WO2009083864A2

WO2009083864A2 - Iterative reconstruction of polyhedral objects from few projections

Info

Publication number: WO2009083864A2
Application number: PCT/IB2008/055348
Authority: WO
Inventors: Klaus Erhard; Dirk Schaefer; Michael Grass
Original assignee: Philips Intellectual Property & Standards Gmbh; Koninklijke Philips Electronics N. V.
Priority date: 2007-12-20
Filing date: 2008-12-16
Publication date: 2009-07-09
Also published as: WO2009083864A3

Abstract

A method for fully automatic modelling of medical structures with homogeneous attenuation function from few projections is provided. The object of interest is modelled as a polyhedron with triangular surface mesh. The optimization of both the coordinates of the polyhedral model and the attenuation value of the polyhedral model is performed during a data reconstruction. This results in an attenuation value together with a three-dimensional surface model of the physical structure imaged. Combining several polyhedral models into one compound model, the method may allow for fully automatic modelling of medical structures with piecewise constant attenuation function.

Description

ITERATIVE RECONSTRUCTION OF POLYHEDRAL OBJECTS FROM FEW PROJECTIONS

The invention relates to the field of medical imaging. In particular, the invention relates to an examination apparatus for modelling a polyhedral model of an object of interest, to a method for modelling a polyhedral model of an object of interest, a computer-readable medium, a program element and an image processing device. High contrast imaging is an important clinical application of X-ray systems providing the physicians with valuable information for diagnosis. Often, the physicians are interested in only a few two-dimensional fluoroscopies acquired from different angles in order to keep the dose applied on the patient as small as possible or due to mechanical restrictions at bed side or in the operating room.

Another example stems from rotational angiography of the vessel tree. Although the number of measured projections may vary from 80 to 200, the projections belonging to one cardiac phase are significantly less, for example in the order of 4 to 10. However, three-dimensional reconstructions from a limited number of projections with standard forward back-projection techniques (FBP) may be blurred. Iterative maximum likelihood (ML) algorithms may improve the signal-to-noise ratio but without additional regularization a reasonable reconstruction may not be possible.

In recent years there has been some progress in reconstructing sparse objects such as bolus filled vessels, see references [1], [2] and [3]. Another technique based on polyhedral object models has been presented in reference [5], in which the attenuation value is a known value. Both techniques make use of a priori knowledge such as "sparseness" or the polyhedral nature of the object in order to stabilize the reconstruction. It would be desirable to have an improved modelling of a polyhedral model of an object of interest.

The invention provides an examination apparatus, a method, a computer- readable medium, a program element and an image processing device with the features according to the independent claims. It should be noted that the following described exemplary embodiments of the invention, which are described with respect to the examination apparatus, apply also for the method, the computer-readable medium, the program element and the image processing device. According to a first aspect of the present invention, an examination apparatus for modelling a polyhedral model of an object of interest is provided, wherein the polyhedral model comprises vertices having coordinates, wherein the polyhedral model comprises a topology connecting at least one of the vertices to a face of a surface of the polyhedral model, and wherein the examination apparatus comprises a calculation unit adapted for performing the steps of optimizing (for example alternately) the coordinates of the polyhedral model and an attenuation function of the polyhedral model during a data reconstruction, resulting in an optimized attenuation value together with a surface model of the object of interest.

It should be noted that the topology may connect each one of the vertices with a corresponding surface of the model. However, for performing the modelling of the polyhedral model, not all of the vertices may have to be connected to a respective surface.

It should be noted that optimizing of coordinates of the polyhedral model may be performed for example alternately or by optimizing of one unknown parameter X = (γ_ι,...,γ_N,μ) which comprises the vertices or coordinates y_{,...,y_N of the surface model and the attenuation value μ .

Furthermore, it should be noted that the polyhedral model may consist of several sub-models, each of which endowed with its own vertices and attenuation value. In this case the unknown vector which is subject to the optimization procedure can be written as X = (X_J VJXJ with χ_; = (V_I '-->V_ΛΓ ' ^ " ^ⁿ ^^{s sense}' ^me *^erm "polyhedral model" comprises compound models and the term "attenuation value" comprises a corresponding attenuation vector describing the attenuation coefficient in the submodels of the compound polyhedral model.

In other words, an examination apparatus is provided in which a polyhedral model of for example a bone or an organ may be modelled. The modelling is performed in an iterative manner in which the coordinates of the vertices of the polyhedral model and the attenuation function or attenuation value of the model are optimized. Thus, the attenuation function or value does not have to be known in advance.

Therefore, the invention provides for a fully automatic three-dimensional modelling of medical structures (in particular high contrast objects) with (piecewise) homogeneous attenuation function from few projections without the need for a three- dimensional volume segmentation. There may be no principle restrictions on the geometrical setup of the data acquisition and the modelling may even be capable of reconstructing severely non-convex objects, such as the objects depicted in Figs. 2, 3, 5A and 5B. Possible clinical applications comprise high-contrast imaging of coronary veins and ventricles in rotational angiography, orthopaedic imaging of bones and joints and the reconstruction of deformable medical devices. Furthermore, this new method may be easily applicable in the field of digital subtraction angiography. Since the underlying reconstruction algorithm is of iterative nature, the invention may be suited for a variety of acquisition geometries such as rotational runs, dual axis movements and acquisitions which are geometrically limited to gather only few projections.

The object of interest may be modelled as a polyhedron with triangular surface mesh. Although the topology of the model does not change during iteration, the method may easily reconstruct even non-convex shapes. Often an application- specific model such as a heart, vessel, or bone model is available to initialize the iterative procedure and to improve the convergence of the algorithm. However, the method may also be capable of reconstructing arbitrary polyhedron structures from simple spherical initial meshes by stabilizing the reconstruction with suitable regularization terms.

The polyhedral object is assumed to be situated in air, alternatively, the projection may be pre-processed in order to remove contributions from surrounding tissue. Alternatively, the background structure may be incorporated into the model as an additional polyhedral object. The algorithm exploits a gradient descent scheme in order to minimize the object function which consists of a data fit term and additional penalty terms to stabilize the reconstruction procedure. Both the vertices of the polyhedral object and its attenuation value are optimized during the algorithm. The result may be an attenuation value together with a 3D surface model of the physical structure that has been imaged with X-rays. Compared to voxel-based reconstruction techniques a further segmentation is not necessary. Hence, the reconstructed model may immediately be used both for visualization and further computations (heart volume, bone thickness, vessel diameter) without any additional image processing.

According to another exemplary embodiment of the present invention, the polyhedral model comprises a triangular surface mesh. It should be noted, that the present invention is not limited to triangular surface meshes. However, such a triangular surface mesh may provide for a fast and efficient modelling.

According to another exemplary embodiment of the present invention, the calculation unit is further adapted for performing the step of stabilizing the reconstruction by adding at least one penalty term to a data mismatch error term. It should be noted that no regularization or stabilization may be required in case the starting model is of sufficient quality. Furthermore, other penalty terms may be used for stabilization or regularization.

Such a stabilization may prevent a degeneracy of the model. For example, according to another exemplary embodiment of the present invention, the at least one penalty term is selected from the group comprising a deviation of vertices from a barycenter of neighbours, a deviation of a face area from an average triangle area in the mesh, a penalty term for kissing triangles, and a deviation from regular triangles.

Furthermore, according to another exemplary embodiment of the present invention, the attenuation value is fixed during the optimization of the coordinates of the polyhedral model, during which a minimization of a residual between measured projection values and calculated forward projection values is performed.

According to another exemplary embodiment of the present invention, the minimization comprises a gradient descent scheme. According to another exemplary embodiment of the present invention, the coordinates of the polyhedral model are fixed during the optimization of the attenuation value, during which a minimum of the following function is determined:

P(μ) = μ²∑>,')² -2μ∑>'_Λ +∑>?

I=I 1=1 1=1

\ι_l-\

According to another exemplary embodiment of the present invention, the examination apparatus is adapted as one of a three-dimensional computed tomography apparatus, a three-dimensional rotational X-ray apparatus, and an orthopaedic X-ray imaging apparatus. For example, the examination apparatus is a C- arm system.

According to another exemplary embodiment of the present invention, the examination apparatus is adapted for being applied in the field of digital subtraction angiography.

According to another exemplary embodiment of the present invention, the attenuation function is piecewise constant.

According to another exemplary embodiment of the present invention, the data reconstruction is performed during an acquisition of projection data of the object of interest, wherein a result of the reconstruction is visualized during the acquisition.

According to another exemplary embodiment of the present invention, the visualized result comprises at least one of an intermediate image and an intermediate attenuation function. For example, the intermediate surface model may be visualized or otherwise analyzed after each or a predetermined number of optimization steps during the iterative reconstruction. Furthermore, or alternatively, the attenuation function or simply the attenuation value may be visualizes or otherwise analyzed, independently from the intermediate surface model. Therefore, the intermediate results may be valuated during the iterative reconstruction, thus allowing for a correction of the reconstruction after having the results analyzed. Such analysis may be performed by comparison of the intermediate result with a projection, thus providing a feedback of the quality of the model. In other words, the convergence quality of the iterative reconstruction may be tracked, for example visually.

According to another exemplary embodiment of the present invention , a user interface for visualization of an intermediate result of a data reconstruction of, for example, the above described polyhedral model of an object of interest is provided, wherein the visualization and the data reconstruction are performed during an acquisition of projection data of the object of interest.

Such a user interface may comprise a display or a monitor for visualizing the intermediate result. After each iteration the surface of the model is displayed such that the convergence of the iterative reconstruction may be graphically (visually) tracked by the user. By projecting the intermediate model in or after each iteration step on a single projection, a visual feedback relating to the quality of the model may be provided.

Thus, contrary to an image segmentation, not only a surface model may be generated which is optimally adapted (to the object of interest), but also a corresponding (intermediate) absorption coefficient or attenuation function is generated, such that all line integrals through the object belonging to a projection have the smallest difference to the measured data. Such a coefficient or function may not be provided by a normal segmentation process.

According to another exemplary embodiment of the present invention, the visualized intermediate result comprises at least one of an intermediate image and an intermediate attenuation function.

According to another exemplary embodiment of the present invention, the data reconstruction is an iterative data reconstruction.

According to another exemplary embodiment of the present invention, a method for modelling a polyhedral model of an object of interest is provided, wherein the polyhedral model comprises vertices having coordinates, wherein the polyhedral model comprises a topology connecting at least one of the vertices to a face of a surface of the polyhedral model, and wherein the method comprises the steps of alternately optimizing the coordinates of the polyhedral model and optimizing an attenuation function of the polyhedral model during a data reconstruction, resulting in an optimized attenuation value together with a surface model of the object of interest.

Furthermore, according to another exemplary embodiment of the present invention, a computer-readable medium is provided, in which a computer program for modelling a polyhedral model of an object of interest is stored, wherein the polyhedral model comprises vertices having coordinates, wherein the polyhedral model comprises a topology connecting at least one of the vertices to a face of a surface of the polyhedral model, and wherein the computer-readable medium, when executed by a processor, causes the processor to carry out the above-mentioned method steps.

According to another exemplary embodiment of the present invention, a program element for modelling a polyhedral model of an object of interest is provided, wherein the polyhedral model comprises vertices having coordinates, wherein the polyhedral model comprises a topology connecting at least one of the vertices to a face of a surface of the polyhedral model, and wherein the program element, when being executed by a processor, causes the processor to carry out the above-mentioned method steps.

According to another exemplary embodiment of the present invention, an image processing device for modelling a polyhedral model of an object of interest is provided, wherein the polyhedral model comprises vertices having coordinates, wherein the polyhedral model comprises a topology connecting at least one of the vertices to a face of a surface of the polyhedral model, and wherein the image processing device is adapted for carrying out the above-mentioned method steps.

It may be seen as a gist of an exemplary embodiment of the present invention, that both an optimization of coordinates of the polyhedral model and an optimization of an attenuation function or attenuation value of the polyhedral model is performed during a data reconstruction of the object of interest. The optimization scheme exploits a gradient descent scheme to minimize the object function, which consists of a data fit term and additional penalty terms to stabilize the reconstruction procedure. This may result in an attenuation value together with a three-dimensional surface model of the X-ray physical structure of the object of interest.

These and other aspects of the present invention will become apparent from and elucidated with reference to the embodiments described hereinafter.

Exemplary embodiments of the present invention will now be described in the following, with reference to following drawings.

Fig. 1 shows an exemplary embodiment of an examination apparatus according to the present invention.

Fig. 2 shows the reconstruction of an L-shaped obstacle with unknown attenuation from 10 projections according to an exemplary method of the present invention. Fig. 3 shows a reconstruction of a stenosis phantom according to an exemplary method of the present invention.

Figs. 4A and 4B show an illustration of a projection of a segmented left ventricle for the cardiac phases 10 and 22 out of 23 phases according to an exemplary method of the present invention. Figs. 5 A and 5B show final reconstructions of the left ventricle for cardiac phases 10 and 22, respectively, according to an exemplary method of the present invention. Fig. 6 shows a flow-chart of an exemplary embodiment according to the present invention. Fig. 7 shows an exemplary embodiment of a compound polyhedral model illustrated in 2D. Fig. 8 shows an exemplary embodiment of an image processing device according to the present invention for executing an exemplary embodiment of a method in accordance with the present invention.

The illustration in the drawings is schematically. In different drawings, similar or identical elements are provided with the same reference numerals.

3D reconstruction from a small number of projections is an active field of research and only partial results are known up to now. In contrast to the work presented in [l]-[3] this invention is not limited to sparse objects and can also be applied to bolus filled heart chambers. In [4] the authors propose to optimize a surface model by forward projecting the model and adapting it to the edge contours in the projections. Since the approach in [4] is focused on the silhouettes of the model rather than the line-integrals, non-convex structures can hardly be reconstructed, whereas the proposed method is fully based on the physical model of attenuated X-rays and easily reconstructs non- convexities. Moreover, the presented modelling scheme reconstructs both the polyhedral shape and the attenuation of a homogeneous obstacle. Compared with voxel- based iterative reconstruction schemes, the polyhedral reconstruction is contour based which may result in a smaller number of unknowns due to the reduction of one dimension. Hence, the contour-based reconstruction may be faster than conventional voxel-based iterative algorithms. The unknown object is modelled with a triangular surface mesh, where a rough first guess initializes the reconstruction procedure. The topology of the model must be known or guessed a priori and is often given together with a good initial mesh by the particular application such as heart, vessel or bone imaging. Then, the coordinates of the vertices are optimized in the reconstruction scheme. Additionally, the constant attenuation of the object is a further unknown, which is optimized alternately with the vertices. To stabilize the reconstruction and to avoid self-intersections and degenerated triangles, a variety of different penalty terms may be added to the data mismatch error term. For the speed up of the modelling scheme, a refinement scheme may be provided which starts with a coarse surface mesh and down-sampled projections. When the decrease of the penalty term is slowing down, the surface mesh may be refined and, if necessary, the projections are resampled. The regularization parameters may be controlled adaptively, too.

In the following a detailed description on an exemplary embodiment of the present invention is provided:

In addition to the contour reconstruction of a high-contrast object, the attenuation coefficient is reconstructed as well. To this end the unknown object is modelled via a triangular surface mesh with vertices V = {v. : i = 1,...,N} and a vertex index list F = {F_jk : j = l,...,M;k = 1,2,3} , which defines M triangular faces T₁ = V_p V_p V_p ordered such that the corresponding face normal

(y, - v_F U(y. -v_F ) ri : = ^■

¹ ll (% -%)^χ(% -V^₁) H points into the exterior of the object for j = \,...,M . Together with the constant coefficient μ , these parameters constitute the model (μ,V,F) . Here the attenuation μ and the vertex positions V are unknown while the ordering of the faces F is known in advance. For the reconstruction of the object model the attenuation μ is optimized alternately with the vertices v. . To this end the residual between the measured projection values P₁ , I = \,...,L and the forward projection values q_t = A₁ {\i,V, F) of the model is computed together with additional penalty terms:

J(μ,K,F) = £(_Λ -^)² + £λΛ(K,F) . (1)

1=1 f=l Since the object is assumed to be situated in air, the forward projection can be calculated via

1, 12

A_ι (μ,V,F) = μ∑\\ w₂ ^ι _i - w₂ ^ι _i__ι \\ (2)

:=1 where the I₁ intersections w_i ^l ,i = \,...,I_l ofthe / -th ray with the surface triangles T_j for j = \,...,M , ofthe polyhedron are ordered by increasing distance to the source location. In case of an uneven number of intersections I₁ , the / -th ray is replaced with a parallel ray in the vicinity ofthe original ray. This situation may occur if the ray hits the object exactly in a vertex or intersects an edge of a triangular face. For fixed vertex positions, the penalty terms are constant and the minimization of (1) boils down to the determination of the minimum of the parabola

P(μ) = μ²∑>,')² -2μ∑>'_Λ +∑>? (3)

I=I I=I 1=1

I₁ 12 with q[ = V Il w₂'_l - w_2l-ι || . In this case, the unique minimizer of ι=l functional (1) is given by the minimizer

of the parabola (3). On the other hand, for a fixed attenuation coefficient μ the functional (1) may be minimized using the following gradient descent scheme:

1. Compute the gradient G = VJ(μ , V, F) of ( 1 ) with respect to the vertices V numerically.

2. Define a one-dimensional optimization problem via the surrogate functional J(s) = J(μ,V - sG,F) .

3. Approximate J(s) by a parabola P(s) such that a. P(O) = J(O) , i.e. P(O) = J(F) , b. P' (0) = J' (0) , i.e. P¹ (0) = - || G ||² c. P(t) = J (f) , for a suitable t ≠ s .

4. Update V = V - sG , where s is the unique minimizer of the parabola P . The following numerical experiments show the applicability of such an alternating minimization of both the attenuation coefficient and the vertices. Fig. 2 shows the reconstruction of a simple but non-convex L-shaped obstacle with unknown attenuation from 10 projections. Reference numerals 201, 202, 203 and 204 reference four different views of the L-shaped object. View 201 is a front view, view 202 is a top view, view 203 is a side view and view 204 is a diagonal top view. Each view depicts individual triangular surface elements which form the overall surface match, such as elements 209, 210, 211, 212.

Reference numerals 205, 206, 207 and 208 reference four different views of the polyhedral reconstruction of the L-shaped obstacle. View 201 corresponds to view 205, view 202 corresponds to view 206, view 203 corresponds to view 207 and view 204 corresponds to view 208. The ten different views have been equally distributed on a circular arc of 220° with 1195mm source to detector distance and 810mm source to center of rotation distance.

The detector resolution was increased during iteration from 32² to 64² detector elements on a detector of 300mm width and 300mm height. The true attenuation value μ = 0.6mm^"1 was reconstructed from the starting value μ₀ = 0.2mm^"1 with a final value of μ^ = 0.601mm^"1. The final reconstructed mesh is illustrated in Fig.

2 and consists of 642 vertices and 1280 triangular faces, while the iteration has been initialized by an icosahedron (20 regular triangles, 12 vertices on a sphere) of 50mm radius. The mesh is successively refined with a 1 to 4 triangle refinement strategy by bisecting each triangle side whenever the relative decrease of the functional (1) is smaller than a given threshold.

For stabilizing the iterative reconstruction procedure the following penalty terms have been chosen: 1. deviation of vertices from barycenter of neighbours (favouring flat surfaces)

1 K, with the barycenter b = V v, of the K₁ neighbours v. , of the

K _{1 k=l} vertex v_} in the mesh (V, F) , 2. deviation of face area from average triangle area in the mesh

-v_FJx(v_Fji -_%) || (6)

3. penalty term for kissing triangles

with the convention that n_{s +l k} = n_{l k} , where J_k is the number of adjacent faces at vertex v_k ,

4. deviation from regular triangles M

R₄( F) = £f l-cos(|-α_Jt)cos(|- β_Jt)cos(|-γ_Jt) (8) k=\

where a_k, $_k,J_k are the three angles of the triangle T_k .

The corresponding regularization parameters X₁,..., λ₄ are controlled during the iteration to guide the optimization procedure. To this end, a first choice of the regularization parameters is made such that the sum of all penalty terms is between 10%-50% of the residual without any additional penalty term. After a fixed number of iterations, the ratio of the penalty terms and the sole residual is checked and adapted if it is out of the range from 10%-50%. Similarly, a regularization parameter is updated if the corresponding penalty term is significantly larger/smaller than the average penalty term. With this parameter choice a self- intersection or degeneration of the polyhedral object model can be avoided. In order to minimize the mismatch in the projection data, the ratio between penalty terms and projection mismatch is successively reduced, whenever the mesh is refined. An example of clinical interest is provided with a stenosis simulation in a

C-arm geometry. Here, a tubular phantom of 15 mm length with proximal vessel diameter 2.5 mm , distal vessel diameter 1.5 mm and a stenosis diameter of 0.8 mm defines the phantom object. The stenosis is forward projected on an FD20 with 1024² detector elements from nine views equally distributed on a full circle. The source- detector distance was assumed to be 1195 mm with 810 mm distance from the source to the isocenter of the rotational movement. For reconstruction, a subarea of 300x100 detector elements has been considered with 0.29 mm detector element spacing.

Fig. 3 illustrates the above mentioned stenosis phantom together with one forward projection and the final reconstruction result. Additionally, the attenuation coefficient μ = 0.45mm^"1 of the stenosis phantom has been reconstructed with a final value of μ^ = 0.447mm^"1 from the starting value μ₀ = 0.2mm^"1 .

The left image 301 shows a projection of the stenosis phantom. The middle image 302 shows a central cut through the stenosis and the right image 303 shows the reconstructed stenosis with a polyhedral surface model, according to an exemplary embodiment of the present invention.

Finally, Figs. 4A, 4B, 5 A and 5B show the reconstruction of a left ventricle which has been segmented from a CT reconstruction and has been forward projected into a similar C-arm geometry as for the L-shaped obstacle above. The illustration 401 relates to a projection of the segmented left ventricle for the cardiac phase 10 and illustration 402 relates to a projection of the segmented left ventricle for the cardiac phase 22 out of 23 phases. From the ECG-signal the projections belonging to one cardiac phase have been identified via the nearest neighbour relation.

Figs. 5 A and 5B show the final reconstruction of phase 10 and 22 out of 23 phases in the cardiac cycle. The illustrations 501, 502, 503 and 504 of Fig. 5 A show four different views of the left ventricle for the cardiac phase 10 and illustrations 505, 506, 507 and 508 show four different views for the left ventricle for cardiac phase 22. The unknown attenuation coefficient has been reconstructed as μ^ = 0.026mm^"1 for cardiac phase 10 and μ^ = 0.027mm^"1 for cardiac phase 22 from the starting value μ₀ = 0.2mm^"1 .

Fig. 1 shows a schematic representation of an exemplary rotational X-ray scanner, adapted as from a C-arm scanner according to an exemplary embodiment of the present invention. It should be noted however, that the present invention is not limited to rotational X-ray scanners.

An X-ray source 100 and a flat detector 101 with a large sensitive area are mounted to the ends of a C-arm 102. The C-arm 102 is held by curved rail, the "sleeve" 103. The C-arm can slide in the sleeve 103, thereby performing a "roll movement" about the axis of the C-arm. The sleeve 103 is attached to an L-arm 104 via a rotational joint and can perform a "propeller movement" about the axis of this joint. The L-arm 104 is attached to the ceiling via another rotational joint and can perform a rotation about the axis of this joint. The various rotational movements are effected by servo motors. The axes of the three rotational movements and the cone-beam axis always meet in a single fixed point, the "isocenter" 105 of the rotational X-ray scanner. There is a certain volume around the isocenter that is projected by all cone beams along the source trajectory. The shape and size of this "volume of projection" (VOP) depend on the shape and size of the detector and on the source trajectory. In Fig. 1, the ball 110 indicates the biggest isocentric ball that fits into the VOP. The object (e.g. a patient or an item of baggage) to be imaged is placed on the table 111 such that the object's volume of interest (VOI) fills the VOP. If the object is small enough, it will fit completely into the VOP; otherwise, not. The VOP therefore limits the size of the VOI. The various rotational movements are controlled by a control unit 112. Each triple of C-arm angle, sleeve angle, and L-arm angle defines a position of the X- ray source. By varying these angles with time, the source can be made to move along a prescribed source trajectory. The detector at the other end of the C-arm makes a corresponding movement. The source trajectory will be confined to the surface of an isocentric sphere.

The C-arm x-ray scanner is adapted for performing an examination method according to the invention.

Fig. 6 shows a flow-chart of an exemplary method according to the present invention. In step 1, an X-ray beam emitted from a radiation source towards a detector is selected and the intersection points between the beam and a polyhedral model are calculated. This calculation results in entry and exit points (in which the beam enters or exists the model).

In general, the number of entry and exit points is an even number, in case no edge or anything similar is hit by the beam.

In step 2, the distance which the beam travels through the object, i.e. the model, is calculated. It should be noted, that more than one entry point into the object of interest and more than one exit point from the object are possible.

In step 3, the line integral of the calculated distances is calculated and a forward projection is performed. This results in calculated data, which correspond to certain data which has been measured during data acquisition.

In step 4, the residual of the calculated and the measured data is determined and in step 5 the residual is minimized, for example on the basis of a gradient descent scheme. However, other minimization schemes may be used.

In order to prevent for degradation or degeneracy, small steps may be performed or/and suitable regularization terms may be implemented.

For the reconstruction of a compound polyhedral model consisting of M sub-models χ_; = (V₁V₅V_{jV >} M-_:) ₅ i⁼l₅— ₅M, the forward projector may be redefined as

M Qi ⁼ ∑^' A (Vⁱ _ι ->V_ι ->F_ι) •> i-^e- ^as the sum of contributions from each sub-model, and hereby

redefines the object function (1). Each of the sub-models may be penalized with one ore more regularization terms (5)-(8). Moreover, additional penalty terms may be introduced in order to control inter-object behavior of the sub-models.

Fig. 7 shows an exemplary embodiment of a compound polyhedral model illustrated in two dimensions (2D). The model 800 comprises five sub-models 801, 802, 803, 804, 805. It should be noted that different sub-models may overlap each other or one may include another. Furthermore, two or more sub-models may share a common interface.

The reconstruction of a compound polyhedral model may be performed by minimizing the residual (1) with respect to the unknowns X = (X_{J VJ}XJ _> f°^r example with a gradient descent scheme. The reconstructed attenuation values μ_; and the physical attenuation value μ of the object under investigation are related via

M μ(x) = ^ M-, (^χ) ■_> where μ_; (x) equals the reconstructed attenuation value μ_; inside and

vanishes outside the i-th sub-model.

Fig. 8 shows an exemplary embodiment of a image processing device 700 according to the present invention for executing an exemplary embodiment of a method in accordance with the present invention. The image processing device 700 depicted in Fig. 8 comprises a central processing unit (CPU) or image processor 701 connected to a memory 702 for storing an image depicting an object of interest, such as a patient or an item of baggage. The image processor 701 may be connected to a plurality of input/output network or diagnosis devices, such as a CT device. The image processor 701 may furthermore be connected to a display device 703, for example, a computer monitor, for displaying information or an image computed or adapted in the image processor 701. An operator or user may interact with the image processor 701 via a keyboard 704 and/or other output devices, which are not depicted in Fig. 8. Furthermore, via the bus system 705, it may also be possible to connect the image processing and control processor 701 to, for example, a motion monitor, which monitors a motion of the object of interest. In case, for example, a lung of a patient is imaged, the motion sensor may be an exhalation sensor. In case the heart is imaged, the motion sensor may be an electrocardiogram. Exemplary embodiments of the invention may be sold as a software option to CT scanner console, imaging workstations or PACS workstations. This invention is in particular applicable, where radiographs of high-contrast objects such as bones and implants or ventricles and vessels which are filled with contrast agent are available without background structure. The clinical relevance of this invention may even be further improved by preprocessing the projection data in order to remove surrounding tissue and/or by application of this method with the presented compound model approach.

It should be noted that the term "comprising" does not exclude other elements or steps and the "a" or "an" does not exclude a plurality. Also elements described in association with different embodiments may be combined.

It should also be noted that reference signs in the claims shall not be construed as limiting the scope of the claims.

Claims

CLAIMS:

1. Examination apparatus for modelling a polyhedral model of an object of interest, wherein the polyhedral model comprises vertices having coordinates; wherein the polyhedral model comprises a topology connecting at least one of the vertices to a face of a surface of the polyhedral model; wherein the examination apparatus comprises a calculation unit adapted for performing the steps of: optimizing the coordinates of the polyhedral model and an attenuation function of the polyhedral model during a data reconstruction, resulting in an optimized attenuation value together with a surface model of the object of interest.

2. Examination apparatus of claim 1, wherein the polyhedral model comprises a triangular surface mesh.

3. Examination apparatus of claim 1, wherein the calculation unit is further adapted for performing the step of: stabilizing the reconstruction by adding at least one penalty term to a data mismatch error term.

4. Examination apparatus of claim 3, wherein the at least one penalty term is selected from the group comprising a deviation of vertices from a barycenter of neighbours, a deviation of a face area from an average triangle area in the mesh, a penalty term for kissing triangles, and a deviation from regular triangles.

5. Examination apparatus of claim 1, wherein during the optimization of the coordinates of the polyhedral model the attenuation value is fixed and a minimization of a residual between measured projection values and calculated forward projection values is performed.

6. Examination apparatus of claim 5, wherein the minimization comprises a gradient descent scheme.

7. Examination apparatus of claim 1, wherein during the optimization of the attenuation value the coordinates of the polyhedral model are fixed and a minimum of the following function is determined:

P(μ) = μ²j>;)² -2μ∑>;_A +∑pf

I1==I1 1=1 1=1

II₁₁ 1122 with qι = ∑\\ w₂'_l - W_2^1 W . ι=l

8. Examination apparatus of claim 1, the examination apparatus (100) being adapted as one of a 3D computed tomography apparatus, a 3D rotational X-ray apparatus, and an orthopaedic X-ray imaging apparatus.

9. Examination apparatus of claim 1, wherein the examination apparatus is adapted for being applied in the field of digital subtraction angiography.

10. Examination apparatus of claim 1, wherein optimizing the coordinates of the polyhedral model and an attenuation value of the polyhedral model during a data reconstruction is performed alternately.

11. Examination apparatus of claim 1 , wherein the attenuation function is piecewise constant.

12. Examination apparatus of claim 1, wherein the data reconstruction is performed during an acquisition of projection data of the object of interest; and wherein a result of the reconstruction is visualized during the acquisition.

13. Examination apparatus of claim 12, wherein the visualized result comprises at least one of an intermediate image and an intermediate attenuation function.

14. User interface for visualization of an intermediate result of a data reconstruction, wherein the visualization and the data reconstruction are performed during an acquisition of projection data of the object of interest.

15. User Interface of claim 14, wherein the visualized intermediate result comprises at least one of an intermediate image and an intermediate attenuation function.

16. User Interface of claim 14, wherein the data reconstruction is an iterative data reconstruction.

17. A method for modelling a polyhedral model of an object of interest, wherein the polyhedral model comprises vertices having coordinates, wherein the polyhedral model comprises a topology connecting at least one of the vertices to a face of a surface of the polyhedral model, and wherein the method comprises the steps of: optimizing the coordinates of the polyhedral model and an attenuation value of the polyhedral model during a data reconstruction, resulting in an optimized attenuation value together with a surface model of the object of interest.

18. A computer-readable medium (702), in which a computer program for modelling a polyhedral model of an object of interest is stored; wherein the polyhedral model comprises vertices having coordinates; wherein the polyhedral model comprises a topology connecting at least one of the vertices to a face of a surface of the polyhedral model; and wherein the computer-readable medium, when executed by a processor (701), causes the processor to carry out the steps of: optimizing the coordinates of the polyhedral model and an attenuation function of the polyhedral model during a data reconstruction, resulting in an optimized attenuation value together with a surface model of the object of interest.

19. A program element for modelling a polyhedral model of an object of interest, wherein the polyhedral model comprises vertices having coordinates, wherein the polyhedral model comprises a topology connecting at least one of the vertices to a face of a surface of the polyhedral model, and wherein the program element, when being executed by a processor (701), causes the processor to carry out the steps of: optimizing the coordinates of the polyhedral model and an attenuation function of the polyhedral model during a data reconstruction, resulting in an optimized attenuation value together with a surface model of the object of interest.

20. An image processing device for modelling a polyhedral model of an object of interest, wherein the polyhedral model comprises vertices having coordinates, wherein the polyhedral model comprises a topology connecting at least one of the vertices to a face of a surface of the polyhedral model, the image processing device being adapted for: optimizing the coordinates of the polyhedral model and an attenuation function of the polyhedral model during a data reconstruction, resulting in an optimized attenuation value together with a surface model of the object of interest.