 7 voxels, the centre frequency π/2, and the bandwidth 2 octaves. The results from measuring the fibre orientation anisotropy are presented in Table 1, where r1 , r2 , and r3 denote the eigenvalues of the outer product matrix, A, sorted in descending order. As can be seen in the table, the estimated anisotropies correspond well to the ground truth. Volume renderings showing the ground truth orientations and the estimated orientations from a part of volume V2 are shown in Figure 1. The colours indicate the orientation in each voxel neighbourhood. As seen in the figure, the correct orientation is estimated for most voxels. The model does not hold for the fibre ends, where the largest errors occur. However, note that this error is too small to be clearly visible in the figure. The HSV colour space is used to construct the colour map used in Figure 1. The projection of the orientation vector on the xy plane is mapped to hue with the
90
XY
V1 V2 V3 V4 V5 V6
Table 1. Anisotropy estimation. Ground truth Estimated anisotropy r1 /r2 r2 /r3 r1 /r2 r2 /r3 1.0329 2.2618 1.0149 2.2475 1.6502 2.3094 1.5572 2.3444 1.0609 1.0159 1.0475 1.0135 1.8047 1.6483 1.8370 1.6974 2.2747 1.0580 2.1827 1.0601 3.6232 1.0466 3.4115 1.0535
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Figure 2. The ground truth and estimated fibre distributions.
Figure 1. (Top) Ground truth orientations. (Bottom) Estimated orientations.
same colour for all pairs of parallel vectors. The absolute value of the z coordinate is mapped to saturation, with full saturation for the equator of the sphere. Further, the certainty of the estimated orientation can be
mapped to the value component. This is not used in the illustration, but is suitable when evaluating the result in 2D slices. The proposed colour map is intended for paper, which is a highly layered structure where the fibres lay mainly in the xy plane. Other colour schemes for visualising orientations are found in [3]. The mean difference in angle between the ground truth orientations and the estimated orientations weighted by the image intensities is approximately 3.8 degrees in all test images. However, this measurement is dependent on the length of the fibres, since the largest errors occur at the fibre ends. If the mean difference is also weighted by the certainties c2 it is reduced to approximately 3.0 degrees. This measurement is more relevant, since voxels with low certainties should not be used as estimates. The correspondence between the estimated fibre orientation distribution and the ground truth for volume V2 is presented in Figure 2. The orientation vectors are projected on the xy and yz image planes and binned based on the angle. Clearly, good correspondence is obtained. The method was also applied to a fibre reinforced
Figure 3. Estimated orientation in a wood fibre based composite material.
composite material that consists of a plastic matrix reinforced with wood fibres that was imaged using X-ray microtomography at the European Synchrotron Radiation Facility in Grenoble. The original volumes are large images of 1024 × 1024 × 1024 voxels with a voxel size of 0.7 × 0.7 × 0.7 µm3 . This high resolution is used to measure other properties of the material, but is not needed to estimate the fibre orientation. Since the method does not assume tubular fibres and the fibre orientation is a relatively large-scale property the samples can be downsampled to reduce processing time. A volume of 342 × 342 × 342 voxels and the same parameters as for the synthetic data were used. The result is presented in Figure 3. Fibre orientation seems to be well estimated by the method since the same orientation is estimated for entire fibres and fibres in similar orientation have similar colour. For visualisation purposes the orientation is shown only in voxels with high image intensities that correspond to the fibre part of the material. The anisotropy is r1 /r2 = 3.1524 and r2/r3 = 6.2838 using the eigenvalues of the outer product matrix A.
4. Discussion and conclusion In this paper, a new method for estimating the 3D fibre orientation in greyscale volume images has been presented. As shown, the method provides estimates of the fibre orientation with good angular resolution using only six quadrature filters. A corresponding certainty measurement is also available for all estimated orienta-
tions. Fibre orientation estimates are available for all voxels in the image and can be averaged for different layers or parts of the material. The fibre distribution and orientation anisotropy are also easily estimated using the results from the proposed method. The method is based on a model of the material where fibres are elongated structures with small signal variation in the local fibre direction. This makes the method applicable to both tubular structures, such as wood fibres, and solid fibres, that locally appear as lines. It also enables the possibility to downsample large volumes from, e.g., X-ray microtomography, and estimate the fibre orientation in the smaller volumes which reduces processing time. However, the model does not apply to the fibre ends and the estimation error is largest for these voxels. If the certainty values, which are low for the same voxels, are used as weights in the calculations even better estimates are obtained. Further evaluation of the method on real data should be performed by correlating the orientation estimates with the mechanical properties of the materials. This requires imaging of many samples of the same material to provide good statistics. For some materials, e.g., with more than one fibre type, it can be useful to estimate the fibre orientation on multiple scales. The largest c2 value in each voxel could be then used to select the orientation from the best scale.
5. Acknowledgement The author acknowledges Gunilla Borgefors, Joakim ¨ Rydell, and Catherine Ostlund for scientific support.
References [1] A.-L. Erkkil¨a, P. Pakarinen, and M. Odell. Sheet forming studies using layered orientation analysis. Pulp and Paper Canada, 99(1):81–85, 1998. [2] G. H. Granlund and H. Knutsson. Signal Processing for Computer Vision. Kluwer Academic Publishers, 1995. ISBN 0-7923-9530-1. [3] S. Pajevic and C. Pierpaoli. Color schemes to represent the orientation of anisotropic tissues from diffusion tensor data: Application to white matter fiber tract mapping in the human brain. Magnetic Resonance in Medicine, 42:526–540, 1999. [4] K. Robb, O. Wirjandi, and K. Schladitz. Fiber orientation estimation from 3D image data: Practical algorithms, visualization, and interpretation. In Proceedings of 7th International Conference on Hybrid Intelligent Systems, pages 320–325, September 2007. [5] E. Samuelsen, Ø. Gregersen, P.-J. Houen, T. Helle, C. Raven, and A. Snigirev. Three-dimensional imaging of paper by use of synchroton X-Ray microtomography. Journal of Pulp and Paper Science, 27(2):50–53, 2001.
Abstract Fibre orientation is an important structural property of fibre-based materials. For example, in paper the orientation of the fibres influences the dimensional strength of the sheet and the tendency of the sheet to curl and twist at moisture changes. Here, we present a threedimensional image analysis method for estimating the fibre orientation and the orientation anisotropy. The proposed method can be applied directly to greyscale volume images and is based on local orientation estimates using quadrature filters and structure tensors. From the tensor field the fibre orientation can be estimated together with a corresponding certainty measurement. Good results are obtained for both synthetic fibre data sets and fibre based materials imaged using X-ray microtomography.
1. Introduction Fibre-based materials are used in many applications. For example, fibre reinforced composites are used as strong lightweight materials and paper is used in many areas of our everyday lives. In structure characterisation of fibre based materials it is important to estimate the fibre orientation, since it determines the mechanical properties of the material. Here, the focus is on wood fibre based materials, such as paper, but the proposed method is directly applicable to other fibre based materials, e.g., glass fibre reinforced composites, fabrics, and press felts. The fibre orientation in paper influences, e.g., the dimensional strength of the sheet and the tendency of the paper to curl and twist at moisture changes. Two-dimensional (2D) methods have been proposed to measure the fibre orientation, e.g., in [1] a sheet splitting and gradient image analysis method is presented for paper. Stereological methods are common, but not suited for natural fibres due to large shape variations. Also, the three-dimensional (3D) fibre orientation can
978-1-4244-2175-6/08/$25.00 ©2008 IEEE
not be estimated using 2D methods. Volume images of materials have become available through advances in imaging techniques and high resolution images can be acquired using X-ray microtomography [5]. In such data sets it is possible to estimate the full 3D fibre orientation. However, there are few previously proposed 3D methods. In [4] a method using anisotropic Gaussian filters is presented. A fixed number of filters with varying orientation is used and the orientation of the filter with the largest filter output is selected as the fibre orientation in each voxel. This approach becomes computationally heavy when good angular resolution is needed. Also, there is no certainty measurement for the estimated orientations. In this paper, the fibre orientation is assumed to correlate directly with the local material orientation. Based on this assumption, the fibre orientation in a local neighbourhood is modeled as the orientation with the least signal variation. We propose a method for estimating the 3D fibre orientation. The method is based on a phase invariant orientation estimate using quadrature filters [2]. It can be directly applied to greyscale volume images and is not dependent on segmentations of individual fibres or segmentations of the image into fibre material and void space. Estimates of the fibre orientation are obtained for each voxel and can be averaged for parts of, or the whole, volume image. A certainty measurement is available for each estimated orientation. The fibre orientation distribution and anisotropy in the material can be calculated directly from the orientation estimates. The method is described in Section 2, experiments and results are presented in Section 3, and a discussion and concluding remarks are found in Section 4.
2. Method The local 3D orientation in a neighbourhood of a voxel can be estimated using a set of quadrature filters, where each filter is sensitive to signal variation in a cer-
tain orientation. The framework for local orientation estimation using quadrature filters is thoroughly described in [2]. In 3D, six quadrature filters are needed to get a symmetric distribution of the filter orientations. The filters are phase invariant and the real part of each filter corresponds to a line detector and the complex part corresponds to an edge detector. The shape of the filters is preferably optimised to have as small error as possible compared to the ideal filters both in the spatial and the Fourier domain [2]. The centre frequency of the radial function of the filters should be chosen to correspond to the size of the structures to be detected. The filters should be sensitive to the dimensions of the fibre walls in the case of tubular fibres and the cross section of the fibres in the case of solid fibres. The input volume image is convolved with the quadrature filters and the filter outputs are used to construct a structure tensor in each voxel. The tensor field is smoothed by convolving each component image with a small Gaussian kernel to obtain better estimates and reduce local errors in the orientation estimates. The fibre orientation is assumed to vary slower than the small scale variations and noise that are suppressed by the smoothing. The eigenvectors and the eigenvalues of the tensor contain information about the local orientations and local structure anisotropy. The eigenvectors of each tensor are calculated and sorted according to the size of the eigenvalues in descending order. The sorted eigenvalues are denoted λ1 , λ2 , and λ3 and the corresponding eigenvectors e1 , e2 , and e3 . We calculate certainty measurements that describe the relative anisotropy of the orientation tensor as, c1 =
λ 1 − λ2 , λ1
c2 =
λ2 − λ3 , λ1
c3 =
λ3 λ1
The fibre orientation is estimated from the smoothed tensor field using the eigenvector e3 of each tensor. The c2 measurement is used as certainty for the estimated orientation. c2 is large in neighbourhoods where the signal varies much more in the e1 and e2 orientations than in the e3 orientation. In the input volume, this corresponds to the case where e3 is parallel to a fibre and e1 and e2 are parallel to the plane of the fibre cross section. If c2 is small, it means that the variation is either found in only one main orientation, indicated by a large c1 value, or in all three orientations, indicated by a large c3 value. c3 is also large if the signal is approximately constant in all orientations. The fibre orientation is estimated in each voxel neighbourhood and estimates for any subvolume can be obtained by averaging the estimates. In the averaging,
the certainty value c2 and the image intensities should be taken into account as weights for each measurement. The orientation distribution and the anisotropy of the distribution are interesting measurements derived from the orientation estimates. These measurements indicate potential differences in material properties in different orientations. When measuring the orientation distribution, each local orientation estimate should be weighted by the image intensity in order to measure the orientation distribution using only voxels that contain fibres. The estimate should also be weighted by the certainty c2 in order to use only the reliable estimates. The anisotropy can be calculated from the sum of outer products of the e3 vectors in the fibre voxels, A, by taking the ratios between the eigenvalues of the matrix. Note that the tensor anisotropy is different from the fibre orientation anisotropy. The tensor anisotropy holds information about the local neighbourhood whereas the fibre orientation anisotropy holds information about the fibres throughout the image.
3. Experiments and results Synthetic test volumes were used to evaluate the method and to find suitable parameters for different fibre dimensions. Volume images with straight synthetic fibres were generated with known dimensions and orientations of the individual fibres and known fibre orientation distribution. Tubular fibres were generated to simulate wood fibres. Here, some results that illustrate the performance of the method are presented for six volumes, V1 – V6 , with different orientation anisotropies. The size of the test volumes was 300 × 300 × 300 voxels, the fibre diameter 5 voxels, and the fibre length 50 voxels. The spatial filter size was 7 × 7 × 7 voxels, the centre frequency π/2, and the bandwidth 2 octaves. The results from measuring the fibre orientation anisotropy are presented in Table 1, where r1 , r2 , and r3 denote the eigenvalues of the outer product matrix, A, sorted in descending order. As can be seen in the table, the estimated anisotropies correspond well to the ground truth. Volume renderings showing the ground truth orientations and the estimated orientations from a part of volume V2 are shown in Figure 1. The colours indicate the orientation in each voxel neighbourhood. As seen in the figure, the correct orientation is estimated for most voxels. The model does not hold for the fibre ends, where the largest errors occur. However, note that this error is too small to be clearly visible in the figure. The HSV colour space is used to construct the colour map used in Figure 1. The projection of the orientation vector on the xy plane is mapped to hue with the
90
XY
V1 V2 V3 V4 V5 V6
Table 1. Anisotropy estimation. Ground truth Estimated anisotropy r1 /r2 r2 /r3 r1 /r2 r2 /r3 1.0329 2.2618 1.0149 2.2475 1.6502 2.3094 1.5572 2.3444 1.0609 1.0159 1.0475 1.0135 1.8047 1.6483 1.8370 1.6974 2.2747 1.0580 2.1827 1.0601 3.6232 1.0466 3.4115 1.0535
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Figure 2. The ground truth and estimated fibre distributions.
Figure 1. (Top) Ground truth orientations. (Bottom) Estimated orientations.
same colour for all pairs of parallel vectors. The absolute value of the z coordinate is mapped to saturation, with full saturation for the equator of the sphere. Further, the certainty of the estimated orientation can be
mapped to the value component. This is not used in the illustration, but is suitable when evaluating the result in 2D slices. The proposed colour map is intended for paper, which is a highly layered structure where the fibres lay mainly in the xy plane. Other colour schemes for visualising orientations are found in [3]. The mean difference in angle between the ground truth orientations and the estimated orientations weighted by the image intensities is approximately 3.8 degrees in all test images. However, this measurement is dependent on the length of the fibres, since the largest errors occur at the fibre ends. If the mean difference is also weighted by the certainties c2 it is reduced to approximately 3.0 degrees. This measurement is more relevant, since voxels with low certainties should not be used as estimates. The correspondence between the estimated fibre orientation distribution and the ground truth for volume V2 is presented in Figure 2. The orientation vectors are projected on the xy and yz image planes and binned based on the angle. Clearly, good correspondence is obtained. The method was also applied to a fibre reinforced
Figure 3. Estimated orientation in a wood fibre based composite material.
composite material that consists of a plastic matrix reinforced with wood fibres that was imaged using X-ray microtomography at the European Synchrotron Radiation Facility in Grenoble. The original volumes are large images of 1024 × 1024 × 1024 voxels with a voxel size of 0.7 × 0.7 × 0.7 µm3 . This high resolution is used to measure other properties of the material, but is not needed to estimate the fibre orientation. Since the method does not assume tubular fibres and the fibre orientation is a relatively large-scale property the samples can be downsampled to reduce processing time. A volume of 342 × 342 × 342 voxels and the same parameters as for the synthetic data were used. The result is presented in Figure 3. Fibre orientation seems to be well estimated by the method since the same orientation is estimated for entire fibres and fibres in similar orientation have similar colour. For visualisation purposes the orientation is shown only in voxels with high image intensities that correspond to the fibre part of the material. The anisotropy is r1 /r2 = 3.1524 and r2/r3 = 6.2838 using the eigenvalues of the outer product matrix A.
4. Discussion and conclusion In this paper, a new method for estimating the 3D fibre orientation in greyscale volume images has been presented. As shown, the method provides estimates of the fibre orientation with good angular resolution using only six quadrature filters. A corresponding certainty measurement is also available for all estimated orienta-
tions. Fibre orientation estimates are available for all voxels in the image and can be averaged for different layers or parts of the material. The fibre distribution and orientation anisotropy are also easily estimated using the results from the proposed method. The method is based on a model of the material where fibres are elongated structures with small signal variation in the local fibre direction. This makes the method applicable to both tubular structures, such as wood fibres, and solid fibres, that locally appear as lines. It also enables the possibility to downsample large volumes from, e.g., X-ray microtomography, and estimate the fibre orientation in the smaller volumes which reduces processing time. However, the model does not apply to the fibre ends and the estimation error is largest for these voxels. If the certainty values, which are low for the same voxels, are used as weights in the calculations even better estimates are obtained. Further evaluation of the method on real data should be performed by correlating the orientation estimates with the mechanical properties of the materials. This requires imaging of many samples of the same material to provide good statistics. For some materials, e.g., with more than one fibre type, it can be useful to estimate the fibre orientation on multiple scales. The largest c2 value in each voxel could be then used to select the orientation from the best scale.
5. Acknowledgement The author acknowledges Gunilla Borgefors, Joakim ¨ Rydell, and Catherine Ostlund for scientific support.
References [1] A.-L. Erkkil¨a, P. Pakarinen, and M. Odell. Sheet forming studies using layered orientation analysis. Pulp and Paper Canada, 99(1):81–85, 1998. [2] G. H. Granlund and H. Knutsson. Signal Processing for Computer Vision. Kluwer Academic Publishers, 1995. ISBN 0-7923-9530-1. [3] S. Pajevic and C. Pierpaoli. Color schemes to represent the orientation of anisotropic tissues from diffusion tensor data: Application to white matter fiber tract mapping in the human brain. Magnetic Resonance in Medicine, 42:526–540, 1999. [4] K. Robb, O. Wirjandi, and K. Schladitz. Fiber orientation estimation from 3D image data: Practical algorithms, visualization, and interpretation. In Proceedings of 7th International Conference on Hybrid Intelligent Systems, pages 320–325, September 2007. [5] E. Samuelsen, Ø. Gregersen, P.-J. Houen, T. Helle, C. Raven, and A. Snigirev. Three-dimensional imaging of paper by use of synchroton X-Ray microtomography. Journal of Pulp and Paper Science, 27(2):50–53, 2001.
