Automatic articulated registration of hand ... - Semantic Scholar

Image and Vision Computing 27 (2009) 1207–1222

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Image and Vision Computing journal homepage: www.elsevier.com/locate/imavis

Automatic articulated registration of hand radiographs Miguel Á. Martín-Fernández *, Rubén Cárdenes, Emma Muñoz-Moreno, Rodrigo de Luis-García, Marcos Martín-Fernández, Carlos Alberola-López Laboratory of Image Processing, ETSI Telecomunicaciones, University of Valladolid, Campus Miguel Delibes s/n, 47011 Valladolid, Spain

a r t i c l e

i n f o

Article history: Received 4 April 2008 Received in revised form 7 November 2008 Accepted 10 November 2008

Keywords: Elastic registration Bone age assessment Landmarks detection Segmentation Deformable geometry Anatomical structures

a b s t r a c t In this paper, we propose a methodology to automatically carry out registration of hands out of conventional X-ray images. The registration method we describe here will be referred to as ‘‘articulated registration”; the method is a landmark-based elastic registration procedure in which individual bones are affinely registered and soft tissues are elastically registered so that long skeletal structures are maintained straight while a continuous and smooth transformation is obtained all over the image. In order for the method to be fully automatic, the landmarks used for the registration are detected using a number of image processing algorithms. An optimization step for the refinement of the landmarks locations is included within the registration algorithm; the algorithm is based on an iterative procedure to maximize a local similarity measure. A final procedure to correct bone width has also been performed. We show that the articulated registration described here is robust and outperforms alternatives based on the thin-plate splines (TPS) algorithm. The algorithm for automatic landmark position finding has been tested using registered images with landmarks manually selected by an expert. Ó 2008 Elsevier B.V. All rights reserved.

1. Introduction and State of the Art Image registration is the determination of a geometric transformation that maps one image into another, aligning objects in both images [1]. Image registration can be seen as a common problem but also as a useful tool in many application fields such as geophysics, robotics and medical imaging. Registration within medical imaging is of particular interest [1,2]; methods proposed within this field have been classified as frame-based, landmark-based, surface-based, and intensity-based. Methods of the first type are the oldest, and make use of some sort of extrinsic object (such as stereotactic frames) to facilitate registration. The other three categories are intrinsic as they only use anatomical characteristics, such as landmarks [3,4], lines or surfaces [5], or directly the image pixel or voxel intensities [6,7]. Some intrinsic methods use a combination of these features, especially landmarks and image intensities [8,9]. With respect to the geometrical transformation, methods can be either rigid (only rotations and translations are used) [7], non-rigid (including scaling, affine transformations, projections and perspective) [8] and elastic transformations [3,4,6,9,10]; the latter are

* Corresponding author. Address: ETSIT, Universidad de Valladolid, Campus Miguel Delibes s/n, 47011 Valladolid, Spain. Tel.: +34 983 423660; fax: +34 983 42367. E-mail addresses: [email protected] (M.Á. Martín-Fernández), [email protected]. uva.es (R. Cárdenes), [email protected] (E. Muñoz-Moreno), [email protected] (R. de Luis-García), [email protected] (M. Martín-Fernández), [email protected] (C. Alberola-López). 0262-8856/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.imavis.2008.11.001

widely used since local features can be matched while continuity and smoothness in the transformation is maintained. Several elastic image registration algorithms based on the exact matching of corresponding landmarks in two images have been reported. Geometrical transformation are usually chosen to be continuous and differentiable, especially when the morphological changes from one view to the other are unknown. Examples of these transformations include polynomials, polynomial splines and thin-plate splines (TPS) [1]. The latter are based on a physical model of a thin metal plate, and has been widely employed for representing anatomical shape changes in medical images [3,4,11]. Some other transformations are volume spline (VS) and elastic body spline (EBS), based on a mechanical model of a 3D elastic body material [10]. However, these models do not seem very appropriate for regions of the body with an articulated bony skeleton and surrounding soft tissues, as, for instance, hand and knee. The registration method that we propose here has been developed to help medical doctors in the bone age assessment. Bone age is a state of skeletal maturity, and its assessment is a frequently employed procedure in pediatric radiology, as many diseases affecting growth result in a significant discrepancy between bone age and chronological age. In pediatric radiology bone age assessment is often carried out by visual inspection of an X-ray of the non-dominant patient hand, which needless to say, is a tedious and time consuming task. There are two standard methods to estimate the bone age. The first one, namely, the Greulich–Pyle (GP) method [12] is an atlas-driven method, and it is based on visually comparing the patient radiograph with a number of atlas patterns

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of representative images for each age. Bone age is assessed on the basis of the pattern which more accurately resembles the patient radiograph according to the physician perception. The second one, called Tanner–Whitehouse (TW3) method [13], uses a detailed shape analysis of individual bones of interest, defining a set of evolution stages for each bone. Scores are derived from each bone stage and summed up to compute the assessment. The subjective nature of the GP method, and the considerable complexity of TW3 method, make the automation of bone age assessment a highly desirable goal, in order to assist the radiologist in performing a more objective, fast and accurate analysis without the intrinsic variability of human activities. Various attempts have been made to develop a fully or partially automatic bone age assessment system. The first approaches can be found in [14,15]. After those some other relevant works have been also published, for instance approaches using Neural Networks [16–19], or fuzzy logic [20–22]. The use of digital atlases is important to solve this problem, as shown in [23,24]. Despite the preprocessing used in those methods, most of them rely on the identification of some regions of interest (ROIs) of the hand, to extract some features that are them compared to the features of a template or set of templates. It is important then, for a fully automatic method to identify specific ROIs, which can be done aligning or registering hand radiographs. We propose a registration approach that can be used to compare directly a test image against a set of template images corresponding to different ages (which is the base of the GP method), or can be used as a segmentation-byregistration [25] procedure that would provide the TW3 with the features needed by the method, (in this case only one gold standard image with the identified ROIs is needed). Finally, the bone ages resulting from the two methods should not differ substantially, providing an autovalidation capability of such a methodology. In this paper, however, we will concentrate only on the registration process that will eventually become a segmentationby-registration recursive paradigm. Very few methods have been proposed in the literature to register images including articulated structures. Little et al. [26] present an intra-subject registration technique for head and neck images. They obtain a rigid transformation for each vertebra, and interpolate them to produce a transformation for the entire volume. The use of rigid-body transformations allows only intra-subject registration. Arsigny et al. [27] propose a method in which local rigid or affine transformations are combined in a way that the overall transformation is invertible. They applied their method to the registration of histological images, arguing that it could also be used for articulated structures but they do not show examples. Papademetris et al. [28] combine several rigid transformations, obtaining a continuous overall transformation. They apply this method to the serial registration of lower-limb mouse images, identifying individual joints and the planes in which the axis of rotation for each joint lies. The authors combine piecewise rotations in an overall transformation that is continuous at the joints. Baiker et al. [29] propose an automatic articulated registration approach for whole-body 3D data of mice out of micro-CT volumes. The method is based on a hierarchical anatomical model of the skeletal system specifying the degrees of freedom of each joint, using ICP registration [30] to minimize the distance between the surface representations of the skeletons. In this work, the registration is performed onto the articulated models, but not onto the original data set. Recently, Li et al. [31] use a two-step algorithm to register whole-body inter-subject volumes. In the first step, they register bone structures and the outside body surfaces using a point-based registration algorithm; in the second step this initial transformation is refined using an intensity-based registration algorithm. They apply their technique to the registration of CT volumes of mice and upper human body. This technique is completely

automatic, but from the results shown it does not seem accurate enough for the purpose of bone age assessment. In other recent work, du Bois d’Aische et al. [32], presented an articulated registration method for the spine column, based on the finite elements method for the elastic deformation outside rigid structures. Arsigny et al. [33] propose a log-Euclidean polyaffine transformation approach where its inverse is always guaranteed, being this inverse another log-Euclidean polyaffine transformation. The authors present a method to merge several affine transformations using normalized weighting functions. Their work focuses on the determination of the global transformation by solving an ODE with special stationary properties which provides its invertibility. This approach consists of merging differential affine transformations and then applying a composition of the resulting transformations. In the following section we describe our methodology, then in Section 3 the wire model and the landmarks used in our method are described. In Section 4, the articulated registration algorithm is described and applied to hand X-rays including the landmark optimization and the bone width correction procedures. After that, in Section 5 we discuss the results obtained by applying our method to several hand X-rays. We also compare it with the TPS algorithm, and finally in Section 6, we show the conclusions. 2. Methodology We present a new automatic landmark-based registration method for anatomical structures based on our preliminary work [8] that bears an inner skeleton, such as the hand bones. Such a method has to deal with many degrees of variability since the structures we are considering are articulated. Rigid registration methods are clearly inappropriate for this problem, and wellknown elastic methods do not usually incorporate the restriction of maintaining long skeletal structures straight. This new method, that we call articulated registration, is therefore suitable to deal with such a situation. The inner bone skeleton is modeled with a wire model, where wires are drawn by connecting landmarks located in the main joints of the skeletal structure to be registered (long bones). The main feature of our registration method is that within the bone axis (specifically, where the wires are located) an exact registration is guaranteed, while for the remaining image points an elastic registration is carried out based on a distance transform (with respect to the model wires); this causes the registration on long bones to be affine to all practical purposes, meanwhile the registration of soft tissues – far from the bones – is elastic. Among the registration techniques proposed in the literature, the ‘‘articulated registration” can be classified as a landmark-based registration method [1,2]. These registration methods start from a point set (the landmarks) whose positions are defined in both images, so that we know how to map the landmarks. For the remaining points of the image, the transformation can be obtained by interpolation, for instance, applying a TPS algorithm [3,11]. In order to fully automate the procedure, anatomical landmarks must be automatically found. This is by no means a simple problem, due to the high variability between the bones at different growth stages. For this reason the first stage of the method is an algorithm for the automatic detection of landmarks [34], using a cascade of image processing techniques. 3. Landmarks and wire model In this paper, we apply the wire model to the human hand, but it is clear that this methodology can be applied to other parts of the body. For images of early bone development stages, landmarks may be difficult to position since epiphyses are very small or even

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a

b

W3 L17 R14 L16

W2 L12 R10

R13

L11

W1 L7

R9

R6

L6

R20

R8 R4

L24 L23

R19

L15

L22

L10

R5 L5

Input W4

R12 R18 L14

L9

R22

R3

L28

L27

L26

L3

L8 L13 R2

R16

R21 L20

Edge detection

Finger axes extraction

L25

L19

L2 R1 L1

R23

L21

R17 R7 R11

L4

Gaussian filtering

W5

R15 L18

Watershed segmemtation

Binary Segmentation

Fig. 1. (a) Landmark positions superimposed on a radiograph; (b) scheme of the wire model together with labels of landmarks and rods.

non-existing, making the automatic extraction of landmarks a challenging issue. Relevant points in the skeleton are those related to bone joints and extremes. For the hand we have chosen five landmarks for each finger (four in the thumb) and two landmarks for ulna and radius, resulting in a total of 28 landmarks. We have numbered them starting from the bottom left of the image at the ulna, in ascending order and from left to right as shown in Fig. 1. All the landmarks are horizontally centered in the bones. Fig. 1a shows landmarks superimposed on a radiograph. A wire model is built by joining with straight segments every pair of consecutive landmarks in each finger, following bones axes. The union of two consecutive landmarks makes up a rod, and the union of all the appropriate rods makes up a wire. For the case of a hand, the model has 23 rods (R1–R23) and 5 wires (W1–W5). Fig. 1b shows the wire model together with the landmark and rod labels. As we will show in Section 4, registration on wires (bones axes) will be affine, while all the other pixels of the image will be elastically registered, depending on the relative distance from each pixel to each wire in the model, in such a way that registration of pixels in bones will be quasi-affine. For carpal bones two rods have been considered, connecting first and fourth fingers with ulna and radius respectively, providing some consistence to the carpal region, but allowing an elastic registration at the same time. This has been carried out because of the difficulty of using a wire model for these short bones as well as finding anatomically-significant age-invariant points. 3.1. Landmarks detection In this section, we describe the steps taken to identify anatomical landmarks in a hand X-ray image. This procedure is described in [34] for finger landmarks and here we extend that work for landmarks in the metacarpals and in the ulna and radius. The procedure in [34] can be summarized as follows: after noise reduction using Gaussian filtering, an edge detection using Canny filter [35] is applied to extract bone edges in order to apply a watershed algorithm [36] to segment the bones. In parallel, finger axes are extracted using [37], and finally the landmarks are detected from the segmented image using the finger axes. A block diagram with the whole process is shown in Fig. 2, and an image example with the intermediate steps is illustrated in Fig. 3. In Fig. 4a, we show the finger axes extracted with the method proposed in [37]. The extension of the calculated axes is not suitable for the metacarpals, because they are not necessarily aligned with the phalanges. Therefore, a more sophisticated method has been employed for that.

Landmarks selection

Landmarks

Fig. 2. Block diagram showing landmarks extraction for image registration.

From the segmented image and the finger axes, it is possible to determine correctly 20 landmarks corresponding to the fingers. That is carried out in [34] seeking intensity changes along the finger axes and computing the average point among nearby intensity changes. The resulting landmarks at this step, are shown in Fig. 4b. The landmarks placed at the ulna and radius can be found easily using the watershed labeled image of Fig. 3c and the binary segmented image of Fig. 3d. The landmarks number 1 and 18 can be extracted tracing a horizontal line from the bottom of the binary image, and selecting the centers of the two lines formed by the segmentation of the ulna and the radius. Then, the ulna and the radius are extracted as the watershed regions connected to the previous landmarks, and landmarks 2 and 19 will be extracted by choosing these regions superior extrema. Landmarks 1 and 18 are moved up or down in the direction defined by the bone (ulna or radius), in order to be located at a distance from landmarks 2 and 19, see Fig. 1b equivalent to the length of the proximal phalanx of the middle finger. The remaining four landmarks are those corresponding to the bottom of the metacarpi aligned with the four finger different from the thumb. To extract them, we trace a horizontal line at some distance below landmark number 4. This distance equals to one fourth of the distance between landmarks 4 and 21. Using that distance, a horizontal line is achieved along the metacarpal bones, which is independent of the size of the hand. We select the regions from the watershed image crossing with this line. These regions correspond to the four metacarpi needed. The desired landmarks are then extracted as the metacarpi inferior extrema. All the landmarks automatically detected are shown in Fig. 4c. Using these 28 landmarks we can carry out the registration, as we will explain in the following section. 4. Articulated registration The articulated registration matches the model (and the underlying image) of the moving image, denoted as IM , onto the model of the fixed image, IF . The result of the registration is an elastic

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Fig. 3. (a) Example hand X-ray image, (b) its extracted contour, (c) a segmentation using Watershed filter (each color represents a region), and (d) the segmented bones after thresholding the previous image.

Fig. 4. (a) Finger axes over the original X-ray; (b) landmarks detected in the first step from the finger axes information; (c) all detected landmarks.

transformation of IM using the obtained transformation, denoted as Ta ðIM Þ, (where subindex ‘‘a” is for ‘‘articulated”) in which models and underlying pixels (bone axes) in both images are matched. The main feature of this geometric transformation is that it allows to perform affine and elastic transformations on the same image. In this way, points corresponding to the wires, i.e., principal axes in the bones, are affinely transformed, while all the other points in the image are elastically transformed, so the transform is continuous and differentiable all over the image. The elasticity of the transform depends on the relative distance between each image pixel and each rod in the model. Pixels near model rods will be quasi-affine transformed, while pixels further from rods will experiment transforms with higher elasticity degrees; so in practical terms, pixels belonging to long bones are affine transformed, maintaining bone rigidity. This is shown in Fig. 5. The articulated registration consists of two main stages: the first one is the calculation of the affine geometric transformation concerning each rod in the model. In the second stage the elastic correspondence between each pixel of the original image to the pixels of the registered image is obtained by a regularization process based on a distance transform to each wire in the model. 4.1. Affine transformations on the model rods As it has been said in previous section, points on wires are affine transformed, allowing only translation, rotation and scaling, but

not elastic transformations. To make this possible, an affine transformation is calculated for each rod in the moving image that aligns it with the corresponding rod in the fixed image. Affine transformations can be represented, as it is well known, with a matrix multiplication in homogeneous coordinates [38]. If we denote as x ¼ ðx; yÞ a point in Cartesian coordinates in the input space (the moving image before transformation), and as x0 ¼ ðx0 ; y0 Þ, the same point in the output space (the moving image after transformation), affine transformation can be expressed as1

0 0 1 x a00 B B C 0 x ¼ @ y A ¼ Ax ¼ @ a10 0 1

a01 a11 0

1 x0 CB 0 C a12 A@ y A: 1 1 a02

10

ð1Þ

Let us call Mi , to the transformation matrix corresponding to rod Ri . To compute Mi , we only have to obtain the geometric transformation that aligns the two nodes on rod Ri in image IM with the corresponding nodes on same rod in image IF . The procedure is carried out by an affine transformation consisting on the composition of several translations, a rotation and a scaling operation. In homogeneous coordinates, the reverse address computation matrix, presented in Eq. (1), for the translation is 1 The reverse address computation method has been used, i.e., for each pixel position in the transformed moving image we find the pixel position of the original moving image. It is used, instead direct address computation to avoid the possible lack of bijection (holes and overlaps) [38].


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Fig. 5. Domain of the articulated transformation. (a) Transformations are affine on the wires and elastic far away from them. (b) In a pixel between two rods, the transformation depends on the relative distance from the pixel considered in each rod. (c) A near rod isolates a pixel from the effect of further rods in the same direction.

Fig. 6. Geometric transformations for the computation of the affine transformation of a single rod, Mi . In continuous blue line is represented IM with its rod Ri and in dashed red line IF together with its rod Ri .

0

1 0

tx

1

B C T ¼ @ 0 1 t y A 0 0 1

ð2Þ

for the scaling

0

1=sx B S¼@ 0 0

0

0

1=sy 0

1

C 0A

ð3Þ

1

cos h

sin h

0

1

B C R ¼ @ sin h cos h 0 A: 0 0 1

ð4Þ

In our case, the scaling is equal for the two dimensions since only nodes in the wire model are being matched.2 To be more specific, let u1 ¼ ðu1 ; v1 Þ and u2 ¼ ðu2 ; v2 Þ be the coordinates of landmarks 1 and 2 of rod Ri , respectively in IM (input space), and x1 ¼ ðx1 ; y1 Þ and x2 ¼ ðx2 ; y2 Þ the coordinates of these nodes in IF (output space). The procedure is as follows:

2

The misregistration in long bone widths will be treated in Section 4.4.

h¼

p 2

arctan

v2 v1 : u2 u1

ð5Þ

(3) Scale IM so that rod Ri has the same length in both images. The scale factor to be inserted in matrix S, given by Eq. (3), is

and for the rotation

0

(1) Translate IM so that node u1 is on the origin of coordinates: ðu1 ; v1 Þ ¼ ð0; 0Þ. Translation matrix is T1 , given by Eq. (2), with parameters tx ¼ u1 ; t y ¼ v1 . (2) Rotate IM about node u1 (origin of coordinates) so that Ri in IM falls onto the vertical orientation. The rotation matrix will be R1 , given by Eq. (4), with rotation angle

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðy2 y1 Þ2 þ ðx2 x1 Þ2 s ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ðv2 v1 Þ2 þ ðu2 u1 Þ2

ð6Þ

(4) Rotate IM about node u1 so that Ri in IM has the same orientation as in IF . The rotation matrix will be R2 , given by Eq. (4), with rotation angle

h ¼ arctan

y2 y1 p : x2 x1 2

ð7Þ

(5) Translate IM so that Ri overlaps in both images. The translation matrix is called T2 , given by Eq. (2), with parameters t x ¼ x1 , ty ¼ x1 .

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Fig. 7. (a) Initial weight map w0i ðx; yÞ for i = 11 (R11 , see Fig. 1b) in logarithmic scale. Numerical labels of the axes are image coordinates in a downsampled image version with decimation factor L ¼ 16; (b) final weight map wi ðx; yÞ after the iterative process described in logarithmic scale.

The overall geometric transformation is therefore:

Mi ¼ T1 R1 S R2 T2

ð8Þ

and is sketched in Fig. 6 4.2. Articulated transformation of the image Once we have the affine transformation matrix Mi for each rod, Ri , in the model, in order to calculate the final articulated transformation a weighting of all the transformations is made, with wi ðx; yÞ a measure of how the transformation of Ri influences pixel (x, y) in the image. These weights are function of the distance of each image pixel to the rod. This operation can be expressed in homogeneous coordinates as

0 1 0 1 " # x u N X B C B C wi ðx; yÞMi @ y A; @vA ¼ i¼1 1 1

ð9Þ

where ðu; vÞ and ðx; yÞ are the Cartesian coordinates of each pixel in the moving image before (input space) and after (output space) the transformation. To obtain the elastic – articulated – transformation that complies with the restriction that the transformation must be affine for pixels nearby the wires, the weight wi ðx; yÞ corresponding to the ith rod transformation must have a value very close to 1, while the other weights, wk ðx; yÞ, k–i, must have values very close to 0. We employ an iterative technique to obtain the weight maps wi ðx; yÞ. Initial weights w0i ðx; yÞ are calculated as a distance transform, with a value of 1 on the rod and 0 at an infinite distance from it, by means of the function

w0i ðx; yÞ ¼ sigm

1 ; 1 þ di ðx; yÞ

ð10Þ

where di ðx; yÞ is the distance from image pixel ðx; yÞ to the rod Ri , and sigmðÞ is a sigmoid function, which smooths the slope of the weight map near the wire. Then, an iterative procedure is done, where each weight diminishes the others when it has a high value (close to 1), fulfilling the condition that only the dominant weight has a value close to 1 near its rod, therefore, dominating the global transformation. To do this, in each iteration step n the updated weights are calculated as

wni ðx; yÞ ¼ wn1 ðx; yÞ ð1 k sigmðr ni ðx; yÞÞÞ; i

ð11Þ

where k is a value near 0 controlling the velocity of change in each step, and ri ðx; yÞ is the sum of the other weights in the previous step

r ni ðx; yÞ ¼

N X

wn1 k ðx; yÞ:

ð12Þ

k¼1;k–i

Finally, weights are normalized so that the sum of all the weights in each pixel is unity

wM ðx; yÞ ; wi ðx; yÞ ¼ P i M 8k wk ðx; yÞ

ð13Þ

where M is the number of iterations steps. Fig. 7a shows an initial weight map w0i ðx; yÞ, where it can be seen the smoothing sigmoid effect. Fig. 7b shows the final weight map wi ðx; yÞ after the iterative process, where each weight map has interacted with the others. wi ðx; yÞ has lower values in points where the other maps have high values. To diminish computational load, weight maps are not obtained on all the pixels of the transformed target image, ðx; yÞ, but on a decimated version – with decimation factor L – ðxL ; yL Þ, obtaining a decimated weight map wi ðxL ; yL Þ, which is finally interpolated. Besides, it is interesting to highlight that weight map wi ðxL ; yL Þ depends only on the wire model pose in the fixed image, IF (being therefore independent of the moving image to be registered, IM ); consequently, it is only calculated once per fixed image.3 Finally, the reverse address computation process results in address coordinates ðu; vÞ lying between known pixel values of the input target image. Thus, it is necessary to estimate the unknown pixel value from its neighbors. This process is made by the wellknown method of bilinear interpolation [38]. For future reference, the articulated registration result is denoted as

ðIM Þ0 ¼ Ta ðIM Þ:

ð14Þ

It is important to highlight that if the models are well constructed in both images, with correct correspondence between landmarks, without crosses and assuring a minimum distance between nodes greater than the decimation factor, considering that the weights are continuous and differentiable, the resulting geometric transform is invertible. In the local neighborhood of the rods, the inverse transform is the articulated registration with the inverse affine matrices. It is possible to directly apply the log-Euclidean polyaffine transformation, as proposed by Arsigny et al. [33], in case an accurate invertibility is a requirement. 4.3. Landmarks optimization Original position of landmarks detected using the algorithm described in Section 3.1 may be inaccurate to a certain extent. After the initial articulated registration, landmarks may be relocated using an optimization procedure to a position such as the neighborhood surrounding each landmark has maximal similarity between Ta ðIM Þ and IF , according to a given similarity measure. This procedure is made independently for each landmark in order to improve a given similarity measure in a local neighborhood. Independency of landmarks is possible because of the partially local domain of the transformation, shown in Fig. 5. 3 For bone age assessment we use only one fixed image, registering all the other images to that one, which is used as a Golden Standard.


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tion procedure is plugged into the articulated registration as shown in Fig. 9. In this scheme, we start with the articulated registration. The landmarks at the moving image (IM ) used for the first time are the manually or automatically detected landmarks. After registration, the transformation obtained together with the moving image are used to obtain by interpolation a new moving image. This new moving image, the fixed image, and the landmarks are used to obtain new optimized moving landmarks, that are plugged into the registration again, to iterate the process. 4.4. Bone width correction

Fig. 8. Similarity measure is obtained in the current landmark position (black small square) and in the eight neighbors (white small squares) at the optimization step. Gray squares represent the region where similarity measure is calculated.

The optimization procedure at every landmark is an iterative operation. Starting from the original landmark position, given a similarity measure and an optimization step (symbolized by the arrows in Fig. 8), we look for the similarity measure between the moving and the template image in the eight possible positions at the given step, as shown in Fig. 8. Given the similarity measure value at the landmark position and at these eight neighbors, three different situations may happen: (1) When the best measure is at the current position (black small square in Fig. 8) and the optimization step is greater than one, the optimization step is reduced by half its value, and the process is repeated in order to obtain a more accurate relocation. (2) When the best measure is at the current position and the optimization step is equal to one, the process is stopped for that landmark. (3) When the best measure is not at the current position, we move to the position with the best measure and start the procedure again from there. The similarity measure used in our experiments has been mutual information [7,39], namely Mattes [40] implementation, that measures the image similarity between two images, A and B. Mutual information, say MIðA; BÞ is a measure of the relation between the intensities of two images, by measuring the Kullback–Leibler distance [41] between the joint probability density function of the two, pAB ða; bÞ, with no further assumptions, and the same density in the extreme case of total independence of the two variables, i.e., pA ðaÞpB ðbÞ. Specifically

MIðA; BÞ ¼

X a;b

pAB ða; bÞ log

pAB ða; bÞ pA ðaÞpB ðbÞ

ð15Þ

with a and b the gray level values of images A and B, respectively. The entire process of optimization is computationally expensive, because several iterations are needed to achieve optimal solutions. In order to reduce the computational cost, the algorithm takes into account several considerations. First, it computes the similarity measure only within a ROI, previously defined for each landmark, which is reasonable because each landmark is affected only by a local neighborhood, as shown in Fig. 8. Second, the measure is computed on a percentage of points of the given ROI, performing a trade-off between computational load and MI accuracy. Finally it reuses the computation of measures at repeated positions when the landmark is moved. This optimization scheme gives a great advantage to the whole process, because now the initial position of the landmarks has not to be accurate, which usually happens for the automatically detected landmarks. The optimiza-

As it has been said in Section 4.1, scale factors for each rod, sx and sy in Eq. (3) are made equal in matrix S, as only nodes in wire model – which has no width – are matched. So, only one scale factor, s, is calculated in the affine transformation of each rod, whose expression is given in Eq. 6. In this way, scale factor is isotropic in each bone, producing bone width misregistration, as it can be seen in Fig. 10. In this figure, we can see that it is necessary to adjust not only bone width (as in left bone), but also axial displacement (as in the other two bones). To solve this issue and improve the registration output, we make a finer registration of the width of each bone. This procedure could also be included inside the articulated transformation described in Section 4.2, but it would increase dramatically the computational load of the whole process because the articulated transformation is performed during the landmark optimization at each iteration, and the results would not be improved significantly. To perform bone width correction it is necessary, in first place, to know the ROI in which computations are to be made, to obtain the optimum width scaling of each bone. These ROIs have been manually labeled in the fixed image,4 as it is shown in Fig. 11. Instead of using directly the X-ray images for this correction, we will use the segmented images obtained using the method described in Section 3.1, as the one shown in Fig. 3d. To be more specific, we use the following images: Segmented fixed image; it will be denoted as SF . Segmented moving image, on which it is applied the geometric transformation of the articulated registration; denoted as ðSM Þ0 ¼ Ta ðSM Þ. Image with labeled ROIs, shown in Fig. 11, that will be denoted as SROI . We perform the axial registration of each bone separately; for this purpose, we obtain a transformation matrix for each rod (bone) in the model. For rod Ri , it will be placed in vertical position with one of its nodes in the coordinates origin, which facilitates the scaling matrix determination, as it has been made in Section 4. Next, horizontal scaling of the bone will be made, accompanied by a displacement in the same direction, solving possible misalignment of the rod with bone longitudinal axis, as it is shown in Fig. 10. The two parameters involved in these two transformations will be obtained by means of an optimization procedure. Finally, the initial displacement will be inverted, replacing the rod in the initial position and orientation (possibly with the axial displacement). As optimization problems are computationally expensive, taking into account that IF and its model are fixed, the initial and final transformations may be precalculated and stored, and the reverse address calculation of all the geometric transformations are only calculated in the bounding box of the corresponding ROI, once placed the rod vertically, using the next three subimages for the optimization problem, that are shown at the left side of Fig. 12: 4 Remember that the fixed image is unique for our purposes, so this labeling has to be made only once.

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Fig. 9. Components scheme of articulated registration including the landmarks optimization procedure.

Segmented fixed image in the bounding box corresponding to the rod Ri in the model; denoted as SFi . All these images are also fixed, so they can be precalculated and stored. Segmented moving subimage articulated registered, in the bounding box corresponding to the rod Ri in the model; denoted 0 M as ðSM i Þ ¼ Ta ðSi Þ. This subimage depends on the optimization problem parameters, so it must be calculated along optimization process, but exclusively inside the bounding box corresponding to rod Ri . Let x1 ¼ ðx1 ; y1 Þ and x2 ¼ ðx2 ; y2 Þ be the coordinates of landmarks 1 and 2 of rod Ri , respectively in Ta ðIM Þ.5 The procedure is as follows: Fig. 10. Detail of error image showing bone width misregistration produced by articulated registration.

0 (1) Translate ðSM i Þ so that node x1 is on the origin of coordinates. Translation matrix T2 , given by Eq. (2), with parameters t x ¼ x1 , ty ¼ y1 . (2) Rotation about node x1 (origin of coordinates) so that Ri falls onto the vertical orientation. Rotation matrix R2 , given by Eq. (4), with rotation angle

h ¼ arctan

y2 y1 : x 2 x1

ð16Þ

For convenience these two geometric transformations are denoted by a single matrix, which as said is precalculated, resulting from the product of both

Oi ¼ T2 R2 :

ð17Þ

(3) Horizontal displacement of the bone: it is necessary, as indicated before, because if the rod is misaligned with the bone axis, a width scaling with center in the rod will not give good results. The parameter involved is z1 – whose value will be obtained during optimization – corresponding to a horizontal displacement in the positive direction. Translation matrix Ti ðz1 Þ with parameters tx ¼ z1 , t y ¼ 0. (4) Width scaling of the bone: instead of using the scale factor as parameter, avoiding a bad conditioned optimization problem, we employ another parameter, z2 , defined as the bone horizontal widening, and related with horizontal scaling as

z2 ¼ Di ðsx 1Þ;

Fig. 11. ROIs used to make bone width correction superimposed over fixed image. They are manually selected for the fixed image only once.

where Di is the width in pixels of the subimages. Scaling matrix Si ðz2 Þ, given by Eq. (3), with parameters

sx ¼

Subimage with the ROI in the bounding box corresponding to the rod Ri in the model; it will be denoted as SROI i . All these images are fixed, so they can be precalculated and stored.

ð18Þ

D i þ z2 ; Di

sy ¼ 1:

ð19Þ

5 Remember that once articulated registration is made on IM , its model coincides with the one in IF .


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Fig. 12. From left to right, extraction of the three subfigures, geometric transformations and optimization process involved in bone width correction.

(5) Inverse rotation and translation of steps 1 and 2. Transformation matrix will be O1 i .

ROI ðz1 ; z2 Þ ¼ arg max JCðSFi ; Tw;i ðSM i ; z1 ; z2 Þ; Si Þ ðz1 ;z2 Þ

ð24Þ

The bone width correction process is schematically shown in Fig. 12, and the overall geometric transformation is computed with the matrix

and thus the optimized transformation matrix for rod Ri is given by

Fi ðz1 ; z2 Þ ¼ Oi Ti ðz1 Þ Si ðz2 Þ O1 i :

To delimit the problem, solutions for z1 are searched in the range x1 2 ½Di =4; Di =4; and for z2 in the scale range sx 2 ½0:75; 1:5, which corresponds to z2 2 ½Di =4; Di =2. The optimization technique employed is a quasi-Newton method implemented in Matlab [43]. Finally, the matrices b F i obtained as result of the optimization process for each rod, Ri are postmultiplied with the articulated registration matrices of the same rod, obtaining the final transformation matrix

ð20Þ

The operation that obtains the affine transformation with an axial displacement and scaling centered in rod Ri on image I is denoted as

I0 ¼ Tw;i ðI; z1 ; z2 Þ:

ð21Þ

The best transformation for each bone is obtained, as said, resolving an optimization problem, in which a similarity function will be maximized. We will employ here the Jaccard coefficient (JC) [42] as similarity function. The JC between two segmented regions X and Y is defined as

jX \ Yj JCðX; YÞ ¼ ; jX [ Yj

ð22Þ

where operator j j denotes the number of pixels. In our case we compute it within SROI i , the ROI corresponding to rod Ri as a mask. We define the JC between two segmented images X and Y within a region defined by the mask M as

JCðX; Y; MÞ ¼

jX \ Y \ Mj ; jðX [ YÞ \ Mj

ð23Þ

where X denotes bone segmentation region in SM i and Y denotes bone segmentation region in SFi . The maximum value of JC is 1, is null (perfect when difference of bone segmentations within SROI i registration), and 0 in case there is not overlapping between X and Y within SROI i . Once the similarity function has been defined, we can state the problem of bone width correction as an optimization problem with two unknown parameters, ðz1 ; z2 Þ, as

b F i ¼ Fi ðz1 ; z2 Þ:

Ai ¼ Mi b Fi:

ð25Þ

ð26Þ

Matrices Ai are used to make the final transformation of the articulated registration with bone width correction using Eq. (9) as described in Section 4.2, repeated here for easier reading

0 1 0 1 " # x u N X B C B C wi ðx; yÞAi @ y A: @vA ¼ i¼1 1 1

ð27Þ

In Section 5, we show several examples of articulated registration with bone width correction. 5. Experiments and results This section describes the experiments carried out in order to show the performance and validity of our method. First we visually compare the articulated registration with the TPS registration, secondly a validation study with 20 different subjects is carried out, showing numerical and visual results for the optimization

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procedure, the bone width correction and the automatically and manually placed landmarks. 5.1. Articulated registration vs TPS registration In this section, we compare our articulated registration with a well-known landmark-based registration method. This kind of geometrical transformation used in landmark-based registration is based on the use of splines. They are elastic transformations that map a fixed image over a moving one, according to some smoothing conditions. They start from the known correspondence between landmarks in the two images [2]. TPS is one particular class of splines, usually employed in registration, that can be understood as a 2D interpolation from a landmark set [1,11]. The implementation of the TPS algorithm is reduced to solve a linear equation system to determine the transformation coefficients [3]. So, the registration procedure is efficient. The drawbacks of this method are that long bones are not assured to remain straight after registration, and the error in areas far from these landmarks is higher than in the nearest areas since it only considers landmark information. However, by setting the landmarks in the ROI it is possible to obtain a correct registration of these areas, as it is shown in Fig. 13b, where the example image, shown in Fig. 13a together with the landmarks used, is registered against

the one shown in Fig. 13d. The error image obtained by subtracting both images is captured in Fig. 13c. In order to compare with our articulated registration we show the articulated registration of the same images in Fig. 13e and the corresponding error image in Fig. 13f. Numerical results of the performance of this registration method are shown in the next section, for comparison with the proposed articulated registration. 5.2. Validation study We have applied our registration algorithm to 20 hand radiographs using as fixed image the one shown in Fig. 14d. The population used in the evaluation is as follows: 20 subjects, 9 females and 11 males, from 4 to 14 years old, with mean age of 8.35 and standard deviation of 2.74. For the 20 radiographs registered we have obtained the mean and standard deviation values from two similarity measures, the MI and the JC defined before, computed between the fixed and the registered images. The JC index is computed using the segmentations obtained with the method described in Section 3.1. These values are shown in Table 1, where we compare our registration method with the TPS method, and the registered results using the automatically detected landmarks with the registered images using manually placed landmarks, selected by an expert (validating our landmark detection method). There are also results using optimized and non-optimized land-

Fig. 13. (a) Automatically detected landmarks overlapped in the image before registration, (b) TPS registered image with respect to (d) the template image or gold standard, (c) TPS registration error image, (e) articulated registered image and (f) articulated registration error image.

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marks, in order to show the improvement of the optimization process, and finally, we show the influence of the bone width correction procedure. We have also made a hypothesis test to assess if differences in means of Table 1 are statistically significant. The hypothesis testing framework employed is the paired t test [44]. In Tables 2 and 3 are shown the results for MI and JC indices, respectively. Using MI as similarity measure it can be seen from Tables 1 and 2 that registration methods can be ordered from best to worst in the following way:

AR Manual Opt: > AR Auto Opt: ’ AR Manual Opt: Corr: ’ AR Auto Opt: Corr: ’ AR Manual > AR Auto > TPS Auto:

Table 1 Mean and standard deviation of the MI and JC values obtained from 20 registered radiographs, for articulated and TPS registration algorithms. AR = articulated registration, Manual = manually placed landmarks, Auto = automatic detected landmarks, Opt. = with optimization procedure, Corr. = with bone width correction. Method

AR Manual Opt. Corr. AR Manual Opt. AR Manual AR Auto Opt. Corr. AR Auto Opt. AR Auto TPS Auto

Mutual information

JC

Mean

Std dev.

Mean

Std dev.

1.55 1.60 1.52 1.53 1.56 1.36 1.26

0.14 0.12 0.13 0.13 0.10 0.10 0.11

0.795 0.747 0.716 0.775 0.729 0.604 0.601

0.04 0.06 0.04 0.03 0.04 0.03 0.04

ð28Þ AR Manual Opt: Corr: > AR Auto Opt: Corr:

Without optimization nor bone width correction, the best method, using MI as similarity measure, is our articulated registration with manual landmarks, followed by articulated registration with automatic landmarks and by TPS registration, being these results extremely significant. The landmark optimization procedure improves results in all the cases (with similar values of significance), while bone width correction does not improve the results using MI, but makes them worse instead. On the other hand, using JC index as similarity measure, comparing Tables 1 and 3, the order from best to worse is as follows:

> AR Manual Opt: > AR Auto Opt: ’ AR Manual > AR Auto ’ TPS Auto:

ð29Þ

It can be seen that TPS and articulated registration with automatic landmarks give similar results, although this result is statistically not significant. Articulated registration with manual landmarks and articulated registration with optimized automatic landmarks produce better results than the two aforementioned methods

Fig. 14. (a) Automatic (green) and manually (red) placed landmarks overlapped in image before registration; (b) articulated registered image using automatic landmarks, and (c) using manually placed landmarks, (d) fixed image, and (e, f) absolute difference with respect to the fixed image.

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Table 2 Results of the paired hypothesis test of means equality. MI is used as similarity measure in 20 registered radiographs. A two sided significant test is performed with a ¼ 0:05. For each couple of registration methods, the result of the test and the p-value are shown. We show in bold the values when they are significant.

Table 3 Results of the paired hypothesis test of means equality. JC is used as similarity measure in 20 registered radiographs. A two sided significant test is performed with a ¼ 0:05. For each couple of registration methods, the result of the test and the p-value are shown. We show in bold the values when they are significant.

(extremely significant), being better the second one (statistically non-significant result), which assesses that optimized automatic landmarks are, at least, as good as manual ones, using JC index as similarity measure. The best results are obtained using both landmarks optimization and bone with correction, being the best one articulated registration with optimized manual landmarks and bone width correction (extremely significant result). From these results it is important to note that bone width correction has a negative influence in the MI values, but not in the JC values. This happens because bone width correction modifies bones thickness and also fingers thickness. This modification in fingers thickness modifies also the hand contour, which arises in negative MI results because the MI is computed in the whole image, but it does not affect the JC values, because JC is computed using only the bones. This is not an inconvenient

for our purposes because bones are the only structures of interest for us. In order to visually compare the landmark detection method, and its influence in the registration method, we show in Fig. 14a a radiograph with the automatic and manual landmarks overlapped, in green and red respectively, and in Fig. 14b and c, we show the results of the registered image using the automatic and the manually placed landmarks respectively. Notice the small difference in the location of landmarks detected automatically compared to those placed manually, and that the registered images are very similar, and differ more in the thumb area, where the automatic landmarks are more difficult to detect correctly. The registration error for this example, shown as the absolute difference between the fixed image with respect to the registered


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Fig. 15. (a) Hand radiograph with automatically detected landmarks before (red) and after (green) the optimization procedure, and articulated registration using these landmarks (b) before and (c) after the optimization, and (d, e) their corresponding absolute difference images.

image, is shown in Fig. 14e, for automatically detected landmarks and in Fig. 14f for manually placed landmarks. It is shown that the errors in both cases are low and it can be visually noticed that higher similarity is achieved using the manual landmarks. In order to show the performance of the optimization procedure, we show in Fig. 15 the landmarks detected automatically over a hand radiograph before deformation, and the landmarks

obtained after the optimization procedure. We also show in that figure the articulated registration for the same image without and with the optimization step. It is easy to notice that the optimization produces better results, and it is visually shown especially in the thumb area (see Fig. 15d and e). We also show in Fig. 16 the effect of the bone width correction, where we show the difference of the bone segmentation of two registered images with the bone segmentation of the fixed image. Error pixels (pixels where both images differ) are represented in black. Fig. 16a is obtained from a registration using articulated registration but without the bone width correction and Fig. 16b is from a registration using bone width correction. It is clear that this latter image presents less error pixels, specially in the ulna and radius. Finally, Fig. 17 shows the articulated registration of three radiographs using optimization and bone width correction; this figure illustrates that the wire model is well adapted to articulations of bones, allowing registration of very different hand poses. All the results have been obtained with a decimation factor L ¼ 8, which reduces significantly computational load while preserving high registration quality. The optimization is executed using an optimization step of 2, and using 3% of the points in the ROI to compute the similarity measure.6 These parameters have

Fig. 16. Difference images between the bone segmentation of the registered image with the segmentation of the fixed image, (a) without using bone width correction and (b) using it.

6 The size of the ROIs used in our images is of the order of 250 500 pixels, so a 3% is of the order of 4000 pixels, large enough to compute the MI with sufficient accuracy at a moderate computational cost.

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Fig. 17. Articulated registration for three radiographs, with respect to the template image of Fig. 14e, using automatically detected landmarks and a decimation factor L ¼ 8. (a) Images before registration; (b) images after registration; (c) absolute difference between the original and the template images after registration.

been chosen empirically, and their selection is a good trade-off between computational load and registration accuracy. Our experiments show that greater optimization steps, lower decimation factors, and greater percentage of points to compute the similarity measure, do not improve significantly the results. The optimization and registration algorithm is implemented using ITK [45] class libraries, and the landmark detection is implemented using MATLABÒ 7 R14. Execution times on an IntelÒ XeonTM Dual, 2.8 GHz with 4 GBytes RAM are 48.9 s for the landmark

detection step, and 26 s, 236 s and 4 s for the weight map calculation, optimization and registration, respectively for fixed image size of 2040 1510 pixels and moving image size of 2044 1516 pixels. Notice that weight map calculation has to be done only once for the template image, and the execution times for the landmark detection algorithm can be extremely reduced with a C or ITK implementation. The real bottleneck in our method is the optimization process, although it can be accelerated using lower percentage of points to compute the similarity measure, or


reducing the number of iterations, if the landmarks are detected more accurately. 6. Conclusions We have developed a fully automatic methodology to perform hand X-rays registration that includes a landmark detection algorithm, and an algorithm for the articulated registration of such images. This registration algorithm can be applied to images with regions of the body with an articulated inner skeleton using a wire model. We have called this algorithm articulated registration because it maintains bones straight and it performs elastic registration of pixels far from them. Numerical results (see Table 1) show that our algorithm is more appropriate for this kind of images than TPS, the most usual landmark-based registration algorithm, in terms of MI and JC. We also show with our experiments the improvement achieved by our optimization process carried out to relocate each landmark. The improvement is noticeable for images registered with automatically detected landmarks (mean MI is improved by 0.20) and for images registered with manually placed landmarks (mean MI increase in 0.08). Finally, Table 1 shows that the bone width correction carried out has a negative influence in the MI values, but not in the JC values; bone width correction modifies all the image, not only bone widths to maintain continuity in geometrical transform, which arises in negative MI results because the MI is computed in the whole image, but it does not affect the JC values, because JC is computed using only in the bones. This is not an inconvenient for our purposes focused only on bones. Graphical results also show that our registration algorithm is adequate for images with very different poses and from very different bone ages (see Fig. 17). We have also validated our landmark detection algorithm with a set of 20 hand radiographs, obtaining visual and numerical results very close to those obtained using manually placed landmarks (see also Table 1). We can say that the landmark detection results are extremely good except for the thumb area (landmarks numbers 25 and 26) where the error is more significant because the finger axis does not fit to the thumb adequately in many cases. With this automatic methodology we can extremely reduce the processing time used by experts to analyze the bone age of a patient. One final step to this methodology would be the implementation of a method to automatically carry out the TW3 method to obtain the bone age of the patient, and compare this result with the results obtained from human experts. Acknowledgements The authors acknowledge the Spanish CICYT for research grants TEC2004-06647-C03-01 and TEC 2007-67073/TCM. Thanks also go to Dr. Andrés de Llano and Dr. B. Viñuela from Hospital Río Carrión, Palencia, Spain, and Dr. S. Alberola, Centro de Salud Jardinillos, Palencia, Spain, for their valuable comments of the medical aspects of this work, and for the radiographs collected in the Radiology Department of the above mentioned hospital for research purposes. References [1] J.M. Fitzpatrick, D.L.G. Hill, C.R. Maurer Jr., Image registration, in: M. Sonka, J.M. Fitzpatrick (Eds.), Handbook of Medical Imaging, Medical Image Processing and Analysis, vol. 2, SPIE Press, Belligham, WA, 2000, pp. 447–513 (Chapter 8). [2] J.B.A. Maintz, M.A. Viergever, A survey of medical image registration, Med. Image Anal. 2 (1998) 1–36. [3] K. Rohr, H. Stiehl, T. Buzug, J. Weese, M. Kuhn, Landmark-based elastic registration using approximating thin-plate splines, IEEE Trans. Med. Imaging 20 (6) (2001) 526–534.

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[4] H.J. Johnson, G.E. Christensen, Consistent landmark and intensity-based image registration, IEEE Trans. Med. Imaging 21 (5) (2002) 450–461. [5] P. Thompson, A.W. Toga, A surface-based technique for warping threedimensional images of the brain, IEEE Trans. Med. Imaging 15 (4) (1996) 402–416. [6] G.E. Christensen, H.J. Johnson, Consistent image registration, IEEE Trans. Med. Imaging 20 (7) (2001) 568–582. [7] F. Maes, A. Collingnon, D. Vandermeulen, G. Marchal, P. Suetens, Multimodality image registration by maximization of mutual information, IEEE Trans. Med. Imaging 16 (2) (1997) 187–198. [8] M.A. Martín-Fernández, E. Muñoz Moreno, M. Martín-Fernández, C. AlberolaLópez, Articulated registration: elastic registration based on a wire-model, in: J.M. Fitzpatrick, J.M. Reinhardt (Eds.), Medical Imaging 2005: Image Processing, SPIE Press, San Diego, CA, USA, 2005, pp. 182–191. [9] B. Fisher, J. Modersitzki, Combination of automatic non-rigid and landmark based registration: the best of both worlds, in: M. Sonka, J.M. Fitzpatrick (Eds.), Medical Imaging 2003: Image Processing, San Diego, CA, USA, 2003, pp. 1037– 1048. [10] M.H. Davis, A. Khotanzad, D.P. Flaming, S.E. Harms, A physics-based coordinate transformation for 3-D image matching, IEEE Trans. Med. Imaging 16 (3) (1997) 317–328. [11] F.L. Bookstein, Principal warps: thin-plate splines and the decomposition of deformations, IEEE Trans. Pattern Anal. Mach. Intell. 11 (6) (1989) 567– 585. [12] W. Greulich, S. Pyle, Radiographic Atlas of Skeletal Development of Hand Wrist, second ed., Stanford University Press, Standford, CA, 1971. [13] J. Tanner, M. Healy, H. Goldstein, N. Cameron, Assessment of skeletal maturity and prediction of adult height (TW3 method), WB Saunders, London, UK, 2001. [14] N.D. Efford, Knowledge-based segmentation and feature analysis of handwrist radiographs, Technical Report Report 94.31, School of Computer Studies, University of Leeds, UK, 1994. [15] E. Pietka, Computer-assisted bone age assessment based on features automatically extracted from a hand radiograph, Computer. Med. Imaging Graph. 19 (3) (1995) 251–259. [16] G.W. Gross, J.M. Boon, D.M. Bishop, Pediatric skeletal age: determination with neural networks, IEEE Trans. Med. Imaging 14 (3) (1995) 689–695. [17] R. de Luis-García, J.I. Arribas, S. Aja-Fernández, C. Alberola-López, A neural architecture for bone age assessment, in: Proceedings of the IASTED International Conference on Signal Processing, Pattern Recognition and Applications, Creta, Greece, 2002, pp. 161–166. [18] L. Bocchi, F. Ferrara, I. Nicoletti, G. Valli, An artificial neural network architecture for skeletal age assessment, in: International Conference on Image Processing, Barcelona, Spain, vol. 1, 2003, pp. I–1077-I–1080. [19] A. Tristan, J. Arribas, A radius and ulna skeletal age assessment system, in: IEEE Workshop on Machine Learning for Signal Processing, Mystic, CT, 2005, pp. 221–226. [20] A. Pathak, S.K. Pal, Fuzzy grammars in syntactic recognition of skeletal maturity from X-rays, IEEE Trans. Syst. Man Cybernet. 16 (5) (1986) 657– 666. [21] S. Aja-Fernández, R. de Luis-García, M. Martín-Fernández, C. Alberola-López, A computational TW3 classifier for skeletal maturity assessment. a computing with words approach, J. Biomed. Inform. 37 (2) (2004) 99–107. [22] C. Hsieh, T. Jong, C. Tiu, Bone age estimation based on phalanx information with fuzzy constrain of carpals, Med. Biol. Eng. Comput. 45 (3) (2007) 283– 295. [23] A. Gertych, A. Zhang, J. Sayre, S. Pospiech-Kurkowska, H. Huang, Bone age assessment of children using a digital hand atlas, Computer. Med. Imaging Graph. 31 (4–5) (2007) 322–331. [24] A. Zhang, A. Gertych, B. Liu, Automatic bone age assessment for young children from newborn to 7-year-old using carpal bones, Computer. Med. Imaging Graph. 31 (4–5) (2007) 299–310. [25] S. Warfield, A. Robatino, J. Dengler, F. Jolesz, R. Kikinis, Nonlinear registration and template-driven segmentation, in: A.W. Toga (Ed.), Brain Warping, Academic Press, San Diego, CA, USA, 1999, pp. 67–84 (Chapter 4). [26] J.A. Little, D.L.G. Hill, D.J. Hawkes, Deformations incorporating rigid structures, Comput. Vis. Image Understand. 66 (2) (1997) 223–232. [27] V. Arsigny, X. Pennec, N. Ayache, Polyrigid and polyaffine transformations: a new class of diffeomorphisms for locally rigid or affine registration, in: Medical Image Computing and Computerized Assisted Intervention, LNCS, Montréal, Canada, 2003, pp. 829–837. [28] X. Papademetris, D.P. Dione, L.W. Dobrucki, L.H. Staib, A.J. Sinusas, Articulated rigid registration for serial lower-limb mouse imaging, in: Medical Image Computing and Computerized Assisted Intervention, LNCS, Palm Springs, CA, USA, 2005, pp. 919–926. [29] M. Baiker, J. Milles, A.M. Vossepoel, I. Que, E.L. Kaijzel, C.W.G.M. Lwik, J.H.C. Reiber, J. Dijkstra, B.P.F. Lelieveldt, Fully automated whole-body registration in mice using an articulated skeleton atlas, in: IEEE International Symposium on Biomedical Imaging, Washington, DC, USA, 2007, pp. 728–731. [30] P.J. Besl, N.D. Mckay, A method for registration of 3D shapes, IEEE Trans. Anal. Mach. Intell. 14 (2) (1992) 871–881. [31] X. Li, T.E. Peterson, J.C. Gore, B.M. Dawant, Automatic registration of whole body serial micro CT images with a combination of point-based and intensitybased registration techniques, in: Third IEEE International Symposium on Biomedical Imaging, Arlington, VA, USA, 2006, pp. 18–25.

1222


[32] A. du Bois d’Aische, M.D. Craene, X. Geets, V. Gregoire, B. Macq, S.K. Warfield, Estimation of the deformations induced by articulated bodies: registration of the spinal column, Biomed. Signal Process. Control 2 (1) (2007) 16–24. [33] V. Arsigny, O. Commonwick, X. Pennec, N. Ayache, A log-Euclidean polyaffine framework for locally rigid or affine registration, in: Third International Workshop on Biomedical Image Registration, LNCS, Utrecht, The Netherlands, 2006, pp. 120–127. [34] E. Muñoz Moreno, R. Cárdenes, R. De Luis-García, M. Martín-Fernández, C. Alberola-López, Image registration based on automatic detection of anatomical landmarks for bone age assessment, WSEAS Trans. Comput. 4 (11) (2005) 1596–1603. [35] J. Canny, A computational approach to edge detection, IEEE Trans. Pattern Anal. Mach. Intell. 8 (6) (1986) 679–698. [36] L. Vincent, P. Soille, Watersheds in digital spaces: an efficient algorithm based on immersion simulations, IEEE Trans. Pattern Anal. Mach. Intell. 13 (6) (1991) 583–598. [37] R. de Luis-García, M. Martín-Fernández, J. Arribas, C. Alberola-López, A fully automatic algorithm for contour detection of bones in hand radiographs using active contours, in: International Conference on Image Processing, Barcelona, Spain, vol. 3, 2003, pp. 421–424.

[38] W.K. Pratt, Digital Image Processing, John Wiley & Sons, New York, NY, 1991. [39] W.M. Wells, P. Viola, H. Atsumi, S. Nakajima, R. Kikinis, Multi-modal volume registration by maximization of mutual information, Med. Image Anal. 1 (1996) 35–51. [40] D. Mattes, D.R. Haynor, H. Vesselle, T. Lewellen, W. Eubank, Nonrigid multimodality image registration, in: SPIE Medical Imaging 2001: Image Processing, San Diego, CA, USA, vol. 4322, 2001, pp. 1609–1620. [41] S. Kullback, R.A. Leibler, On information and sufficiency, Ann. Math. Stat. 22 (1951) 79–86. [42] P. Jaccard, étude comparative de la distribuition florale dans une portion des alpes et de jura, Bull. Soc. Voudoise Sci. Naturelles 37 (1901) 547– 579. [43] R. Fletcher, Practical Methods of Optimization, second ed., John Wiley & Sons, New York, NY, 1987. [44] B. Rosner, Fundamentals of Biostatistics, sixth ed., Brooks/Cole, Belmont, CA, 2005. [45] L. Ibañez, W. Schroeder, L. Ng, J. Cates, The ITK Software Guide, Kitware, Inc., second ed., 2005. ISBN 1-930934-15-7. .