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Computer vision tools to optimize reconstruction parameters in x-ray in-line phase tomography
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Institute of Physics and Engineering in Medicine Phys. Med. Biol. 59 (2014) 7767–7775
Physics in Medicine & Biology doi:10.1088/0031-9155/59/24/7767
Computer vision tools to optimize reconstruction parameters in x-ray in-line phase tomography H Rositi, C Frindel, M Wiart, M Langer, C Olivier, F Peyrin and D Rousseau Université de Lyon, Laboratoire CREATIS, CNRS UMR5220, INSERM U1044, Université Lyon 1, INSA-Lyon, 69621 Villeurbanne, France E-mail:
[email protected] Received 9 May 2014, revised 31 August 2014 Accepted for publication 23 September 2014 Published 24 November 2014 Abstract
In this article, a set of three computer vision tools, including scale invariant feature transform (SIFT), a measure of focus, and a measure based on tractography are demonstrated to be useful in replacing the eye of the expert in the optimization of the reconstruction parameters in x-ray in-line phase tomography. We demonstrate how these computer vision tools can be used to inject priors on the shape and scale of the object to be reconstructed. This is illustrated with the Paganin single intensity image phase retrieval algorithm in heterogeneous soft tissues of biomedical interest, where the selection of the reconstruction parameters was previously made from visual inspection or physical assumptions on the composition of the sample. Keywords: x-ray in-line phase tomography, phase retrieval, computer vision, biomedical imaging (Some figures may appear in colour only in the online journal) 1. Introduction In x-ray in-line phase contrast imaging, the refraction of a partially coherent x-ray beam by an object of interest slightly modifies the original wavefront profile. These variations result in changes in the locally transmitted intensity of the wave which contains quantitative information on the phase shift induced by the object. Due to its enhanced contrast over standard attenuation imaging, x-ray in-line phase contrast imaging is receiving more and more attention in medecine and biology (for a review and recent examples in this journal, see Bravin et al (2013), Cedola et al (2014) and Willner et al (2014)). There are a number of phase sensing 0031-9155/14/247767+9$33.00
© 2014 Institute of Physics and Engineering in Medicine
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computational techniques used in coherent x-ray optics (Paganin 2006). In this article, we focus on one of the most popular for its practical interest in terms of acquisition duration. Using the Paganin’s method (Paganin et al 2004) which we briefly recall, phase retrieval at each projection angle can be performed from a single intensity image Iω(x, y, z = ∆) acquired at a distance ∆ from the exit-surface of the object. In the near-field condition, the phase of the wave field ψω(x, y, z) at the exit-surface of the object z = 0 can be retrieved with ψω (x, y, 0) = −
⎛ ⎡ F [Iω (x, y, Δ) / Iin ] ⎤ ⎞ δω ln ⎜⎜F−1 ⎢ ⎥ ⎟⎟ , 2βω ⎝ ⎣ 1 + (πΔλδω / βω ) (k 2 ) ⎦ ⎠
(1)
where F−1 and F stand for the inverse and direct Fourier transforms, k 2 = kx2 + k y2 with kx and ky the Fourier-space coordinates reciprocal to (x, y), Iin is the intensity image captured in the absence of the object, ω the angular frequencies of the x-ray wave assumed to be coherent and the refractive index n = 1 −δω + iβω which depends on ω and on the material composition of the object. As shown in equation (1), the computation of the phase requires knowledge of the ratio δω/βω. This is possible for a homogeneous sample of known composition (Chen et al 2011). Biological tissues are in general heterogeneous, so the exact composition cannot be used as a prior (Langer et al 2014). Consequently, the selection of the δω/βω in Paganin’s method for biological heterogeneous tissues is an open problem of current interest. Various approaches have been proposed to inject priors for the selection of δω/βω, or to introduce several δω/βω for specific applications (Beltran et al 2010, Beltran et al 2011, Sidky et al 2010, Langer et al 2012, Langer et al 2014, Rositi et al 2013). However, the selection of δω/βω is generally made by choosing a range of values from the expected material composition of the imaged object, and then by heuristically selecting the best image from the visual point of view of the expert in charge of producing the 3D images. In this article, we demonstrate the possibility of replacing the expert’s eye to optimize δω/βω with non subjective computer vision tools. 2. Material and methods We used images of a mouse brain recently acquired on the synchrotron radiation beamline ID19 of the European Synchrotron Radiation Facility (ESRF) in Grenoble, France, with the experimental image acquisition setup described in detail in Rositi et al (2013). It was charach terized by a beam of energy E = ω = 17.6 keV, an object-to-detector distance ∆ = 1 meter 2π and produced after reconstruction a 3D volume of 2000 slices of 2048 × 2048 voxels with a slice thickness equal to pixel size (isotropic voxels) 8 µm. A qualitative illustration of the influence of the reconstruction parameter δω/βω is provided on one slice of these samples in figure 1. In this section, we describe the three computer vision tools that we introduce to quantify the quality of the reconstructed images when selecting δω/βω in the Paganin algorithm of equation (1). 2.1. Optimizing similarity to a reference
First we consider the case where a pre-existing atlas image of the object to be imaged can be used as a reference to tune the δω/βω in Paganin’s method. X-ray in-line phase tomography produces stacks of images with spatial resolutions similar to those obtained in standard optical 2D histology. 2D histological atlases of the usual structures of interest in medicine do exist (Paxinos and Franklin 2001). Such a histological view of reference is given for a mouse brain in figure 1(a) and we consider in figures 1(b)–(f) a virtual ‘slice’ 7768
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(a)
(b)
(c)
(d)
(e)
(f) Figure 1. Six representations of our data. (a) Is a histological slice of a mouse brain
reproduced with permission from Paxinos and Franklin (copyright 2001) used as a reference. (b)–(f) are the x-ray in-line phase contrast reconstructed images of a mouse brain at a slice corresponding to the view of (a) using different δω/βω = 50, 200, 321, 600, 1250 in the reconstruction method of equation (1). The area inside the green dashed rectangle represents the structure of interest considered for the number of SIFT matches of section 2.1 and the focus measure FM of equation (3) in section 2.2. Sub-figures 1 and 2 are used for the tractography of section 2.3 and each corresponds to a zoom on a square of size 80 × 80 µm2 respectively, located in a region without neural fibres and in a region of white matter with bundles of neural fibres.
of our sample similar to the view in histology but reconstructed by x-ray in-line phase tomography with various values of the reconstruction parameters δω/βω of equation (1). From a visual inspection of figure 1, one can determine that figure 1(a) has a greater similarity to figures 1(d) or (e) than to figures 1(b), (c), or (f). Large values of δω/βω produce blurry images with loss of contrast while small values of δω/βω yield noisy images. This is particularly visible in the sub-figures of figure 1 where a zoom on a bundle of neural fibres in white matter is depicted for comparison with a zoom on a region free from any perceptible structure. We propose to quantify this qualitative observation with a measure of similarity between figure 1(a) and images like figures 1(b)–(d) and (f ) reconstructed with various δω/βω. The reference atlas image can be misaligned with the sample image. Also, due to mechanical stresses during manipulation, the sample can suffer from deformations compared to the atlas. Consequently, a pixel to pixel registration of the reference atlas with the sample image can be considered a difficult task and a pixel-based similarity measure is not adapted to tune the δω/βω. Instead, we propose a measure of similarity based on local descriptors. A large variety of strategies for the extraction of local features have been reported in the literature, from the early corner detectors up to more recent developments (see Li and Allison (2008) for a recent review). We used SIFT, for 7769
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scale invariant feature transform (Lowe 2004), implemented under the didactic version of Vedaldi and Fulkerson (2008). SIFT first searches for keypoints corresponding to scale invariant extrema in the gradient of a series of smoothed and resampled versions of the original image. For each keypoint, a description vector is computed through measures in the spatial neighborhood of the gradient. The number of pairs of keypoints with similar description vectors, called SIFT matches, are here, and as in Delahaies et al (2012), are used as a measure of local similarity between the atlas image and the sample image. The matching procedure includes two filtering steps. First, to ensure that the distance between the two closest description vectors Sa from the reconstructed image and Sb from the reference image is small enough compared to the distances from all other keypoints, a threshold ν is applied following ∀i, D (Sa, Sb ) * ν ⩽ D (Sa, Si ) , Sa ≡ Sb
(2)
where D is the Euclidian distance between description vectors and i denotes all the other keypoints from the reconstructed image. Secondly, for each match (Sa,Sb) another threshold T expressed in pixels is applied on the Euclidian distance between the position of the keypoints. We tolerate a small spatial shift of the associated keypoints found below T but we filter locally similar structures that would correspond to anatomically different structures when the distance between the keypoints is larger than T. 2.2. Optimizing the focus in 2D
We introduce a second computer vision tool where we assume that the scale σ of the structures of interest in the reconstructed image is known. In this case, the optimal selection of δω/βω in equation (1) is expected to produce sharp structures at this scale. This can be quantified objectively with measures of focus. There are a wide variety of such measures of focus (see Pertuz et al (2013) for a recent review). In this work, we use for illustration a simple one introduced in Groen et al (1985). We first compute for each pixel (i, j) FM (i, j ) =
1 (2N + 1)2
i+N
j+N
∑ ∑
[I (x, y ) − I (i, j )]2 ,
(3)
x=i−N y=j−N
with I (i, j ) the average intensity value around pixel (i, j) of intensity I(i, j) on a neighborhood N = round(σ/Spix) with Spix = 8 µm the pixel size. FM (i, j) is then averaged over the whole image to produce the average local standard deviation of the measured intensity FM considered as our focus measure. In this article, we choose to apply our second computer vision tool FM to enhance the neural fibres depicted in the zoom sub-figures 2 of figure 1 as the structure of interest. In figure 1, these fibres are tubular structures with their main elongated direction oriented perpendicularly to the image. The scale corresponding to their radius can be estimated at σ = 32 µm. 2.3. Optimizing the focus in 3D
As a third computer vision tool, we extend the approach of the previous subsection in 3D. We consider the same structure of interest, i.e. the bundle of neural fibres in white matter, and add the prior knowledge that they are tubular structures in 3D. In this case, the optimal selection of δω/βω in equation (1) is expected to produce continuous fibres. This can be quantified objectively with measures based on fibre tractography. In this work, for illustration we follow the approach of Cetin et al (2013) based on the seminal work of Frangi et al (1998), where the 7770
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Hessian matrix is computed on each voxel (i, j, k) after convolution by a Gaussian 3D kernel with standard deviation corresponding to the expected scale N = (σ/Spix) of the radius of the neural fibres. The fractional anisotropy index FA is then computed at each voxel (i, j, k) as FA (i, j, k ) =
1 2
(λ1 − λ2 )2 + (λ1 − λ3)2 + (λ2 − λ3)2 λ12 + λ22 + λ32
,
(4)
where λ1, λ2, λ3 are the eigenvalues of the Hessian matrix at voxel (i, j, k). FA is an extension of the concept of conic sections represented by a scalar value normalized to the unit range that describes the degree of anisotropy in 3D. A value of FA equal to one is obtained when the filtered Hessian is mainly oriented only along one axis while an isotrope filtered Hessian provides a FA equal to zero. The tractography is then based on the 3D fields of the principal eigenvector (corresponding to the largest eigenvalue) with a continuity of high FA values expected along the neural fibres. Two voxels are considered to be connected by a fibre if their fractional anisotropy FA is larger than a threshold arbitrarily (but without qualitative influence of the following results) taken as 0.1. 3. Results We present the evolution observed with the three computer vision tools described in the previous section for various δω/βω in equation (1) when applied to the heterogeneous biological sample of figure 1 taken for illustration. Figure 2 gives the number of SIFT matches as a function of δω/βω computed in the region of interest inside the dashed rectangle of figure 1. This shows a nonmonotonic evolution culminating for an optimal δopt/βopt = 200 of the same order of magnitude to the one picked up by the eye of the expert from visual inspection of figure 1. The influence of thresholds ν and T is illustrated in figure 2. The number of SIFT matches decreases when the thresholds ν or T in the matching procedure are increased. However, the nonmonotonic evolution is preserved and δopt/βopt does not critically depend on ν or T. As shown in figure 2, FM of equation (3) also shows a nonmonotonic evolution culminating for an optimal δopt/βopt = 600. In figure 3, this nonmonotonic evolution is observed for various sizes of neighborhood N corresponding to the typical scale of neural fibres. This optimal value corresponds only to a local optimal for the SIFT measure. However, here we place the focus on the scale of the neural fibres specifically whereas the SIFT was not applied on a predefined scale. Moreover the optimal δopt/βopt found by the focus measure FM is still in good agreement with the visual inspection provided by the focus on neural fibres given in figure 1. Then we applied the fibre tractography procedure of the previous section to the volumes corresponding to sub-figures 2 of figure 1, centred on a bundle of neural fibres. The results are depicted in figure 4 for three different values of δω/βω. We then computed the mean fibre length and the number of fibres stopped because of low FA as a function of δω/βω. As shown in figure 5, these two descriptors of 3D focus show a nonmonotonic evolution culminating for an optimal δopt/βopt around 600, i.e. the same value found for the focus measure applied in 2D. This is qualitatively illustrated in figure 4 where a low δω/βω yields multiple small noisy fibres in various directions, while for δω/βω that is too large, some bundles of fibres disappear completely because of reconstructed images that are too blurry, as is visible in sub-figures 2 of figure 1(f). Figure 5 also demonstrates that the nonmonotonic evolution is not found to be critically dependent on the choice of the threshold applied to detect the bundle of fibres with a low FA. 7771
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Figure 2. Number of SIFT matches as a function of δω/βω of equation (1) with various
distance threshold values (left) ν = 1.15, 1.23, 1.35, (right) T = 80, 120, 170 both in solid, dashed and dashed–dotted lines, respectively.
Figure 3. Focus measure FM of equation (3) as a function of δω/βω of equation (1) with
various sizes of neighborhood N = 4, 5, 6 pixels in solid, dashed and dashed–dotted lines, respectively.
4. Discussion The previous section demonstrated that it is possible to automatically select an optimal reconstruction parameter δω/βω in equation (1) with the help of computer vision tools built up to visually enhance the structures of interest in the images. We have systematically illustrated the relative robustness of the optimal selection obtained from these computer vision tools with the tuning of their own parameters. Optimization was performed here on a single image, however it would be straightforward from the work presented here to search for average optimal reconstruction conditions on multiple images. Furthermore, some computer vision tools used in this article gave more or less pronounced variations around the optimal value of the reconstruction 7772
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Figure 4. Three representations of the result of the fibre tractography procedure based
on FA of equation (4) applied on the same isotropic sub-volume of size 80 × 80 × 80μ m3 for various δω/βω = 50, 600, 1250 respectively from (a) to (c).
Figure 5. Tractography measures based on the fractional anisotropy FA of equation (4)
as a function of δω/βω of equation (1) with the number of fibres stopped by low FA on the left and the mean fibre length in microns on the left. The various curves correspond to various values of threshold on FA.
parameter and it would be important to search for the most sensitive metrics for a given application. Instead, we would like to stress the originality of the image processing approach developed here by comparing it to the usual physical approach based on material composition for the selection of the parameters of equation (1). In the experiments presented in figure 1, iron oxide nanoparticles were injected into the mouse brains. Following the physical approach of Paganin et al (2004), the δω/βω would be advised at 321 (see figure 1(c)) assuming that the brain contains iron. This allows one to have a physical measurement of the phase at the output of the object, i.e. gray levels which are approximately proportional to radians. This physical approach is important when images are acquired for metrological applications on the phase. But, when structures in the image are to be detected rather than radiometrically analysed, an image processing approach may also be useful (for other recent demonstrations of this point in x-ray in-line phase contrast imaging, see Chou and Anastasio (2009), Garson et al (2013), Rositi et al (2013), Nesterets and Gureyev (2014) and Scholkmann et al (2014)). From an image processing point of view the transformation of equation (1) in Paganin’s reconstruction method is equivalent to a low-pass filter on the spatial frequencies of the single intensity image followed by a logarithmic operator. Therefore, from this perspective, the selection of the reconstruction parameter δω/βω in equation (1) is a matter of the selection of the range of spatial frequencies expected to visually appear in the reconstructed image, rather than a 7773
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physical material composition of the sample. Automated or semi-automated selection of the reconstruction parameter is therefore possible when this range of spatial frequency is a prior. This is a situation found in medecine and biology thanks to atlases of anatomy or structural biology. This change of perspective from phase metrology to image processing opens a variety of possible image processing metrics to select the reconstruction parameter δω/βω. As a proof of feasibility, we have demonstrated the interest of this image processing point of view with three different metrics based on common computer vision tools in 2D or 3D and various types of priors on the expected structures that one expects to visualize in the reconstructed images. 5. Conclusion The selection of the reconstruction parameters in x-ray in-line phase tomography of heterogeneous samples is an important task which is usually performed by visual inspection of the reconstructed images. In this work, we have demonstrated that computer vision tools can be used to replace the eye of the expert in the optimization of these parameters based on quantitative objective measures. This was illustrated with images of a mouse brain with a focus on the neural fibres that are captured by x-ray in-line phase tomography and reconstructed from the Paganin’s reconstruction method. Following this approach, we have proposed three different strategies to inject, in the design of computer vision tools, priors on the shape and scale of the structures to be extracted. These results could be extended to other heterogeneous tissues including various shapes of structures of biological interest where the selection of the reconstruction parameters cannot be determined theoretically. These results could also be extended to other computer vision tools that mimic human vision such as saliency maps (Itti et al 1998) or models of observers (Anastasio et al 2010). Acknowledgments This work was performed within the framework of the LABEX PRIMES (ANR-11LABX-0063) of Université de Lyon, within the program ‘Investissements d’Avenir’ (ANR11-IDEX-0007) conducted by the French National Research Agency (ANR). This work was supported by the European Synchrotron Research Facility (ESRF, project MD-499) through the allocation of beam time. This work was also performed in the context of the French GdR Stic Sante (GdR CNRS 2647). References Anastasio M A, Chou C Y, Zysk A M and Brankov J G 2010 Analysis of ideal observer signal detectability in phase-contrast imaging employing linear shift-invariant optical systems J. Opt. Soc. Am. A 27 2648–59 Beltran M, Paganin D, Siu K K W, Fouras A, Hooper S B, Reser D H and Kitchen M J 2011 Interfacespecific x-ray phase retrieval tomography of complex biological organs Phys. Med. Biol. 56 7353–69 Beltran M, Paganin D, Uesugi K and Kitchen M J 2010 2D and 3D X-ray phase retrieval of multimaterial objects using a single defocus distance Opt. Express 18 6423–36 Bravin A, Coan P and Suortti P 2013 X-ray phase-contrast imaging: from pre-clinical applications towards clinics Phys. Med. Biol. 58 R1–35 Cedola A et al 2014 Three dimensional visualization of engineered bone and soft tissue by combined x-ray micro-diffraction and phase contrast tomography Phys. Med. Biol. 59 189–201
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