B-spline Free-form Deformation Research Articles

Automated shape and thickness optimization for non-matching isogeometric shells using free-form deformation

AbstractIsogeometric analysis (IGA) has emerged as a promising approach in the field of structural optimization, benefiting from the seamless integration between the computer-aided design (CAD) geometry and the analysis model by employing non-uniform rational B-splines (NURBS) as basis functions. However, structural optimization for real-world CAD geometries consisting of multiple non-matching NURBS patches remains a challenging task. In this work, we propose a unified formulation for shape and thickness optimization of separately parametrized shell structures by adopting the free-form deformation (FFD) technique, so that continuity with respect to design variables is preserved at patch intersections during optimization. Shell patches are modeled with isogeometric Kirchhoff–Love theory and coupled using a penalty-based method in the analysis. We use Lagrange extraction to link the control points associated with the B-spline FFD block and shell patches, and we perform IGA using the same extraction matrices by taking advantage of existing finite element assembly procedures in the FEniCS partial differential equation (PDE) solution library. Moreover, we enable automated analytical derivative computation by leveraging advanced code generation in FEniCS, thereby facilitating efficient gradient-based optimization algorithms. The framework is validated using a collection of benchmark problems, demonstrating its applications to shape and thickness optimization of aircraft wings with complex shell layouts.

Inter-pass motion correction for whole-body dynamic PET and parametric imaging.

Whole-body dynamic FDG-PET imaging through continuous-bed-motion (CBM) mode multi-pass acquisition protocol is a promising metabolism measurement. However, inter-pass misalignment originating from body movement could degrade parametric quantification. We aim to apply a non-rigid registration method for inter-pass motion correction in whole-body dynamic PET. 27 subjects underwent a 90-min whole-body FDG CBM PET scan on a Biograph mCT (Siemens Healthineers), acquiring 9 over-the-heart single-bed passes and subsequently 19 CBM passes (frames). The inter-pass motion correction was executed using non-rigid image registration with multi-resolution, B-spline free-form deformations. The parametric images were then generated by Patlak analysis. The overlaid Patlak slope Ki and y-intercept Vb images were visualized to qualitatively evaluate motion impact and correction effect. The normalized weighted mean squared Patlak fitting errors (NFE) were compared in the whole body, head, and hypermetabolic regions of interest (ROI). In Ki images, ROI statistics were collected and malignancy discrimination capacity was estimated by the area under the receiver operating characteristic curve (AUC). After the inter-pass motion correction was applied, the spatial misalignment appearance between Ki and Vb images was successfully reduced. Voxel-wise normalized fitting error maps showed global error reduction after motion correction. The NFE in the whole body (p = 0.0013), head (p = 0.0021), and ROIs (p = 0.0377) significantly decreased. The visual performance of each hypermetabolic ROI in Ki images was enhanced, while 3.59% and 3.67% average absolute percentage changes were observed in mean and maximum Ki values, respectively, across all evaluated ROIs. The estimated mean Ki values had substantial changes with motion correction (p = 0.0021). The AUC of both mean Ki and maximum Ki after motion correction increased, possibly suggesting the potential of enhancing oncological discrimination capacity through inter-pass motion correction.

Improvement of FFD parametric approach in the application of a lifting body

AbstractFFD (free-form deformation method) is one of the most commonly used parameterisation methods at present. It places the parameterised objects inside the control volume through coordinate system transformation, and controls the control volume through control points, thus realising the deformation control of its internal objects. Firstly, this paper systematically analyses and compares the characteristics and technical requirements of Bernstein, B-spline and NURBS (non-uniform rational b-splines) basic functions that can be adopted by FFD, and uses the minimum number of control points required to achieve the specified control effect threshold to express the control capability. Aiming at the problem of discontinuity at the right end in the actual calculation of B-spline basis function, a method of adding a small epsilon is proposed to solve it. Then, three basic functions are applied to the FFD parameterisation method, respectively, and the differences are compared from two aspects of the accurate expression of the model and the ability of deformation control. It is found that the BFFD (b-spline free-form deformation) approach owns better comprehensive performance when the control points are distributed correctly. In this paper, the BFFD method is improved, and a p-BFFD (reverse solution points based BFFD) method based on inverse solution is proposed to realise the free distribution of control points under the specified topology. Further, for the lifting body configuration, the control points of the p-BFFD method are brought closer to the airframe forming the EDGE-p-BFFD (edge constraints based p-BFFD) method. For the case in this paper, the proposed EDGE-p-BFFD method not only has fairly high parameterisation accuracy, but also reduces the expression error from 1.01E-3 to 1.25E-4, which is nearly ten times. It can also achieve effective lifting body guideline constraints, and has the ability of local deformation adapting to the configuration characteristics. In terms of the proportion of effective control points, the EDGE-p-BFFD method increases the proportion of effective control points from 36.7% to 50%, and the more control points, the more obvious the proportion increase effect. The new method also has better effect on the continuity of geometric deformation. At the same time, this paper introduces the independent deformation method of the upper and lower surfaces based on the double control body frames, which effectively avoids the deformation coupling problem of the simultaneous change of the upper and lower surfaces caused by the movement of control points in the traditional single control framework.

Comparison of an in‐house hybrid DIR method to NiftyReg on CBCT and CT images for head and neck cancer

An in‐house hybrid deformable image registration (DIR) method, which combines free‐form deformation (FFD) and the viscous fluid registration method, is proposed. Its results on the planning computed tomography (CT) and the day 1 treatment cone‐beam CT (CBCT) image from 68 head and neck cancer patients are compared with the results of NiftyReg, which uses B‐spline FFD alone. Several similarity metrics, the target registration error (TRE) of annotated points, as well as the Dice similarity coefficient (DSC) and Hausdorff distance (HD) of the propagated organs at risk are employed to analyze their registration accuracy. According to quantitative analysis on mutual information, normalized cross‐correlation, and the absolute pixel value differences, the results of the proposed DIR are more similar to the CBCT images than the NiftyReg results. Smaller TRE of the annotated points is observed in the proposed method, and the overall mean TRE for the proposed method and NiftyReg was 2.34 and 2.98 mm, respectively (p < 0.001). The mean DSC in the larynx, spinal cord, oral cavity, mandible, and parotid given by the proposed method ranged from 0.78 to 0.91, significantly higher than the NiftyReg results (ranging from 0.77 to 0.90), and the HD was significantly lower compared to NiftyReg. Furthermore, the proposed method did not suffer from unrealistic deformations as the NiftyReg did in the visual evaluation. Meanwhile, the execution time of the proposed method was much higher than NiftyReg (96.98 ± 11.88 s vs. 4.60 ± 0.49 s). In conclusion, the in‐house hybrid method gave better accuracy and more stable performance than NiftyReg.

A Statistical Shape-Based Patient-Specific Anatomical Structure Model

A patient-specific anatomical structure model has been widely used in many medical applications. However, in practical applications, to effectively construct a patient-specific anatomical structure model is a challenge, the reasons are: 1) the manual marking process for landmark points is time-consuming and is prone to have false points; 2) the correspondence establishment is difficult; and 3) the performance of the model is limited. Therefore, the purpose of this study is to automatically construct a patient-specific anatomical structure model to solve these difficulties. Firstly, the input data are preprocessed to enhance the region of interest in CT scan images. Then, the region of interest is regarded as a training specimen, and the triangle is used to mesh the training specimen. Meanwhile, vertices contraction strategy is introduced to iteratively contract triangle meshes, and the correspondences are established through improved B-spline free-form deformation. Finally, principal component analysis is used to generate the final patient-specific anatomical structure model. Experimental results on 30 pelvic CT scan images verify that the proposed method outperforms the compared methods.

Cardiovascular magnetic resonance virtual tagging with B-spline-based free-form deformation

PurposeWe developed a virtual tagging technique that reconstructs tagging images using the displacement field obtained by applying B-spline free-form deformation (FFD) between diastolic images and images of other cardiac phases in cardiac cine MRI. The purpose of this study was to validate its characteristics and usefulness in phantom and patient studies. MethodsDigital phantoms simulating uniform and non-uniform wall motion models were created, and virtual tagging images were reconstructed with various matrix sizes and tag resolutions to evaluate the accuracy of FFD and the characteristics of the tags. In the patient study, FFD's accuracy was assessed at three levels (base, middle, and apex) in healthy patients. In patients with heart failure, virtual tagging images were compared with strain maps obtained by feature tracking and virtual tagging. ResultsIn the phantom study, blurring of tags was observed when tags were reconstructed with high resolution using a small matrix size. In the patient study, the accuracy of FFD was lower in the base than in the apex. Patients with heart failure had decreased distortion of the displacement field vector and virtual tags, indicating decreased local wall motion, consistent with areas of abnormalities found in strain maps. ConclusionThe virtual tagging technique does not require additional imaging and can visualize regional LV motion abnormalities via deformation of the tag as well as conventional cardiovascular magnetic resonance tagging.

Spatiotemporal Free-Form Registration Method Assisted by a Minimum Spanning Tree During Discontinuous Transformations.

The sliding motion along the boundaries of discontinuous regions has been actively studied in B-spline free-form deformation framework. This study focusses on the sliding motion for a velocity field-based 3D+t registration. The discontinuity of the tangent direction guides the deformation of the object region, and a separate control of two regions provides a better registration accuracy. The sliding motion under the velocity field-based transformation is conducted under the [Formula: see text]-Rényi entropy estimator using a minimum spanning tree (MST) topology. Moreover, a new topology changing method of the MST is proposed. The topology change is performed as follows: inserting random noise, constructing the MST, and removing random noise while preserving a local connection consistency of the MST. This random noise process (RNP) prevents the [Formula: see text]-Rényi entropy-based registration from degrading in sliding motion, because the RNP creates a small disturbance around special locations. Experiments were performed using two publicly available datasets: the DIR-Lab dataset, which consists of 4D pulmonary computed tomography (CT) images, and a benchmarking framework dataset for cardiac 3D ultrasound. For the 4D pulmonary CT images, RNP produced a significantly improved result for the original MST with sliding motion (p<0.05). For the cardiac 3D ultrasound dataset, only a discontinuity-based registration indicated activity of the RNP. In contrast, the single MST without sliding motion did not show any improvement. These experiments proved the effectiveness of the RNP for sliding motion.

Robustness of Whole Spine Reconstruction using Anterior-Posterior and Lateral Planar X-ray Images

The X-ray-based reconstruction methods using anterior-posterior (AP) and lateral (LAT) images assumes that the angle between the AP and LAT images is perpendicular. However, it is difficult to maintain the perfect perpendicular angle between the AP and LAT images when taking those two images sequentially in real situations. In this study, the robustness of a three-dimensional (3D) whole spine reconstruction method using AP and LAT planar X-ray images was analyzed by investigating cases in which the AP and LAT images were not taken perpendicularly. 3D models of the patient-specific spine from five subjects were reconstructed using AP and LAT X-Ray images and the 3D template models of C1 to L5 vertebrae based on B-spline free-form deformation (FFD) technology. The shape error, projected area error, and relative error in length were quantified by comparing the reconstructed model (FFD model) to the reference model (CT model). The results indicated that the reconstruction method might be considered robust in case that there is a small angular error, such as 5°, between the AP and LAT images. This study suggested that simple technical indications to obtain the perpendicular AP and LAT images in real situations can improve accuracy of 3D spine reconstruction.

Automatic motion correction for in vivo human skin optical coherence tomography angiography through combined rigid and nonrigid registration.

When using optical coherence tomography angiography (OCTA), the development of artifacts due to involuntary movements can severely compromise the visualization and subsequent quantitation of tissue microvasculatures. To correct such an occurrence, we propose a motion compensation method to eliminate artifacts from human skin OCTA by means of step-by-step rigid affine registration, rigid subpixel registration, and nonrigid B-spline registration. To accommodate this remedial process, OCTA is conducted using two matching all-depth volume scans. Affine transformation is first performed on the large vessels of the deep reticular dermis, and then the resulting affine parameters are applied to all-depth vasculatures with a further subpixel registration to refine the alignment between superficial smaller vessels. Finally, the coregistration of both volumes is carried out to result in the final artifact-free composite image via an algorithm based upon cubic B-spline free-form deformation. We demonstrate that the proposed method can provide a considerable improvement to the final en face OCTA images with substantial artifact removal. In addition, the correlation coefficients and peak signal-to-noise ratios of the corrected images are evaluated and compared with those of the original images, further validating the effectiveness of the proposed method. We expect that the proposed method can be useful in improving qualitative and quantitative assessment of the OCTA images of scanned tissue beds.

Non-rigid free-form 2D–3D registration using a B-spline-based statistical deformation model

This paper presents a non-rigid free-from 2D–3D registration approach using a B-spline-based statistical deformation model (SDM). In our approach the B-spline-based SDM is first constructed from a set of training data using a non-rigid registration algorithm based on B-spline free-form deformation to encode a priori information about the underlying anatomy. A novel intensity-based non-rigid 2D–3D registration algorithm is then presented to iteratively fit the 3D B-spline-based SDM to the 2D X-ray images of an unseen subject, which requires a computationally expensive inversion of the instantiated deformation in each iteration. In this paper, we propose to solve this challenge with a fast 3D B-spline pseudo-inversion algorithm that is implemented on graphics processing unit (GPU). Comprehensive experiments on simulated images of patient data sets, on calibrated C-arm and on X-ray images of cadaveric femurs are designed and conducted to evaluate the performance of the proposed approach. The experimental results demonstrate the efficacy of the present approach.

A Parallel Nonrigid Registration Algorithm Based on B-Spline for Medical Images

The nonrigid registration algorithm based on B-spline Free-Form Deformation (FFD) plays a key role and is widely applied in medical image processing due to the good flexibility and robustness. However, it requires a tremendous amount of computing time to obtain more accurate registration results especially for a large amount of medical image data. To address the issue, a parallel nonrigid registration algorithm based on B-spline is proposed in this paper. First, the Logarithm Squared Difference (LSD) is considered as the similarity metric in the B-spline registration algorithm to improve registration precision. After that, we create a parallel computing strategy and lookup tables (LUTs) to reduce the complexity of the B-spline registration algorithm. As a result, the computing time of three time-consuming steps including B-splines interpolation, LSD computation, and the analytic gradient computation of LSD, is efficiently reduced, for the B-spline registration algorithm employs the Nonlinear Conjugate Gradient (NCG) optimization method. Experimental results of registration quality and execution efficiency on the large amount of medical images show that our algorithm achieves a better registration accuracy in terms of the differences between the best deformation fields and ground truth and a speedup of 17 times over the single-threaded CPU implementation due to the powerful parallel computing ability of Graphics Processing Unit (GPU).

Atlas-Based Automatic Generation of Subject-Specific Finite Element Tongue Meshes.

Generation of subject-specific 3D finite element (FE) models requires the processing of numerous medical images in order to precisely extract geometrical information about subject-specific anatomy. This processing remains extremely challenging. To overcome this difficulty, we present an automatic atlas-based method that generates subject-specific FE meshes via a 3D registration guided by Magnetic Resonance images. The method extracts a 3D transformation by registering the atlas' volume image to the subject's one, and establishes a one-to-one correspondence between the two volumes. The 3D transformation field deforms the atlas' mesh to generate the subject-specific FE mesh. To preserve the quality of the subject-specific mesh, a diffeomorphic non-rigid registration based on B-spline free-form deformations is used, which guarantees a non-folding and one-to-one transformation. Two evaluations of the method are provided. First, a publicly available CT-database is used to assess the capability to accurately capture the complexity of each subject-specific Lung's geometry. Second, FE tongue meshes are generated for two healthy volunteers and two patients suffering from tongue cancer using MR images. It is shown that the method generates an appropriate representation of the subject-specific geometry while preserving the quality of the FE meshes for subsequent FE analysis. To demonstrate the importance of our method in a clinical context, a subject-specific mesh is used to simulate tongue's biomechanical response to the activation of an important tongue muscle, before and after cancer surgery.

Evaluation of deformable image registration methods for dose monitoring in head and neck radiotherapy.

In the context of head and neck cancer (HNC) adaptive radiation therapy (ART), the two purposes of the study were to compare the performance of multiple deformable image registration (DIR) methods and to quantify their impact for dose accumulation, in healthy structures. Fifteen HNC patients had a planning computed tomography (CT0) and weekly CTs during the 7 weeks of intensity-modulated radiation therapy (IMRT). Ten DIR approaches using different registration methods (demons or B-spline free form deformation (FFD)), preprocessing, and similarity metrics were tested. Two observers identified 14 landmarks (LM) on each CT-scan to compute LM registration error. The cumulated doses estimated by each method were compared. The two most effective DIR methods were the demons and the FFD, with both the mutual information (MI) metric and the filtered CTs. The corresponding LM registration accuracy (precision) was 2.44 mm (1.30 mm) and 2.54 mm (1.33 mm), respectively. The corresponding LM estimated cumulated dose accuracy (dose precision) was 0.85 Gy (0.93 Gy) and 0.88 Gy (0.95 Gy), respectively. The mean uncertainty (difference between maximal and minimal dose considering all the 10 methods) to estimate the cumulated mean dose to the parotid gland (PG) was 4.03 Gy (SD = 2.27 Gy, range: 1.06–8.91 Gy).

A pre-processing scheme for real-time registration of dynamic contrast-enhanced magnetic resonance images

The paper proposes a fully automatic pre-processing scheme for real-time registration of the breast dynamic contrast-enhanced magnetic resonance images (DCE-MRIs). It consists of an intensity inhomogeneity correction followed by a breast region of interest segmentation. To correct the intensity inhomogeneity in the breast images, a combination of a greyscale morphological closing with the Gaussian filtering is used. The breast region of interest segmentation is based on fitting the human chest by an ellipse. The main advantages of the proposed pre-processing scheme are a low computational complexity and robustness to the breast density. The scheme was evaluated on 50 T1-weighted DCE-MRIs demonstrating a significant decrease in the time taken for registration (almost a factor 2 for the affine registration and almost a factor 9 for a non-linear B-spline free-form deformation).

Hyperspectral Imaging for Cancer Surgical Margin Delineation: Registration of Hyperspectral and Histological Images.

The determination of tumor margins during surgical resection remains a challenging task. A complete removal of malignant tissue and conservation of healthy tissue is important for the preservation of organ function, patient satisfaction, and quality of life. Visual inspection and palpation is not sufficient for discriminating between malignant and normal tissue types. Hyperspectral imaging (HSI) technology has the potential to noninvasively delineate surgical tumor margin and can be used as an intra-operative visual aid tool. Since histological images provide the ground truth of cancer margins, it is necessary to warp the cancer regions in ex vivo histological images back to in vivo hyperspectral images in order to validate the tumor margins detected by HSI and to optimize the imaging parameters. In this paper, principal component analysis (PCA) is utilized to extract the principle component bands of the HSI images, which is then used to register HSI images with the corresponding histological image. Affine registration is chosen to model the global transformation. A B-spline free form deformation (FFD) method is used to model the local non-rigid deformation. Registration experiment was performed on animal hyperspectral and histological images. Experimental results from animals demonstrated the feasibility of the hyperspectral imaging method for cancer margin detection.

Toward adaptive radiotherapy for head and neck patients: Feasibility study on using CT‐to‐CBCT deformable registration for “dose of the day” calculations

The aim of this study was to evaluate the appropriateness of using computed tomography (CT) to cone-beam CT (CBCT) deformable image registration (DIR) for the application of calculating the "dose of the day" received by a head and neck patient. NiftyReg is an open-source registration package implemented in our institution. The affine registration uses a Block Matching-based approach, while the deformable registration is a GPU implementation of the popular B-spline Free Form Deformation algorithm. Two independent tests were performed to assess the suitability of our registrations methodology for "dose of the day" calculations in a deformed CT. A geometric evaluation was performed to assess the ability of the DIR method to map identical structures between the CT and CBCT datasets. Features delineated in the planning CT were warped and compared with features manually drawn on the CBCT. The authors computed the dice similarity coefficient (DSC), distance transformation, and centre of mass distance between features. A dosimetric evaluation was performed to evaluate the clinical significance of the registrations errors in the application proposed and to identify the limitations of the approximations used. Dose calculations for the same intensity-modulated radiation therapy plan on the deformed CT and replan CT were compared. Dose distributions were compared in terms of dose differences (DD), gamma analysis, target coverage, and dose volume histograms (DVHs). Doses calculated in a rigidly aligned CT and directly in an extended CBCT were also evaluated. A mean value of 0.850 in DSC was achieved in overlap between manually delineated and warped features, with the distance between surfaces being less than 2 mm on over 90% of the pixels. Deformable registration was clearly superior to rigid registration in mapping identical structures between the two datasets. The dose recalculated in the deformed CT is a good match to the dose calculated on a replan CT. The DD is smaller than 2% of the prescribed dose on 90% of the body's voxels and it passes a 2% and 2 mm gamma-test on over 95% of the voxels. Target coverage similarity was assessed in terms of the 95%-isodose volumes. A mean value of 0.962 was obtained for the DSC, while the distance between surfaces is less than 2 mm in 95.4% of the pixels. The method proposed provided adequate dose estimation, closer to the gold standard than the other two approaches. Differences in DVH curves were mainly due to differences in the OARs definition (manual vs warped) and not due to differences in dose estimation (dose calculated in replan CT vs dose calculated in deformed CT). Deforming a planning CT to match a daily CBCT provides the tools needed for the calculation of the "dose of the day" without the need to acquire a new CT. The initial clinical application of our method will be weekly offline calculations of the "dose of the day," and use this information to inform adaptive radiotherapy (ART). The work here presented is a first step into a full implementation of a "dose-driven" online ART.

TU‐C‐141‐02: Development of a Hyrbid Biomechanical Model Based Deformable Image Registration; Application in Lung

Purpose: To investigate the reduction in maximum errors in 4D CT lung Deformable Image Registration (DIR) using a hybrid biomechanical model based algorithm with an intensity based registration refinement. Methods: Morfeus, a biomechanical modeling algorithm for deformable image registration was used to register 3D inhale to exhale images of the lungs acquired via 4DCT. The predicted exhale image was obtained by interpolation. An intensity based registration algorithm based on Sum of Absolute Differences (SAD) and BSpline free form deformations (drop3D, Munich, Germany) was then employed to refine the results by a second registration from the Morfeus predicted exhale to the actual exhale image. The BSpline grid point spacing for the refinement was set to 9mm. To measure the accuracy, Target Registration Error (TRE) was calculated based on the Euclidean distance of corresponding bifurcation points on both images. Improvements in the overall TRE as well as reduction in maximum errors were evaluated. Results: The hybrid method has been evaluated on 5 patients to date with 45 to 76 identified bifurcation points per patient. The Morfeus TRE ranged from 2.2±1.0 mm (mean±SD) to 3.6±1.9 mm. This was reduced to 1.4±0.8 mm up to 1.6±2.0 mm with the hybrid method. More importantly, the number of bifurcation points (% of total points) with an error greater than 5 mm was reduced from 54 (22%) points (across all patients, range 3 to 24 points per patient) with Morfeus to 10 (4%) points (range 0 to 4 points per patient) with the hybrid method. Conclusion: The proposed hybrid method combines the flexibility of the biomechanical model based algorithm, including the ability to model the sliding interface between the lungs and the chest wall, with the advantages of intensity based registration for the contrast rich lung images to provide a highly accurate DIR algorithm. This research is supported by the NIH 5RO1CA124714‐02. KK Brock has financial interest in the deformable registration technology through a licensing agreement with RaySearch Laboratories.

Automatic Classification of Left Ventricular Regional Wall Motion Abnormalities in Echocardiography Images Using Nonrigid Image Registration

Identification and classification of left ventricular (LV) regional wall motion (RWM) abnormalities on echocardiograms has fundamental clinical importance for various cardiovascular disease assessments especially in ischemia. In clinical practice, this evaluation is still performed visually which is highly dependent on training and experience of the echocardiographers and therefore suffers from significant interobserver and intraobserver variability. This paper presents a new automatic technique, based on nonrigid image registration for classifying the RWM of LV in a three-point scale. In this algorithm, we register all images of one cycle of heart to a reference image (end-diastolic image) using a hierarchical parametric model. This model is based on an affine transformation for modeling the global LV motion and a B-spline free-form deformation transformation for modeling the local LV deformation. We consider image registration as a multiresolution optimization problem. Finally, a new regional quantitative index based on resultant parameters of the hierarchical transformation model is proposed for classifying RWM in a three-point scale. The results obtained by our method are quantitatively evaluated to those obtained by two experienced echocardiographers visually as gold standard on ten healthy volunteers and 14 patients (two apical views) and resulted in an absolute agreement of 83 % and a relative agreement of 99 %. Therefore, this diagnostic system can be used as a useful tool as well as reference visual assessment to classify RWM abnormalities in clinical evaluation.

Real-time B-spline Free-Form Deformation via GPU acceleration

Accurate Free-Form Deformation is an analytical solution of deformation sampling, where a polygonal object is deformed as a set of trimmed Bézier surfaces. However, the operation is far from being interactive due to its high computational cost. In this paper, we proposed a real-time B-spline Free-Form Deformation of polygonal objects via GPU acceleration. Various time-consuming evaluations are designed and performed by means of highly parallel processing on GPGPU, such as evaluations of points of B-spline volume, calculations of control points of Bézier surfaces, tessellations of trimmed Bézier surfaces, evaluations of normals of tessellated triangles, etc. The adoption of vertex buffer object for rendering the tessellated trimmed Bézier surfaces greatly saves data I/O throughput. Experimental results show that the proposed GPU algorithm gains more than 200 times acceleration than its CPU counterpart.

B-Spline Free-Form Deformation with Improved PN Triangles

B-Spline Free-Form Deformation with Improved PN Triangles