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Quantitative Assessment of Variational Surface Reconstruction from Sparse Point Clouds in Freehand 3D Ultrasound Imaging during Image-Guided Tumor Ablation
Quantitative Assessment of Variational Surface Reconstruction from Sparse Point Clouds in...
Deng, Shuangcheng;Li, Yunhua;Jiang, Lipei;Liang, Ping
2016-04-19 00:00:00
applied sciences Article Quantitative Assessment of Variational Surface Reconstruction from Sparse Point Clouds in Freehand 3D Ultrasound Imaging during Image-Guided Tumor Ablation 1 , 1 , † 2 , † 3 , † Shuangcheng Deng *, Yunhua Li , Lipei Jiang and Ping Liang School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China; yhli@buaa.edu.cn Opto-Mechatronic Equipment Technology Beijing Area Major Laboratory, Beijing Institute of Petrochemical Technology, Beijing 102617, China; lipei@bipt.edu.cn Department of Interventional Ultrasound, General Hospital of PLA, Beijing 100853, China; liangping301@hotmail.com * Correspondence: dengshuangcheng@bipt.edu.cn; Tel.: +86-10-8129-2220; Fax: +86-10-8129-2221 † These authors contributed equally to this work. Academic Editor: Chien-Hung Liu Received: 13 December 2015; Accepted: 12 April 2016; Published: 19 April 2016 Abstract: Surface reconstruction for freehand 3D ultrasound is used to provide 3D visualization of a VOI (volume of interest) during image-guided tumor ablation surgery. This is a challenge because the recorded 2D B-scans are not only sparse but also non-parallel. To solve this issue, we established a framework to reconstruct the surface of freehand 3D ultrasound imaging in 2011. The key technique for surface reconstruction in that framework is based on variational interpolation presented by Greg Turk for shape transformation and is named Variational Surface Reconstruction (VSR). The main goal of this paper is to evaluate the quality of surface reconstructions, especially when the input data are extremely sparse point clouds from freehand 3D ultrasound imaging, using four methods: Ball Pivoting, Power Crust, Poisson, and VSR. Four experiments are conducted, and quantitative metrics, such as the Hausdorff distance, are introduced for quantitative assessment. The experiment results show that the performance of the proposed VSR method is the best of the four methods at reconstructing surface from sparse data. The VSR method can produce a close approximation to the original surface from as few as two contours, whereas the other three methods fail to do so. The experiment results also illustrate that the reproducibility of the VSR method is the best of the four methods. Keywords: freehand 3D ultrasound; surface reconstruction; variational surface reconstruction; ball pivoting; Power Crust; Poisson reconstruction; Hausdorff distance 1. Introduction In recent years, image-guided percutaneous tumor ablation has become more and more widespread in clinical applications because of its minimally invasive characteristics [1–3]. Three imaging modalities are commonly used for guidance of intra-operative tumor ablation: computed tomography (CT), magnetic resonance imaging (MRI), and ultrasound imaging (US). CT and MRI provide large or adequate fields of view, and good contrast and resolution, but they are limited by costs, the need to adapt equipment for a magnetic field, availability, and radiation doses in clinical applications. US can provide the unique advantages of low cost, real-time monitoring, and a lack of ionizing radiation, but lacks a large field of view and adequate resolution at increasing lesion Appl. Sci. 2016, 6, 114; doi:10.3390/app6040114 www.mdpi.com/journal/applsci Appl. Sci. 2016, 6, 114 2 of 22 Appl. Sci. 2016, 6, 114 2 of 22 depth [4–6]. It is clear that no single modality is superior in all areas, and in fact, the modalities are often complementary to each other. However, because image-guided tumor ablation is usually [4–6]. It is clear that no single modality is superior in all areas, and in fact, the modalities are often performed using physical tools, such as forceps, it would be ideal for the imaging system to clearly complementary to each other. However, because image‐guided tumor ablation is usually performed using physical tools, such as forceps, it would be ideal for the imaging system to clearly and accurately image those tools as they move inside the patient. US can accomplish this task, but and accurately image those tools as they move inside the patient. US can accomplish this task, but other modalities with poor real-time imaging are not as effective. other modalities with poor real‐time imaging are not as effective. Conventional two-dimensional (2D) US is extensively used for a variety of clinical applications in Conventional two‐dimensional (2D) US is extensively used for a variety of clinical applications tumor ablation surgery. Although conventional 2D US is a highly flexible imaging modality, its images in tumor ablation surgery. Although conventional 2D US is a highly flexible imaging modality, its represent only 2D thin planes of the patient’s three-dimensional (3D) anatomy, and the physician images represent only 2D thin planes of the patient’s three‐dimensional (3D) anatomy, and the must mentally integrate many 2D images to form an impression of the 3D anatomy and pathology. physician must mentally integrate many 2D images to form an impression of the 3D anatomy and This practice is inefficient, and may lead to inconsistency and incorrect diagnoses and treatment. pathology. This practice is inefficient, and may lead to inconsistency and incorrect diagnoses and This problem can be solved by using 3D US. Through 3D US, arbitrarily oriented planes within treatment. This problem can be solved by using 3D US. Through 3D US, arbitrarily oriented planes the patient, including planes inaccessible by 2D US, can be viewed. In addition, 3D US also allows within the patient, including planes inaccessible by 2D US, can be viewed. In addition, 3D US also a 3D volume-rendering or surface-rendering view and 3D segmentations of the VOI (Volume of allows a 3D volume‐rendering or surface‐rendering view and 3D segmentations of the VOI Interest). In recent years, many researchers have attempted to reconstruct 3D volumes or surfaces from (Volume of Interest). In recent years, many researchers have attempted to reconstruct 3D volumes conventional 2D ultrasound data. Among the several currently developed 3D ultrasound techniques, or surfaces from conventional 2D ultrasound data. Among the several currently developed 3D freehand 3D ultrasound may easily gain widespread clinical acceptance because it allows physicians ultrasound techniques, freehand 3D ultrasound may easily gain widespread clinical acceptance to move the ultrasound probe unrestrictedly [7–9]. because it allows physicians to move the ultrasound probe unrestrictedly [7–9]. 1.1. Freehand 3D Ultrasound Imaging 1.1. Freehand 3D Ultrasound Imaging Freehand 3D ultrasound imaging adopts conventional ultrasound technology to build up 3D Freehand 3D ultrasound imaging adopts conventional ultrasound technology to build up 3D volumes or surfaces from a number of 2D B-scans acquired in succession (see Figure 1). It consists volumes or surfaces from a number of 2D B‐scans acquired in succession (see Figure 1). It consists of of tracking a standard 2D ultrasound probe using a 3D spatial localizer (magnetic, mechanical, or tracking a standard 2D ultrasound probe using a 3D spatial localizer (magnetic, mechanical, or optical). The localizer is attached to the probe, and can continuously measure the 3D position and optical). The localizer is attached to the probe, and can continuously measure the 3D position and orientation of the probe while the physician moves the probe slowly and steadily over a particular orientation of the probe while the physician moves the probe slowly and steadily over a particular anatomical region of the patient. The measured outputs of the 3D position and orientation are used anatomical region of the patient. The measured outputs of the 3D position and orientation are used for the localization of B-scans in the coordinate system of the spatial localizer. In order to establish for the localization of B‐scans in the coordinate system of the spatial localizer. In order to establish the transformation between the B-scan coordinates and the 3D position and orientation of the probe, the transformation between the B‐scan coordinates and the 3D position and orientation of the probe, a calibration procedure is necessary [10–12]. a calibration procedure is necessary [10–12]. Figure 1. Freehand 3D US system. This is also the setup for the experiments. The spatial localizer is Figure 1. Freehand 3D US system. This is also the setup for the experiments. The spatial localizer is an an electromagnetic tracking device called AURORA from Northern Digital Inc. (Waterloo, ON, electromagnetic tracking device called AURORA from Northern Digital Inc. (Waterloo, ON, Canada). Canada). IGS (image‐guided surgery) software serves as the 3D surface reconstruction and IGS (image-guided surgery) software serves as the 3D surface reconstruction and visualization program. visualization program. In order to obtain a 3D volume or surface model of the VOI (e.g., the lesion to be ablated), the In order to obtain a 3D volume or surface model of the VOI (e.g., the lesion to be ablated), the 2D B-scans of the VOI should be acquired first, and a 3D reconstruction method should be applied to 2D B‐scans of the VOI should be acquired first, and a 3D reconstruction method should be applied these B-scans. to these B‐scans. Appl. Sci. 2016, 6, 114 3 of 22 Freehand 3D ultrasound has a noteworthy characteristic: the recorded B-scans of freehand 3D ultrasound are non-parallel and may intersect mutually, because the movement of the ultrasound probe is free and unrestricted in space. These non-parallel B-scans are different from the parallel slices of CT or MRI imaging. When the B-scans are segmented into contours, the contours are also non-parallel, which increases the difficulty of 3D reconstruction for freehand 3D ultrasound. Presently, the 3D reconstruction methods for freehand 3D ultrasound can be divided into two categories: volume reconstruction and surface reconstruction. 1.2. Volume Reconstruction for Freehand 3D Ultrasound Imaging Volume reconstruction methods interpolate the input data to a regular 3D array (voxel array) to obtain the 3D volume model of the VOI. Please refer to [13,14] for a review of freehand 3D US volume reconstruction algorithms. The different volume reconstruction algorithms have been classified into three groups in [13]: Voxel-Based Methods (VBM), Pixel-Based Methods (PBM), and Function-Based Methods (FBM). More papers about volume reconstruction methods can be found in references [15–18]. Because the volume reconstruction is performed without segmentation of the 2D US images in a prerequisite step, it is not able to display the VOI distinct from the rest of the reconstructed volume. This can only be achieved by surface reconstruction of the VOI. In order to achieve this goal, a 3D segmentation step followed by a triangulation algorithm (e.g., the Marching Cubes algorithm proposed by Lorensen and Cline [19,20]) should be performed. Because the 3D volume model is in a regular 3D voxel array, it is somewhat similar to the voxel lattice in CT or MRI imaging. If the reconstructed 3D volume is dense enough (i.e., no holes or gaps in between), many segmentation methods could be used to segment the VOI from the rest of the volume in the same way as they are used in CT or MRI imaging, such as level sets [21], 3D live-wire [22], image-adaptive radial basis functions [23], Random Walker [24], and Graph Cuts [25]. However, due to the wave interference phenomenon inherent in any coherent imaging process, US will suffer from speckle noise that makes the segmentation of US difficult, whether it is carried out by humans or using automatic segmentation methods. Moreover, if the input data are sparse, all the volume reconstruction methods cannot reconstruct a volume efficiently without introducing geometrical artifacts or distorting the images, which may lead to incorrect segmentation of the VOI by automatic segmentation methods. In clinical applications, manual segmentation is still the only universally reliable method for ultrasound data [9]. 1.3. Surface Reconstruction for Freehand 3D Ultrasound Imaging To avoid a voxel-based intermediate step, i.e., volume reconstruction, we can reconstruct the 3D surface of the VOI directly from contours (cross-sections) segmented from the original 2D B-scans in a prerequisite step. Because the contours are non-parallel, a method that does not assume parallel contours is required. Such reconstruction methods are scarce. Moreover, because manual segmentation is still the only universally reliable method for ultrasound data in clinical applications [9], it is one of the most time-consuming processes during a tumor ablation surgery. It would be preferable to reconstruct surfaces from only a few ultrasound contours, which would be valuable in a clinical setting as it reduces manual work and reduces time spent on manual segmentation. Therefore, only a small number of B-scans are recorded and manually segmented to guarantee short segmentation time in clinical applications. However, a small number of contours provide only sparse input data to surface reconstruction. Thus, a surface reconstruction method that can handle both non-parallel and sparse contours is necessary. These two requirements (i.e., non-parallel and sparse) make the surface reconstruction of freehand 3D US a challenging task. Candidate surface reconstruction methods for freehand 3D US usually fall into two groups: contour-based methods, and point-based methods. Contour-Based methods try to construct a triangle mesh directly from the segmented ultrasound contours and are usually ineffective when applied on non-parallel contours. Point-based methods first convert all contours into a point cloud and then construct triangle meshes from this point cloud. Hence, point-based methods can address non-parallel contours, but they are still ineffective when applied on sparse input data. Appl. Sci. 2016, 6, 114 4 of 22 Cook et al. [26] proposed an algorithm for volume estimation based on polyhedral approximation, and several researchers [27] made use of this method for surface reconstruction. Two contours were sampled as a set of lines joining boundary points, and then the volume was calculated from these points by forming tetrahedra with a common central point on each plane. The surface was formed by triangulating each boundary point in turn between the contours. This would only be accurate for fairly similar contours; the contours are not allowed to intersect each other. Hodges et al. [28] used a CAD package to triangulate between cross sections of saphenous vein phantoms. The cross sections were nearly parallel or nearly circular, and consequently the resulting surface was conical or cylindrical. Liu et al. [29] proposed a method to reconstruct non-parallel contours. It was performed directly in the triangle-mesh domain and required dense contours for input. It resulted in disconnected geometry when the projections of the curves were not overlapping between planes; thus, it is incapable of reconstructing sparse data in our case. Most of the above surface reconstruction methods could not address arbitrarily oriented contours. The orientation of each contour was typically restricted so that the contours were nearly parallel or did not intersect mutually. In fact, methods that can handle mutually intersected contours are uncommon. Although the method by Liu et al. [29] could handle non-parallel contours, it could not tackle sparse data. After the conversion of contours to a 3D point cloud, the orientation of the contours becomes irrelevant and we can disregard the non-parallel contours. Currently, there are many surface reconstruction methods that can be used to reconstruct a surface from point clouds [30–36]. In this paper, we have chosen three well-known and widely accepted methods: the Ball Pivoting Algorithm [37], Power Crust [38], and Poisson reconstruction [39]. These are also representative of three types of point cloud reconstruction methods. The Ball-Pivoting Algorithm (BPA) directly forms a triangle mesh from a given point cloud. The Power Crust method uses the medial axis transform (MAT) of the object to be reconstructed and an inverse MAT to produce the surface of the object. Poisson reconstruction is an implicit-function-based method; it casts the surface reconstruction from oriented points as a spatial Poisson problem. After finding an implicit function by solving the spatial Poisson problem, the reconstructed surface is obtained by extracting an appropriate iso-surface from the implicit function. Deng et al. [40] provided a surface reconstruction method for freehand 3D US, which is an implicit-function-based method similar to Poisson reconstruction. The surface reconstruction method was based on variational interpolation, which was used by Greg Turk for shape transformation [41]; hence, it was named the variational surface reconstruction (VSR) method. The VSR method can effectively address both sparse and mutually intersected contours. Variational interpolation is commonly used in the surface reconstruction of dense point clouds from ranger scanners [42–45], and also in the segmentation of medical data [46], but has not been used in the surface reconstruction of freehand 3D US. Although [46] was focused on the 3D segmentation of medical images, the variational interpolation technique used was the same as that in [40]. Because this paper was not dedicated to ultrasound imaging, the specific characteristics of surface reconstruction in freehand 3D US were not addressed. However, there are three drawbacks to the previous VSR work [40]. First, the experiments were simplistic, as they only used the VSR method to reconstruct a seashell and a plastic apple. Second, the quality of the reconstructed surface was assessed by visual inspection and two simple metrics: area difference and volume difference between the reconstructed surface and the original surface. Third, the VSR method was not compared with other surface reconstruction methods. These three drawbacks make the conclusions regarding the prior VSR work [40] less convincing. 1.4. Contribution and Structure of this Paper The three drawbacks to the work by Deng et al. [40] are ameliorated in this paper. More experiments are conducted, quantitative metrics are introduced to assess the quality of surface Appl. Sci. 2016, 6, 114 5 of 22 Appl. Sci. 2016, 6, 114 5 of 22 The three drawbacks to the work by Deng et al. [40] are ameliorated in this paper. More experiments are conducted, quantitative metrics are introduced to assess the quality of surface reconstruction, and the surface reconstruction quality of the VSR method is compared with three reconstruction, and the surface reconstruction quality of the VSR method is compared with three other other methods. The major contribution of this paper is that it provides many quantitative metrics, methods. The major contribution of this paper is that it provides many quantitative metrics, such as such as the Hausdorff distance, to evaluate the quality of surface reconstruction using four methods, the Hausdorff distance, to evaluate the quality of surface reconstruction using four methods, especially especially in cases where the input data are extremely sparse point clouds from freehand 3D ultrasound. in cases where the input data are extremely sparse point clouds from freehand 3D ultrasound. The rest of the paper is organized as follows: Section 2 describes the framework of the surface The rest of the paper is organized as follows: Section 2 describes the framework of the surface reconstruction for freehand 3D US using the VSR method. Section 3 describes the VSR method and reconstruction for freehand 3D US using the VSR method. Section 3 describes the VSR method and three other methods. Section 4 introduces the quantitative metrics used to assess the quality of three other methods. Section 4 introduces the quantitative metrics used to assess the quality of surface surface reconstruction. Section 5 describes the four experiments conducted. Finally, Section 6 reconstruction. Section 5 describes the four experiments conducted. Finally, Section 6 summarizes the summarizes the conclusions of the paper. conclusions of the paper. 2. Framework of Surface Reconstruction for Freehand 3D US Using VSR 2. Framework of Surface Reconstruction for Freehand 3D US Using VSR The framework of surface reconstruction for freehand 3D ultrasound using the VSR method [40] The framework of surface reconstruction for freehand 3D ultrasound using the VSR method [40] is shown in Figure 2. is shown in Figure 2. First, we perform a spatial transformation to convert the 2D pre-segmented ultrasound contours First, we perform a spatial transformation to convert the 2D pre‐segmented ultrasound into a 3D point cloud. Thus the orientation of the contours will become irrelevant and the non-parallel contours into a 3D point cloud. Thus the orientation of the contours will become irrelevant and the contours can be disregarded. non‐parallel contours can be disregarded. Second, we define constraint conditions for the surface reconstruction. All the boundary points of Second, we define constraint conditions for the surface reconstruction. All the boundary points the contours are defined as on-surface constraints. Moreover, additional off-surface constraints are of the contours are defined as on‐surface constraints. Moreover, additional off‐surface constraints defined to indicate which points should be located inside the surface. are defined to indicate which points should be located inside the surface. Third, the variational interpolation is invoked to find a function that has the minimum bending Third, the variational interpolation is invoked to find a function that has the minimum bending energy and satisfies the defined constraints. The solution to the variational interpolation is a single 3D energy and satisfies the defined constraints. The solution to the variational interpolation is a single implicit function and is at least C -continuous, i.e., smooth. 3D implicit function and is at least C ‐continuous, i.e., smooth. Figure 2. Framework of surface reconstruction of freehand 3D US using VSR. Figure 2. Framework of surface reconstruction of freehand 3D US using VSR. Finally, we perform an iso‐surface extraction step using the Marching Cubes algorithm Finally, we perform an iso-surface extraction step using the Marching Cubes algorithm proposed proposed by Lorensen et al. [19]. The implicit function of VSR is then evaluated to extract the by Lorensen et al. [19]. The implicit function of VSR is then evaluated to extract the zero-valued surface zero‐valued surface as the reconstruction result. as the reconstruction result. To assess the quality of the reconstructed surface, an extra assessment step is performed. To assess the quality of the reconstructed surface, an extra assessment step is performed. Quantitative metrics [47], e.g., the Hausdorff distance, are calculated to compare the quality of Quantitative metrics [47], e.g., the Hausdorff distance, are calculated to compare the quality of reconstructed surface by four point‐based surface reconstruction methods: Ball Pivoting, Power reconstructed surface by four point-based surface reconstruction methods: Ball Pivoting, Power Crust, Poisson, and VSR. Crust, Poisson, and VSR. 3. Four Methods of Surface Reconstruction from Point Clouds 3. Four Methods of Surface Reconstruction from Point Clouds 3.1. VSR 3.1. VSR In the framework described in Section 2, the VSR method consisted of two steps: constraint In the framework described in Section 2, the VSR method consisted of two steps: constraint definition and variational interpolation. definition and variational interpolation. Appl. Sci. 2016, 6, 114 6 of 22 The mathematics about the VSR method was previously explained by [40]. Here, only the step-by-step process of constraint definition and variational interpolation without explanation are described. 3.1.1. Constraint Definition Step 1: Begin with the first contour. Mark it as the current contour. Step 2: For each point c of the current contour, assign a scalar value 0 to c . After the assignment, c i i i on becomes an on-surface constraint c . off Step 3: For each point c of the current contour, calculate its corresponding off-surface constraint c i i according to Equations (1) and (2). off N on c c c n (1) i i i C on on C on on n c c n c c i i1 i 1 i n (2) on on on on C C k n c c n c c k i i1 i 1 i Step 4: When all the points of the current contour are finished, go to the next contour. Mark it as the current contour. Step 5: Loop from step 2 to step 4 until all contours are finished. 3.1.2. Variational Interpolation Step 1: For any pair of constraints c and c , calculate fpc c q k c c k . i j i j i j Step 2: After all pairs of c and c pi 1 . . . n, j 1 . . . nq are calculated, form the coefficient matrix A i j in Equation (3): x z f f . . . f 1 c c c 11 12 1n 1 1 1 x z f f . . . f 1 c c c 21 22 2n 2 2 2 . . . . . . . . . . . . . . . . . . . . . . . . x z f f . . . f 1 c c c n1 n2 nn n n n A (3) 1 1 . . . 1 0 0 0 0 x x x c c . . . c 0 0 0 0 1 2 y y y c c . . . c 0 0 0 0 1 2 z z z c c . . . c 0 0 0 0 1 2 Step 3: Calculate vector B using Equation (4). Each h denotes the corresponding value assigned to the constraint point c . B h h . . . h 0 0 0 0 (4) 1 2 n Step 4: Solve Equation (5). The solution vector v contains the weights d of fpxq and the coefficients of Ppxq. Av B (5) v d d . . . d p p p p (6) 1 2 0 1 2 3 fpxq d fpxc q Ppxq (7) j j j1 Ppxq p p x p y p z (8) 0 1 2 3 Appl. Sci. 2016, 6, 114 7 of 22 3.2. Ball Pivoting The Ball-Pivoting Algorithm (BPA) [37] directly forms a triangle mesh from a given point cloud. Its principle is very simple: starting with a seed triangle, a ball of a user-specified radius pivots around an edge of the seed triangle until it reaches another point. The edge and the new point form another triangle to start with. The process continues until all reachable edges have been finished, and then starts from another seed triangle, until all points have been considered. BPA is intuitive, flexible, efficient, and robust. If given sufficiently dense input data, it is capable of reconstructing a surface homeomorphic to and within a bounded distance from the original manifold. However, BPA cannot necessarily reconstruct a closed surface from a sparse point cloud. 3.3. Power Crust The Power Crust method [38] is divided into two stages: First, the medial axis transform (MAT) of the object to be reconstructed is approximated from the point cloud. Then, an inverse transform is invoked to produce the surface from the MAT. The Power Crust method does not depend on the quality of the input point sample. It can produce watertight surfaces when the input sample is sufficiently dense. When the sample is not sufficiently dense, the Power Crust method may not produce a perfect manifold surface but its output is watertight. However, the Power Crust method may fail if the noise of the input data is above a certain threshold. 3.4. Poisson Reconstruction Poisson reconstruction [39] casts the surface reconstruction from oriented points as a spatial Poisson problem: The computed Laplace operator of the scalar function is equal to the divergence of a vector field V , as follows: D r r r V (9) where the vector field V is defined by the oriented points and is a 3D indicator function. After finding by solving Equation (9), the reconstructed surface is obtained by extracting an appropriate iso-surface from . The time and space complexities of Poisson reconstruction are proportional to the size of the reconstructed model, and it is highly resilient to data noise. One drawback of Poisson reconstruction is that it requires oriented normal vectors at the input points to define the vector field V . 4. Quantitative Metrics for Assessment of Surface Reconstruction In most papers about 3D surface reconstruction, the quality of the reconstructed surface is assessed by visual inspection, rather than quantitative assessment, for two reasons. First, the original 3D surface model of VOI is unavailable before the reconstruction. Second, there is a lack of proper quantitative metrics for assessing surface reconstruction. In this paper, we prepare two types of original surface models to assess the quality of surface reconstruction. The first type is synthetic data (such as an acorn model). The second type is original surface models gained by other image modalities of the same VOI, e.g., we prepare a kidney surface model from dense and parallel contours of CT imaging, and then compare it with its reconstructed ultrasound surface model. For the quantitative metrics, we compute several distances between the original surface and the reconstructed surface. For each vertex of a surface, we compute its distance to the closest point on another surface. From the histogram of these values the following metrics are computed: mean distance (mean) standard deviation from the mean distance (std) Appl. Sci. 2016, 6, 114 8 of 22 mean distance (mean) standard deviation from the mean distance (std) root mean square distance (rms) Appl. Sci. 2016, 6, 114 8 of 22 maximum distance (Hausdorff) medial distance (median). root mean square distance (rms) These metrics measure the shape difference between two triangulated surfaces, and their maximum distance (Hausdorff) meanings are self‐explanatory and intuitive. For example, the Hausdorff distance measures the medial distance (median). maximum difference between two triangulated surfaces, and std and rms detail statistical information about the difference between the two triangulated surfaces. Moreover, we also These metrics measure the shape difference between two triangulated surfaces, and their meanings compute the area difference and volume difference between the original and the reconstructed are self-explanatory and intuitive. For example, the Hausdorff distance measures the maximum surfaces. For these metrics, a greater value indicates a greater difference between the original and difference between two triangulated surfaces, and std and rms detail statistical information about the the reconstructed surface. difference between the two triangulated surfaces. Moreover, we also compute the area difference and volume difference between the original and the reconstructed surfaces. For these metrics, a greater For a set P R and a point x R , let dx,P denote the Euclidean distance of x from value indicates a greater difference between the original and the reconstructed surface. P , i.e., k k For a set P R and a point x P R , let dpx, Pq denote the Euclidean distance of x from P, i.e., dx (,P) inf p x (10) pP dpx, Pq inf tk p x ku (10) pPP The Hausdorff distance [48] between two sets XY , R is given by The Hausdorff distance [48] between two sets X, Y R is given by # + max supdx ( ,Y ), supd (y,X ) (11) xXy Y max sup dpx, Yq, sup dpy, Xq (11) xPX yPY 5. Experiments 5. Experiments Four experiments were conducted to evaluate the quality of surface reconstruction. The first Four experiments were conducted to evaluate the quality of surface reconstruction. The first experiment used the synthetic data as the original surface model. The second experiment used the experiment used the synthetic data as the original surface model. The second experiment used the CT CT data to extract the original surface model. The third experiment used the real ultrasound data of data to extract the original surface model. The third experiment used the real ultrasound data of two two liver tumors, without the original surface model of the tumors. The fourth experiment assessed liver tumors, without the original surface model of the tumors. The fourth experiment assessed the the reproducibility of the VSR method and three other methods using the kidney data in reproducibility of the VSR method and three other methods using the kidney data in Experiment 2 Experiment 2 and the first liver tumor data in Experiment 2. and the first liver tumor data in Experiment 2. For ultrasound image acquisition, we used a ZK‐3000 ultrasound machine with a 3.5 MHz For ultrasound image acquisition, we used a ZK-3000 ultrasound machine with a 3.5 MHz conventional 2D ultrasound probe (Beijing Zhongke‐Tianli Tech. Co., Ltd., Beijing, China). The conventional 2D ultrasound probe (Beijing Zhongke-Tianli Tech. Co., Ltd., Beijing, China). electromagnetic tracking device was the AURORA from Northern Digital Inc. (Waterloo, ON, The electromagnetic tracking device was the AURORA from Northern Digital Inc. (Waterloo, ON, Canada). The digital ultrasound image was acquired through an image‐grabbing card. As noted Canada). The digital ultrasound image was acquired through an image-grabbing card. As noted previously, the position and orientation of the ultrasound probe was also recorded simultaneously previously, the position and orientation of the ultrasound probe was also recorded simultaneously using the tracking device. The experiment system setup is shown in Figure 1. using the tracking device. The experiment system setup is shown in Figure 1. The 3D image reconstruction and visualization were performed using a personal computer The 3D image reconstruction and visualization were performed using a personal computer with with a 2.66 GHz Intel Core2™ quad CPU (Lenovo, Beijing, China). We developed an IGS a 2.66 GHz Intel Core2™ quad CPU (Lenovo, Beijing, China). We developed an IGS (image-guided (image‐guided surgery) software for microwave and radio‐frequency ablation of hepatic tumors [2,3] surgery) software for microwave and radio-frequency ablation of hepatic tumors [2,3] (Figure 3), which (Figure 3), which was used as the surface reconstruction and visualization program. was used as the surface reconstruction and visualization program. Figure 3. Reconstruction and visualization software. Appl. Sci. 2016, 6, 114 9 of 22 Appl. Sci. 2016, 6, 114 9 of 22 Figure 3. Reconstruction and visualization software. 5.1. Experiment 1—Reconstruction of Acorn 5.1. Experiment 1—Reconstruction of Acorn In Experiment 1, we reconstructed an acorn from several non-parallel contours. The original 3D surface model was from the Amira Demos 3.1 CD (version 3.1, FEI Corporate, Hillsboro, OR, In Experiment 1, we reconstructed an acorn from several non‐parallel contours. The original 3D USA, 2005). Between two and seven mutually intersected contours were re-sampled from the original surface model was from the Amira Demos 3.1 CD (version 3.1, FEI Corporate, Hillsboro, OR, USA, surface 2005 model ). Betwe and en wer two eand used seven to r econstr mutually uct inthe tersected acorn contours surface. were The experiment re‐sampled from results the ar or eigin shown al in surface model and were used to reconstruct the acorn surface. The experiment results are shown in Figures 4 and 5 and Table 1. Figures 4 and 5 and Table 1. In Figure 4, each row shows the results of surface reconstruction by four methods: BPA, Power In Figure 4, each row shows the results of surface reconstruction by four methods: BPA, Power Crust, Poisson, and VSR, respectively. From left to right, each row shows the contours used, the Crust, Poisson, and VSR, respectively. From left to right, each row shows the contours used, the original surface, the surface reconstructed by BPA, the surface reconstructed by Power Crust, the original surface, the surface reconstructed by BPA, the surface reconstructed by Power Crust, the surface reconstructed by Poisson, and the surface reconstructed by VSR. surface reconstructed by Poisson, and the surface reconstructed by VSR. Table 1 shows the quantitative differences between the original surface and the reconstructed Table 1 shows the quantitative differences between the original surface and the reconstructed surface using the metrics we described in Section 4. Figure 5 is the same Hausdorff distance shown in surface using the metrics we described in Section 4. Figure 5 is the same Hausdorff distance shown Tablein 1 Tab butle expr 1 but essed expressed graphically graphi.cally. Figure 4. Surface reconstruction of an acorn by four methods. For each row, from left to right: the contours used, the original surface, the surface reconstructed by BPA, the surface reconstructed by Power Crust, the surface reconstructed by Poisson, and the surface reconstructed by VSR. Appl. Sci. 2016, 6, 114 10 of 22 Figure 4. Surface reconstruction of an acorn by four methods. For each row, from left to right: the contours used, the original surface, the surface reconstructed by BPA, the surface reconstructed by Appl. Sci. 2016, 6, 114 10 of 22 Power Crust, the surface reconstructed by Poisson, and the surface reconstructed by VSR. Figure 5. Hausdorff distance between the reconstructed surface and the original surface (ACORN). Figure 5. Hausdorff distance between the reconstructed surface and the original surface (ACORN). Table 1. Difference between the reconstructed surface and the original surface (ACORN). Table 1. Difference between the reconstructed surface and the original surface (ACORN). Number Surface Area Volume Mean Hausdorff Medial Number Surface Area Volume of Cross Reconstruction std rms Difference Difference Mean Hausdorff Medial Distance Distance Distance of Cross Reconstruction std rms Difference Difference Sections Method (%) (%) Distance Distance Distance Sections Method (%) (%) BPA 2.6412 2.19942 3.43703 11.6783 2.16232 −48 −60 PowerCrust 9.50837 6.57579 11.5607 23.4187 9.52517 −55 −81 BPA 2.6412 2.19942 3.43703 11.6783 2.16232 48 60 2 Poisson 2.74878 2.24832 3.55114 11.1559 2.15936 99 −35 PowerCrust 9.50837 6.57579 11.5607 23.4187 9.52517 55 81 VSR 0.710704 0.709221 1.00403 3.40886 0.477692 −12 −4 Poisson 2.74878 2.24832 3.55114 11.1559 2.15936 99 35 BPA 1.65938 1.34633 2.13684 6.20571 1.36133 −27 −35 VSR 0.710704 0.709221 1.00403 3.40886 0.477692 12 4 PowerCrust 1.68578 1.57828 2.30927 6.71023 1.23402 −12 −32 3 BPA 1.65938 1.34633 2.13684 6.20571 1.36133 27 35 Poisson 2.13187 1.64198 2.69089 7.98914 1.73299 73 −77 PowerCrust 1.68578 1.57828 2.30927 6.71023 1.23402 12 32 VSR 0.926695 1.01773 1.37641 4.35121 0.537287 −19 −17 Poisson 2.13187 1.64198 2.69089 7.98914 1.73299 73 77 BPA 1.41748 1.34832 1.95631 6.20571 1.02828 −22 −24 VSR 0.926695 1.01773 1.37641 4.35121 0.537287 19 17 PowerCrust 1.51101 1.66481 2.24826 6.71748 0.778636 −17 −26 4 BPA 1.41748 1.34832 1.95631 6.20571 1.02828 22 24 Poisson 2.59704 2.0811 3.32798 9.51066 2.05565 45 −6 PowerCrust 1.51101 1.66481 2.24826 6.71748 0.778636 17 26 VSR 0.876429 1.12105 1.42297 4.59235 0.350886 −18 −15 Poisson 2.59704 2.0811 3.32798 9.51066 2.05565 45 6 BPA 0.888351 0.79243 1.19042 4.05483 0.658741 −11 −4 VSR 0.876429 1.12105 1.42297 4.59235 0.350886 18 15 PowerCrust 0.639664 0.675445 0.930259 3.68787 0.394742 2 −9 5 Poisson 1.96806 1.64165 2.56285 7.70361 1.4991 67 30 BPA 0.888351 0.79243 1.19042 4.05483 0.658741 11 4 VSR 0.411081 0.476312 0.629169 3.02812 0.227498 −8 −1 PowerCrust 0.639664 0.675445 0.930259 3.68787 0.394742 2 9 BPA 0.865362 0.801298 1.17937 4.05483 0.608282 −10 −5 Poisson 1.96806 1.64165 2.56285 7.70361 1.4991 67 30 PowerCrust 0.656822 0.805945 1.03968 4.19336 0.321317 1 −13 VSR 0.411081 0.476312 0.629169 3.02812 0.227498 8 1 6 Poisson 1.91001 1.51941 2.44062 7.74737 1.53498 79 43 BPA 0.865362 0.801298 1.17937 4.05483 0.608282 10 5 VSR 0.456283 0.632145 0.779608 3.41585 0.191891 −9 −6 PowerCrust 0.656822 0.805945 1.03968 4.19336 0.321317 1 13 BPA 0.876748 0.854095 1.22399 4.54538 0.561955 −11 −2 Poisson 1.91001 1.51941 2.44062 7.74737 1.53498 79 43 PowerCrust 0.498742 0.591533 0.773721 4.13959 0.285986 5 −11 7 VSR 0.456283 0.632145 0.779608 3.41585 0.191891 9 6 Poisson 1.83339 1.63492 2.45646 7.77791 1.34775 98 26 BPA 0.876748 0.854095 1.22399 4.54538 0.561955 11 2 VSR 0.274174 0.315076 0.417661 2.12882 0.154248 −4 −2 PowerCrust 0.498742 0.591533 0.773721 4.13959 0.285986 5 11 Poisson 1.83339 1.63492 2.45646 7.77791 1.34775 98 26 All the contours used in this experiment are non‐parallel to each other and intersected mutually. VSR 0.274174 0.315076 0.417661 2.12882 0.154248 4 2 VSR’s performance is excellent when the input data are sparse, such that two contours can lead to a close approximation to the original surface. This indicates that the VSR method can process both All the contours used in this experiment are non-parallel to each other and intersected mutually. arbitrarily oriented and sparse contours. VSR’s performance is excellent when the input data are sparse, such that two contours can lead to We can conclude that: a close approximation to the original surface. This indicates that the VSR method can process both In all cases, the surface reconstructed by the VSR method best visually resembles the original arbitrarily oriented and sparse contours. surface compared with the other methods. Most metrics show that the VSR‐reconstructed We can conclude that: surface shows fewer differences from the original surface compared with the other methods. In all cases, the surface reconstructed by the VSR method best visually resembles the original The only exception is observed in volume difference for four contours, where the Poisson method surface compared with the other methods. Most metrics show that the VSR-reconstructed surface produces a volume that is more similar to the original surface than that of the VSR method. shows fewer differences from the original surface compared with the other methods. The only exception is observed in volume difference for four contours, where the Poisson method produces a volume that is more similar to the original surface than that of the VSR method. Appl. Sci. 2016, 6, 114 11 of 22 Appl. Sci. 2016, 6, 114 11 of 22 For the other three methods, the quality of the reconstructed surfaces drops dramatically as the For the other three methods, the quality of the reconstructed surfaces drops dramatically as the number of contours decreases (as seen for two or three contours in Figures 4 and 5). However, number of contours decreases (as seen for two or three contours in Figures 4 and 5). However, as the number of contours decreases, both the quality and the Hausdorff distance for the VSR as the number of contours decreases, both the quality and the Hausdorff distance for the VSR method show little variation. Even with only two contours, the VSR surface reconstruction method show little variation. Even with only two contours, the VSR surface reconstruction closely closely approximates the original surface. approximates the original surface. Visually and quantitatively, the Poisson method performs the worst, except when V reconstructed isually and quantitatively with two co , the ntoPoi urssson . Alth method ough P po erforms isson and the worst, VSR are except impl when icit‐fu rncti econstr on‐ba ucted sed with methods, two contours. the Poisson Although methodPoisson fails to rec andonstruct VSR ar eaimplicit-function-based satisfactory surface in al methods, l cases. the Poisson method Although fails the to Power reconstr Crust uct a method satisfactory claims surface to be in ca allpa cases. ble of producing watertight surfaces, it fails to do so when it is performed on two contours. This makes it the worst method for surface Although the Power Crust method claims to be capable of producing watertight surfaces, it fails reconstruction with only two contours. to do so when it is performed on two contours. This makes it the worst method for surface As an intuitive and efficient method, BPA succeeds in constructing a triangle mesh for all cases. reconstruction with only two contours. When the input data are dense (i.e., seven contours), it can produce a satisfactory surface of the As an intuitive and efficient method, BPA succeeds in constructing a triangle mesh for all cases. acorn. However, its performance decreases dramatically as the number of contours decreases. When the input data are dense (i.e., seven contours), it can produce a satisfactory surface of the acorn. However, its performance decreases dramatically as the number of contours decreases. 5.2. Experiment 2—Reconstruction of Human Kidney 5.2. Experiment 2—Reconstruction of Human Kidney In Experiment 2, we reconstructed a human kidney from non‐parallel contours re‐sampled fromIn an Experiment original sur 2,fac we e mo reconstr del ofucted the kidney. a human The kidney original from sunon-paralle rface modell of contours the kidney re-sampled was created from an original surface model of the kidney. The original surface model of the kidney was created from its from its CT counterpart as follows: First, CT images of a volunteer were acquired in our cooperative CT hospital. counterpart Second, as follows: the kidne First, y was CT se images gmented of a slice volunteer by slice wer efro acquir m the ed CT in our ima cooperati ge data,ve usin hospital. g the Second, the kidney was segmented slice by slice from the CT image data, using the semi-automatic semi‐automatic segmentation function in Amira software (version 3.1, FEI Corporate, Hillsboro, OR, segmentation USA, 2005) [49 function ], which in wa Amira s based softwar on Act ei(version ve Contour 3.1, Model. FEI Corporate, Third, the Hillsbor Amira o, so OR, ftware USA, was 2005) used [49 to ], which was based on Active Contour Model. Third, the Amira software was used to reconstruct the reconstruct the surface of the segmented kidney to produce the original surface model of this surface experiment. of the segmented kidney to produce the original surface model of this experiment. The differences between the four reconstructed surfaces and the original surface are illustrated in The differences between the four reconstructed surfaces and the original surface are illustrated Figur in Figes ur6esand 6 and 7 and 7 and Table Tab 2l.e 2. The results of this experiment conformed to those of Experiment 1. When the input data were The results of this experiment conformed to those of Experiment 1. When the input data were dense dense enough enough ((i.e i.e.,., 14 14 or or 12 12 contours), contours), all all four four methods methods obtained obtained satisfactory satisfactory surfaces. surfaces. All All the the quantitative differences between the reconstructed surfaces and original surface were small for all four quantitative differences between the reconstructed surfaces and original surface were small for all methods. four methods. However However , even, with even awith dense a point dense cloud, point the clou VSR d, the method VSR method still remained still remained the best whether the best evaluated visually or quantitatively. Although the Hausdorff distance of the Power Crust method was whether evaluated visually or quantitatively. Although the Hausdorff distance of the Power Crust slightly method less wasthan slightly that of less the tha VSR n tha method t of the when VSR the method number when of used the contours number was of used greater contour thans 10, waits behaved poorly when the number of contours used decreased. greater than 10, it behaved poorly when the number of contours used decreased. For For the theBP BPA, A, Power Power Cr ust, Crust, and and Poisson Poisso methods, n methods, the quality the qu ofal the ity reconstr of the ucted reconstructed surface dr su opped rface dramatically when the number of contours used decreased. However, the performance of the VSR dropped dramatically when the number of contours used decreased. However, the performance of method the VSR was method still excellent was still ,eboth xcellent visually , both and visua quantitatively lly and quan.ti Even tatively. when Even six, when four, and six, four, three and contours three were used, the VSR reconstruction was a close approximation to the original surface, while the other contours were used, the VSR reconstruction was a close approximation to the original surface, while thr theee other methods three failed methods to cr fa eate iledcomplete to create or compl closed ete surfaces. or closed surfaces. The BPA method was the worst when reconstructions were produced with more than six contours, The BPA method was the worst when reconstructions were produced with more than six as contours, seen in as Table seen 2. in When Tablthe e 2.input When data thewer input e extr dat emely a were sparse extreme (i.e.,ly six, spar four se ,(and i.e., six thr,ee fou contours), r, and three the Poisson method performed worst. contours), the Poisson method performed worst. Figure 6. Cont. Appl. Sci. 2016, 6, 114 12 of 22 Appl. Sci. 2016, 6, 114 12 of 22 Figure Figure 6. 6. Surface Surface rreconstruction econstruction of of aa human human kidney kidney by by four four methods. methods. For For ea each ch row, row, fr from om left left to to right: right: the the contours contours used, used,the theoriginal original surface, surface, the the surface surface reconstr reconstructed ucted by by BP BPA A, the , the surface surface reconstr reconstructed ucted by Power by Power Crust, Crus the t, the surface surface reconstr reconstructed ucted by Poisson, by Poisson, and and the th surface e surface reconstr recon ucted structed by VSR. by VSR. Appl. Sci. 2016, 6, 114 13 of 22 Appl. Sci. 2016, 6, 114 13 of 22 Figure 7. Hausdorff distance between the reconstructed surface and the original surface Figure 7. Hausdorff distance between the reconstructed surface and the original surface (HUMAN KIDNEY). (HUMAN KIDNEY). Table 2. Difference between the reconstructed surface and the original surface (HUMAN KIDNEY). Table 2. Difference between the reconstructed surface and the original surface (HUMAN KIDNEY). Number of Surface Area Volume Mean Hausdorff Medial Cross Reconstruction std rms Difference Difference Distance Distance Distance Number Surface Area Volume Sections Method (%) (%) Mean Hausdorff Medial of Cross Reconstruction std rms Difference Difference BPA 5.40364 4.37961 6.9542 18.9081 4.27574 −48.10 −86.11 Distance Distance Distance Sections Method (%) (%) Power Crust 2.89954 2.96081 4.14305 14.0522 1.87489 −12.56 −26.25 3 BPA 5.40364 4.37961 6.9542 18.9081 4.27574 48.10 86.11 Poisson 5.95107 5.20793 7.90635 23.448 4.37537 110.84 −61.14 Power Crust 2.89954 2.96081 4.14305 14.0522 1.87489 12.56 26.25 VSR 2.57361 2.91418 3.88682 14.539 1.37939 −7.49 −11 Poisson 5.95107 5.20793 7.90635 23.448 4.37537 110.84 61.14 BPA 3.45712 3.51744 4.93067 17.8971 2.35204 −26.14 −54.27 VSR 2.57361 2.91418 3.88682 14.539 1.37939 7.49 11 Power Crust 2.54163 2.37519 3.47789 9.96287 1.76471 −9.91 −17.72 4 BPA 3.45712 3.51744 4.93067 17.8971 2.35204 26.14 54.27 Poisson 6.06799 4.58929 7.60663 24.0824 5.02371 26.70 30.68 Power Crust 2.54163 2.37519 3.47789 9.96287 1.76471 9.91 17.72 VSR 1.7113 1.8289 2.504 8.05269 0.974841 −6.73 −6.71 Poisson 6.06799 4.58929 7.60663 24.0824 5.02371 26.70 30.68 BPA 2.30932 2.67167 3.53038 13.9329 1.20211 −14.57 −34.66 VSR 1.7113 1.8289 2.504 8.05269 0.974841 6.73 6.71 Power Crust 10.4462 12.8452 16.5517 42.5017 3.72379 −39.17 −52.38 6 BPA 2.30932 2.67167 3.53038 13.9329 1.20211 14.57 34.66 Poisson 2.80168 2.66679 3.86704 12.9421 2.04536 4.74 8.25 Power Crust 10.4462 12.8452 16.5517 42.5017 3.72379 39.17 52.38 VSR 1.08773 1.32213 1.71156 7.36376 0.530696 −3.39 −3.05 Poisson 2.80168 2.66679 3.86704 12.9421 2.04536 4.74 8.25 BPA 1.73677 2.15556 2.76733 11.2383 0.88484 −7.93 −39.12 VSR 1.08773 1.32213 1.71156 7.36376 0.530696 3.39 3.05 Power Crust 1.32654 1.87959 2.29979 10.6975 0.663859 −5.69 −8.62 8 BPA 1.73677 2.15556 2.76733 11.2383 0.88484 7.93 39.12 Poisson 1.54515 1.47949 2.13873 9.09251 1.11002 0.37 3.42 Power Crust 1.32654 1.87959 2.29979 10.6975 0.663859 5.69 8.62 VSR 0.862839 1.15736 1.44313 7.16898 0.33444 −2.70 −2.65 Poisson 1.54515 1.47949 2.13873 9.09251 1.11002 0.37 3.42 BPA 1.49347 2.22578 2.67947 13.3502 0.690223 −3.80 −7.80 VSR 0.862839 1.15736 1.44313 7.16898 0.33444 2.70 2.65 Power Crust 0.789766 0.929686 1.2195 5.72804 0.446035 −3.83 −4.75 10 BPA 1.49347 2.22578 2.67947 13.3502 0.690223 3.80 7.80 Poisson 1.33454 1.48037 1.99255 9.15823 0.822045 0.78 3.51 Power Crust 0.789766 0.929686 1.2195 5.72804 0.446035 3.83 4.75 10 VSR 0.675437 0.675437 1.0794 1.27285 7.16558 0.25 −1.56 Poisson 1.33454 1.48037 1.99255 9.15823 0.822045 0.78 3.51 BPA 1.29316 1.86346 2.26743 12.2463 0.553634 −2.97 −5.67 VSR 0.675437 0.675437 1.0794 1.27285 7.16558 0.25 1.56 Power Crust 0.682934 0.819252 1.06625 4.56298 0.375812 −3.02 −5.14 11 BPA 1.29316 1.86346 2.26743 12.2463 0.553634 2.97 5.67 Poisson 1.25069 1.12206 1.67987 6.08752 0.949839 −1.88 1.43 Power Crust 0.682934 0.819252 1.06625 4.56298 0.375812 3.02 5.14 11 VSR 0.654974 1.08766 1.26917 7.19773 0.213162 2.03 −3.20 Poisson 1.25069 1.12206 1.67987 6.08752 0.949839 1.88 1.43 BPA 1.25462 1.86828 2.24967 12.2463 0.503546 −2.84 −5.09 VSR 0.654974 1.08766 1.26917 7.19773 0.213162 2.03 3.20 Power Crust 0.639903 0.820045 1.03984 4.96253 0.343724 −2.80 −4.90 12 BPA 1.25462 1.86828 2.24967 12.2463 0.503546 2.84 5.09 Poisson 1.03552 1.01728 1.45125 6.10591 0.725931 0.12 3.43 Power Crust 0.639903 0.820045 1.03984 4.96253 0.343724 2.80 4.90 VSR 0.538622 0.905817 1.05347 6.72458 0.181576 1.79 −2.23 Poisson 1.03552 1.01728 1.45125 6.10591 0.725931 0.12 3.43 BPA 1.29556 2.19755 2.55007 13.8442 0.457283 −2.85 −3.76 VSR 0.538622 0.905817 1.05347 6.72458 0.181576 1.79 2.23 Power Crust 0.5029 0.591473 0.776141 3.61966 0.289134 −2.52 −3.58 BPA 1.29556 2.19755 2.55007 13.8442 0.457283 2.85 3.76 14 Poisson 0.883901 1.0258 1.35369 6.70781 0.554642 −1.83 1.42 Power Crust 0.5029 0.591473 0.776141 3.61966 0.289134 2.52 3.58 Poisson 0.883901 1.0258 1.35369 6.70781 0.554642 1.83 1.42 VSR 0.473663 0.758462 0.893891 6.0587 0.16908 1.91 −1.43 VSR 0.473663 0.758462 0.893891 6.0587 0.16908 1.91 1.43 5.3. Experiment 3—Reconstruction of Liver Tumor 5.3. Experiment 3—Reconstruction of Liver Tumor In Experiment 3, we reconstructed two human liver tumors from real ultrasound data. In this In Experiment 3, we reconstructed two human liver tumors from real ultrasound data. In this case, we did not have the original surface model for comparison. We could not perform a quantitative case, we did not have the original surface model for comparison. We could not perform a quantitative assessment of the reconstructed surface, and thus we inspected the visual differences instead. assessment of the reconstructed surface, and thus we inspected the visual differences instead. The visual differences between the reconstructed surfaces and the original surface are The visual differences between the reconstructed surfaces and the original surface are illustrated illustrated in Figures 8 and 9. As we did not have the original surface, the reconstructions are as in Figures 8 and 9. As we did not have the original surface, the reconstructions are as follows, from follows, from left to right: the contours used, the surface reconstructed by BPA, the surface Appl. Sci. 2016, 6, 114 14 of 22 Appl. Sci. 2016, 6, 114 14 of 22 Appl. Sci. 2016, 6, 114 14 of 22 reconstructed by Power Crust, the surface reconstructed by Poisson, and the surface reconstructed left to right: the contours used, the surface reconstructed by BPA, the surface reconstructed by Power reconstructed by Power Crust, the surface reconstructed by Poisson, and the surface reconstructed by VSR. Cr by ust, VSR. the surface reconstructed by Poisson, and the surface reconstructed by VSR. Figure 8. Surface reconstruction of one human liver tumor using four methods. For each row, from Figure 8. Surface reconstruction of one human liver tumor using four methods. For each row, from Figure 8. Surface reconstruction of one human liver tumor using four methods. For each row, from left to right: the contours used, the surface reconstructed by BPA, the surface reconstructed by Power left to right: the contours used, the surface reconstructed by BPA, the surface reconstructed by Power left to right: the contours used, the surface reconstructed by BPA, the surface reconstructed by Power Crust, the surface reconstructed by Poisson, and the surface reconstructed by VSR. Crust, the surface reconstructed by Poisson, and the surface reconstructed by VSR. Crust, the surface reconstructed by Poisson, and the surface reconstructed by VSR. Figure 9. Surface reconstruction of another human liver tumor by four methods. For each row, from Figure 9. Surface reconstruction of another human liver tumor by four methods. For each row, from Figure left to ri 9.gSurface ht: the contou reconstr rs used uction , the of surface another reconstructed human liver by tumor BPAby , the four surface methods. recons For tructed each by row Power , from left to right: the contours used, the surface reconstructed by BPA, the surface reconstructed by Power left Crus tot,right: the surf theace contours reconstructed used, the by surface Poisson, reconstr and the ucted surface by reconstructed BPA, the surface by VSR. reconstr ucted by Power Crust, the surface reconstructed by Poisson, and the surface reconstructed by VSR. Crust, the surface reconstructed by Poisson, and the surface reconstructed by VSR. Figure 8 also indicated that VSR was the best compared to three other methods, similar to the Figure 8 also indicated that VSR was the best compared to three other methods, similar to the results of Experiment 1 and Experiment 2. When two contours were used, the VSR method still Figure 8 also indicated that VSR was the best compared to three other methods, similar to the results of Experiment 1 and Experiment 2. When two contours were used, the VSR method still reconstructed a satisfactory surface while the other three methods failed to create closed surfaces. results of Experiment 1 and Experiment 2. When two contours were used, the VSR method still reconstructed a satisfactory surface while the other three methods failed to create closed surfaces. In Figure 9, only two contours were used to reconstruct the surface of another hepatic tumor. reconstructed a satisfactory surface while the other three methods failed to create closed surfaces. In Figure 9, only two contours were used to reconstruct the surface of another hepatic tumor. Visually, this figure also indicated that VSR was the best method compared with the other methods, In Figure 9, only two contours were used to reconstruct the surface of another hepatic tumor. Visually, this figure also indicated that VSR was the best method compared with the other methods, consistent with the results of the previous experiments. Visually, this figure also indicated that VSR was the best method compared with the other methods, consistent with the results of the previous experiments. consistent with the results of the previous experiments. 5.4. Experiment 4—Assessing the Reproducibility of Four Methods 5.4. Experiment 4—Assessing the Reproducibility of Four Methods Appl. Sci. 2016, 6, 114 15 of 22 Appl. Sci. 2016, 6, 114 15 of 22 5.4. Experiment 4—Assessing the Reproducibility of Four Methods Appl. Sci. 2016, 6, 114 15 of 22 Finally, we investigated the reproducibility of the four methods. Four different users drew the Finally, we investigated the reproducibility of the four methods. Four different users drew the Finally, we investigated the reproducibility of the four methods. Four different users drew the same number but different contours from the kidney in Experiment 2 and the first hepatic tumor in same number but different contours from the kidney in Experiment 2 and the first hepatic tumor in same number but different contours from the kidney in Experiment 2 and the first hepatic tumor in Experiment 3, and surfaces were reconstructed from these contours and the quality of the reconstructed Experiment 3, and surfaces were reconstructed from these contours and the quality of the Experiment 3, and surfaces were reconstructed from these contours and the quality of the surfaces were evaluated. reconstructed surfaces were evaluated. reconstructed surfaces were evaluated. In the case of the kidney in Experiment 2, each user drew four different contours from the original In the case of the kidney in Experiment 2, each user drew four different contours from the In the case of the kidney in Experiment 2, each user drew four different contours from the surface model of the kidney. The differences between the four reconstructed surfaces and the original original surface model of the kidney. The differences between the four reconstructed surfaces and original surface model of the kidney. The differences between the four reconstructed surfaces and surface are illustrated in Figures 10 and 11 and Table 3. the original surface are illustrated in Figures 10 and 11 and Table 3. the original surface are illustrated in Figures 10 and 11 and Table 3. (a) (a) (b) (b) (c) (c) (d) (d) Figure 10. Surface reconstruction of a human kidney by four methods. The number of contours are Figure 10. Surface reconstruction of a human kidney by four methods. The number of contours are Figure 10. Surface reconstruction of a human kidney by four methods. The number of contours are the same in different rows, but the contours were drawn by different users: (a) User 1; (b) User 2; the same in different rows, but the contours were drawn by different users: (a) User 1; (b) User 2; the same in different rows, but the contours were drawn by different users: (a) User 1; (b) User 2; (c) User 3; (d) User 4. For each row, from left to right: the contours used, the original surface, the (c) User 3; (d) User 4. For each row, from left to right: the contours used, the original surface, the (c) User 3; (d) User 4. For each row, from left to right: the contours used, the original surface, the surface reconstructed by BPA, the surface reconstructed by Power Crust, the surface reconstructed surface reconstructed by BPA, the surface reconstructed by Power Crust, the surface reconstructed by surface reconstructed by BPA, the surface reconstructed by Power Crust, the surface reconstructed by Poisson, and the surface reconstructed by VSR. Poisson, and the surface reconstructed by VSR. by Poisson, and the surface reconstructed by VSR. Figure 11. Hausdorff distances between the reconstructed surfaces and the original surface Figure 11. Hausdorff distances between the reconstructed surfaces and the original surface Figure 11. Hausdorff distances between the reconstructed surfaces and the original surface (HUMAN (HUMAN KIDNEY). The contours were drawn by four different users. (HUMAN KIDNEY). The contours were drawn by four different users. KIDNEY). The contours were drawn by four different users. Table 3. Difference between the reconstructed surface and the original surface (HUMAN KIDNEY; Table 3. Difference between the reconstructed surface and the original surface (HUMAN KIDNEY; the contours were drawn by different users). the contours were drawn by different users). Surface Mean Hausdorff Medial Area Volume User Surface Mean std rms Hausdorff Medial Area Volume User Reconstruction Distance std rms Distance Distance Difference Difference Reconstruction Distance Distance Distance Difference Difference Appl. Sci. 2016, 6, 114 16 of 22 Table 3. Difference between the reconstructed surface and the original surface (HUMAN KIDNEY; the contours were drawn by different users). Surface Area Volume Mean Hausdorff Medial User Reconstruction std rms Difference Difference Distance Distance Distance Method (%) (%) BPA 3.45712 3.51744 4.93067 17.8971 2.35204 26.14 54.27 Power Crust 2.54163 2.37519 3.47789 9.96287 1.76471 9.91 17.72 User 1 Poisson 6.06799 4.58929 7.60663 24.0824 5.02371 26.70 30.68 VSR 1.7113 1.8289 2.504 8.05269 0.974841 6.73 6.71 BPA 4.48905 5.10272 6.79433 22.0837 2.09925 39.04 95.96 Power Crust 3.2693 3.37085 4.69462 14.7158 1.9469 13.94 29.77 User 2 Poisson 3.61952 3.42344 4.98086 15.594 2.48503 37.66 26.67 Appl. Sci. 2016, 6, 114 16 of 22 VSR 1.99204 2.78775 3.42519 14.6415 0.946489 7.72 7.85 BP Method A 4.40708 4.73455 6.4665 21.115 2.68107 (%) 50.46 (%) 71.71 Power Crust 2.72485 2.63747 3.79131 11.5233 1.96596 10.82 23.71 BPA 3.45712 3.51744 4.93067 17.8971 2.35204 −26.14 −54.27 User 3 Poisson 3.3612 2.8271 4.39114 13.3416 2.59396 47.83 22.53 Power Crust 2.54163 2.37519 3.47789 9.96287 1.76471 −9.91 −17.72 User 1 VSR 1.34403 1.71604 2.17905 9.10527 0.616535 5.71 5.89 Poisson 6.06799 4.58929 7.60663 24.0824 5.02371 26.70 30.68 VSR 1.7113 1.8289 2.504 8.05269 0.974841 −6.73 −6.71 BPA 4.91483 4.93414 6.96252 23.0959 3.22832 49.22 46.76 BPA 4.48905 5.10272 6.79433 22.0837 2.09925 −39.04 −95.96 Power Crust 2.8246 2.99256 4.11397 12.8419 1.85294 13.05 20.85 User 4 Power Crust 3.2693 3.37085 4.69462 14.7158 1.9469 −13.94 −29.77 Poisson 6.73954 5.39803 8.63311 24.1895 5.40374 80.34 26.53 User 2 Poisson 3.61952 3.42344 4.98086 15.594 2.48503 37.66 26.67 VSR 2.06118 2.59825 3.3155 12.6043 0.963157 10.40 11 VSR 1.99204 2.78775 3.42519 14.6415 0.946489 −7.72 −7.85 BPA 4.40708 4.73455 6.4665 21.115 2.68107 −50.46 −71.71 Power Crust 2.72485 2.63747 3.79131 11.5233 1.96596 −10.82 −23.71 User 3 We can conclude that: Poisson 3.3612 2.8271 4.39114 13.3416 2.59396 47.83 22.53 VSR 1.34403 1.71604 2.17905 9.10527 0.616535 −5.71 −5.89 The contours used greatly affect the quality of the reconstructed surfaces by all four methods. BPA 4.91483 4.93414 6.96252 23.0959 3.22832 −49.22 −46.76 Power Crust 2.8246 2.99256 4.11397 12.8419 1.85294 −13.05 −20.85 Visually, the contours drawn by User 3 produce the best approximation to the original surface, and User 4 Poisson 6.73954 5.39803 8.63311 24.1895 5.40374 80.34 −26.53 the contours drawnVSR by User2.40611 pr8oduce 2.5982the 5 poor 3.3155 est. 12. Figur 6043 e 11 0.9631 confirms 57 −10.this 40 conclusion −11 graphically. This result indicates that the sparse input contours should cover the key contours of the original We can conclude that: surface to produce a close approximation to it. The contours used greatly affect the quality of the reconstructed surfaces by all four methods. In all cases, the surface reconstructed by the VSR method best visually resembles the original Visually, the contours drawn by User 3 produce the best approximation to the original surface, surface compar and the ed contours with the drawn other by methods. User 4 produce All the metrics poorest. show Figurethat 11 confirms the VSR-r this econstr conclusioucted n surface graphically. This result indicates that the sparse input contours should cover the key contours shows fewer differences from the original surface compared with the other methods, which of the original surface to produce a close approximation to it. indicates that the reproducibility of the VSR method is the best of the four methods. In all cases, the surface reconstructed by the VSR method best visually resembles the original surface compared with the other methods. All metrics show that the VSR‐reconstructed surface In the case of the first hepatic tumor in Experiment 3, each user manually segmented four different shows fewer differences from the original surface compared with the other methods, which contours from the freehand ultrasound data of the tumor. The reconstructed surfaces are illustrated indicates that the reproducibility of the VSR method is the best of the four methods. in Figure 12. As we did not have the original surface of the hepatic tumor, in order to evaluate the In the case of the first hepatic tumor in Experiment 3, each user manually segmented four reproducibility different of each contours method, from the the freeh reconstr and ultrasoun uctedd surface data of the from tumor. the The contours reconstructed drawn surfac byes user are 1 using that illustrated in Figure 12. As we did not have the original surface of the hepatic tumor, in order to method was temporarily taken as the original surface to evaluate the quality of surface reconstruction evaluate the reproducibility of each method, the reconstructed surface from the contours drawn by from the contours drawn by Users 2, 3, and 4, e.g., the quantitative assessment of the quality of the user 1 using that method was temporarily taken as the original surface to evaluate the quality of reconstructed surfaces from the contours drawn by Users 2, 3, and 4 using the VSR method were surface reconstruction from the contours drawn by Users 2, 3, and 4, e.g., the quantitative assessment of the quality of the reconstructed surfaces from the contours drawn by Users 2, 3, and 4 performed based on the reconstruction from the contours drawn by User 1 using the VSR method. using the VSR method were performed based on the reconstruction from the contours drawn by The quantitative differences are shown in Figure 13 and Table 4. User 1 using the VSR method. The quantitative differences are shown in Figure 13 and Table 4. (a) (b) (c) Figure 12. Cont. Appl. Sci. 2016, 6, 114 17 of 22 Appl. Sci. 2016, 6, 114 17 of 22 Appl. Sci. 2016, 6, 114 17 of 22 (d) (d) Figure 12. Surface reconstruction of a hepatic tumor by four methods. The numbers of contours are Figure 12. Surface reconstruction of a hepatic tumor by four methods. The numbers of contours are Figure 12. Surface reconstruction of a hepatic tumor by four methods. The numbers of contours are the same in different rows, but the contours were drawn by different users: (a) User 1; (b) User 2; the same in different rows, but the contours were drawn by different users: (a) User 1; (b) User 2; the same in (dif c) Us fer er ent 3; (d)r User ows, 4. but For each the row, contours from left wer to right: e drawn the contours by dif used fer , the ent surface users: reco(nstructed a) User by 1; (b) User 2; BPA, the surface reconstructed by Power Crust, the surface reconstructed by Poisson, and the surface (c) User 3; (d) User 4. For each row, from left to right: the contours used, the surface reconstructed by (c) User 3; (d) User 4. For each row, from left to right: the contours used, the surface reconstructed by reconstructed by VSR. BPA, the surface reconstructed by Power Crust, the surface reconstructed by Poisson, and the surface BPA, the surface reconstructed by Power Crust, the surface reconstructed by Poisson, and the surface reconstructed reconstructed by by VSR. VSR. Figure 13. Hausdorff distances between the reconstructed surfaces from the contours drawn by Users 2, 3, and 4 and the reconstructed surface from the contours drawn by User 1 (LIVER TUMOR). The contours were drawn by four different users. Table 4. Difference between the reconstructed surfaces from the contours drawn by Users 2, 3, and 4 and the reconstructed surface from the contours drawn by User 1 (LIVER TUMOR). Surface Area Volume Figure 13. Hausdorff distances between the reconstructed surfaces from the contours drawn by Mean Hausdorff Medial Figure 13. Hausdorff distances between the reconstructed surfaces from the contours drawn by Users User Reconstruction std rms Difference Difference Distance Distance Distance Method (%) (%) Users 2, 3, and 4 and the reconstructed surface from the contours drawn by User 1 (LIVER TUMOR). 2, 3, and 4 and the reconstructed surface from the contours drawn by User 1 (LIVER TUMOR). The BPA 0.606759 0.709767 0.932945 2.85121 0.304864 −2.72 119.87 The contours were drawn by four different users. contours were drawn Power Crust by four 0.56dif 5943 fer ent 0.58 users. 6972 0.814723 2.37748 0.383221 11.60 −25.79 User 2 Poisson 1.04422 0.845751 1.34295 3.33231 0.737621 44.81 13.57 VSR 0.199237 0.253187 0.321874 1.31926 0.0916052 19.80 ‐9 Table 4. Difference BP between A 0. the 1790 38reconstructe 0.311993 d su 0.35rface 9301 s fro 1.m 5673 the 2 conto0u rs drawn 6.94 by Users −94. 2, 72 3, and 4 Table 4. Difference between the reconstructed surfaces from the contours drawn by Users 2, 3, and 4 Power Crust 0.196016 0.31971 0.374599 1.2938 0.0102139 16.88 −25.98 User 3 and the reconstructed surface from the contours drawn by User 1 (LIVER TUMOR). and the reconstructed surface from the contours drawn by User 1 (LIVER TUMOR). Poisson 0.66439 0.61721 0.906199 3.37821 0.492152 90.58 23.22 VSR 0.0703013 0.123795 0.142199 0.718006 0.019699 23.09 −6 Surface Area Volume BPA 0.542069 1.06084 1.18987 5.79188 0.153002 1.63 345.95 Mean Hausdorff Medial Surface Area Volume Power Crust 1.68e + 009 3.75e + 009 4.10e + 009 1e + 010 13.5631 −100 −100 User Reconstruction std rms Difference Difference User 4 Mean Hausdorff Medial Distance Distance Distance Poisson 1.85742 1.97787 2.71109 8.48791 0.996965 91.69 1140.05 User Reconstruction std rms Difference Difference Method (%) (%) Distance Distance Distance VSR 0.166469 0.265851 0.313325 1.31816 0.0547988 18.85 −11 Method (%) (%) BPA 0.606759 0.709767 0.932945 2.85121 0.304864 −2.72 119.87 BPA 0.606759 0.709767 0.932945 2.85121 0.304864 2.72 119.87 Power Crust 0.565943 0.586972 0.814723 2.37748 0.383221 11.60 −25.79 The results of this experiment conform to those of the previous reproducibility experiment, so User 2 Power Crust 0.565943 0.586972 0.814723 2.37748 0.383221 11.60 25.79 Poisson 1.04422 0.845751 1.34295 3.33231 0.737621 44.81 13.57 we can conclude that: User 2 Poisson 1.04422 0.845751 1.34295 3.33231 0.737621 44.81 13.57 VSR 0.199237 0.253187 0.321874 1.31926 0.0916052 19.80 ‐9 The contours used greatly affect the quality of the reconstructed surfaces by all four methods. VSR 0.199237 0.253187 0.321874 1.31926 0.0916052 19.80 -9 BPA 0.179038 0.311993 0.359301 1.56732 0 6.94 −94.72 Visually, the contours drawn by User 3 produce the best approximation to the reconstructed Power Crust 0.196016 0.31971 0.374599 1.2938 0.0102139 16.88 −25.98 BPA 0.179038 0.311993 0.359301 1.56732 0 6.94 94.72 User 3 surface from the contours drawn by User 1 using the BPA, Power Crust, and VSR methods. Poisson Power Crust 0.66 0.196016 439 0.61 0.31971 721 0.90 0.374599 6199 3.37 1.2938 821 0.0102139 0.492152 16.88 90.58 25.98 23.22 User 3 The contours drawn by User 4 produce the poorest approximation, especially when the Power VS Poisson R 0.0703 0.66439 013 0.12 0.61721 3795 0.14 0.906199 2199 0.71 3.37821 8006 0.492152 0.019699 90.58 23.09 23.22−6 Crust method is used; it fails to produce a surface. Figure 13 confirms this conclusion VSR 0.0703013 0.123795 0.142199 0.718006 0.019699 23.09 6 BPA 0.542069 1.06084 1.18987 5.79188 0.153002 1.63 345.95 graphically. It should be noted that the Hausdorff distance of the Power Crust method is Power Crust 1.68e + 009 3.75e + 009 4.10e + 009 1e + 010 13.5631 −100 −100 BPA 0.542069 1.06084 1.18987 5.79188 0.153002 1.63 345.95 User 4 missing in Figure 13 since its value is much greater than that of the other methods. Poisson 1.85742 1.97787 2.71109 8.48791 0.996965 91.69 1140.05 Power Crust 1.68e + 009 3.75e + 009 4.10e + 009 1e + 010 13.5631 100 100 User 4 VSR 0.166469 0.265851 0.313325 1.31816 0.0547988 18.85 −11 Poisson 1.85742 1.97787 2.71109 8.48791 0.996965 91.69 1140.05 VSR 0.166469 0.265851 0.313325 1.31816 0.0547988 18.85 11 The results of this experiment conform to those of the previous reproducibility experiment, so we can conclude that: The results of this experiment conform to those of the previous reproducibility experiment, so we can conclude that: The contours used greatly affect the quality of the reconstructed surfaces by all four methods. Visually, the contours drawn by User 3 produce the best approximation to the reconstructed The contours used greatly affect the quality of the reconstructed surfaces by all four methods. surface from the contours drawn by User 1 using the BPA, Power Crust, and VSR methods. Visually, the contours drawn by User 3 produce the best approximation to the reconstructed The contours drawn by User 4 produce the poorest approximation, especially when the Power surface from the contours drawn by User 1 using the BPA, Power Crust, and VSR methods. Crust method is used; it fails to produce a surface. Figure 13 confirms this conclusion The contours drawn by User 4 produce the poorest approximation, especially when the Power graphically. It should be noted that the Hausdorff distance of the Power Crust method is Crust method is used; it fails to produce a surface. Figure 13 confirms this conclusion graphically. missing in Figure 13 since its value is much greater than that of the other methods. Appl. Sci. 2016, 6, 114 18 of 22 It should be noted that the Hausdorff distance of the Power Crust method is missing in Figure 13 since its value is much greater than that of the other methods. In all cases, the surface reconstructed by the VSR method best visually resembles the original surface compared with the other methods. All metrics show that the VSR-reconstructed surfaces show fewer differences from the surface reconstructed from the contours drawn by User 1 compared with the other methods, which indicates that the reproducibility of the VSR method is the best of four methods. 6. Discussion The main goal of this paper is to evaluate the quality of surface reconstruction from point clouds using four point-based methods, especially when the inputs are extremely sparse point clouds from freehand 3D ultrasound. The BPA method is intuitive and robust. If given sufficiently dense input data, it is capable of reconstructing a satisfactory surface from the original manifold. However, the BPA method failed to produce a closed surface from sparse point clouds, as observed in Experiment 2 and Experiment 3. In Experiment 2, the BPA method was the worst method when more than six contours were used. The Power Crust method produced watertight surfaces when the input sample was sufficiently dense. However, when the input data were extremely sparse, as in Experiment 2 and Experiment 3 (i.e. two contours), it failed completely. The Poisson method also created satisfactory surfaces when given sufficiently dense input data. However, the Poisson method required oriented normal vectors at the input points to define the vector field. When the input data were extremely sparse, the vector field could not be correctly defined, and the Poisson method failed to reconstruct the correct surfaces. Hence, the Poisson method was the worst of the four methods in all experiments when reconstructed from extremely sparse contours. In all cases, the surface reconstructed by the VSR method showed the greatest visual resemblance to the original surface, compared with the other three methods. In most cases, the metrics also indicated that the quantitative differences between the VSR reconstructed surface and the original surface were the smallest. Even given extremely sparse input data, the VSR method could reconstruct a close approximation to the original surface. Therefore, the strengths of the VSR method were as follows: (1) the VSR method could reconstruct satisfactory surfaces from sparse data; (2) the surfaces reconstructed by the VSR method were smooth; (3) although the VSR and Poisson methods were both implicit-function-based methods, the VSR method did not need to compute the oriented normal vectors at the input point clouds to define a vector field. However, the second strength of VSR was also a weakness. Because the reconstructed surfaces by VSR method were smooth, the reconstruction of objects with high curvature might fail if the surface was sampled too sparsely, i.e., with not enough contours. The VSR method has another limitation concerning contradictory inputs as shown by Heckel et al. [46], i.e., contours that define different surfaces, although they should be located on the same surface. This issue was demonstrated in Figure 14. In Figure 14, we used three same contours of the kidney as in the first row of Figure 6, except that the horizontal contour was edited manually to form contradictory contours. In row (a) of Figure 14, we could see that the original horizontal contour (yellow) should intersect with the vertical contour (blue), but the edited version of the horizontal contour (red) did not intersect with the vertical contour (blue). The quantitative assessments of the quality of surface reconstruction from the contradictory contours are shown in Table 5. Appl. Sci. 2016, 6, 114 19 of 22 Appl. Sci. 2016, 6, 114 19 of 22 (a) (b) (c) (d) Figure 14. Surface reconstruction of contradictory contours of the kidney. (a) Contradictory contour: Figure 14. Surface reconstruction of contradictory contours of the kidney. (a) Contradictory contour: the original horizontal contour (yellow) should intersect with the vertical contour (blue), but the the original horizontal contour (yellow) should intersect with the vertical contour (blue), but the edited edited version of the horizontal contour (red) did not intersect with the vertical contour (blue); (b) version of the horizontal contour (red) did not intersect with the vertical contour (blue); (b) surface surface reconstruction from the original contours; (c) surface reconstruction from the contradictory reconstruction from the original contours; (c) surface reconstruction from the contradictory contours. contours. For row b and c, from left to right: the original surface of the kidney, the surface For row b and c, from left to right: the original surface of the kidney, the surface reconstructed by reconstructed by BPA, the surface reconstructed by Power Crust, the surface reconstructed by BPA, the surface reconstructed by Power Crust, the surface reconstructed by Poisson, and the surface Poisson, and the surface reconstructed by VSR. (d) Detail of the surface reconstruction near the intersection of the contradictory contours. From left to right: the original surface of the kidney and reconstructed by VSR. (d) Detail of the surface reconstruction near the intersection of the contradictory the contradictory contours, the surface reconstructed by BPA, the surface reconstructed by Power contours. From left to right: the original surface of the kidney and the contradictory contours, the Crust, the surface reconstructed by Poisson, and the surface reconstructed by VSR. surface reconstructed by BPA, the surface reconstructed by Power Crust, the surface reconstructed by Poisson, and the surface reconstructed by VSR. Table 5. Difference between the reconstructed surfaces from contradictory contours and the original surface (HUMAN KIDNEY). Table 5. Difference between the reconstructed surfaces from contradictory contours and the original Surface Area Volume surface (HUMAN KIDNEY). Mean Hausdorff Medial Case Reconstruction std rms Difference Difference Distance Distance Distance Method (%) (%) Surface BPA 5.4036 4.3796 6.9542 18.9081 4.2757 −48.10 Area −86. V11 olume Mean Hausdorff Medial Case Reconstruction std rms Difference Difference Non‐contradictory Power Crust 2.8995 2.9608 4.1430 14.0522 1.8748 −12.56 −26.25 Distance Distance Distance contours Method Poisson 5.9510 5.2079 7.9063 23.448 4.3753 110.84 (%) −61.14 (%) VSR 2.5736 2.9142 3.8868 14.539 1.3793 −7.49 −11 BPA 5.4036 4.3796 6.9542 18.9081 4.2757 48.10 86.11 BPA 6.1428 6.0208 8.5992 29.3689 3.9117 −54.21 −89.94 Power Crust 2.8995 2.9608 4.1430 14.0522 1.8748 12.56 26.25 Non-contradictory Contradictory Power Crust 3.7181 3.3860 5.0277 15.5735 2.7963 −17.79 −35.19 Poisson 5.9510 5.2079 7.9063 23.448 4.3753 110.84 61.14 contours contours Poisson 7.8823 6.5397 10.239 26.9386 5.9497 40.99 −62.38 VSR 2.5736 2.9142 3.8868 14.539 1.3793 7.49 11 VSR 2.6500 2.8585 3.8968 14.6123 1.6135 −8.74 −13 BPA 6.1428 6.0208 8.5992 29.3689 3.9117 54.21 89.94 Power Crust 3.7181 3.3860 5.0277 15.5735 2.7963 17.79 35.19 Contradictory Poisson 7.8823 6.5397 10.239 26.9386 5.9497 40.99 62.38 contours VSR 2.6500 2.8585 3.8968 14.6123 1.6135 8.74 13 Appl. Sci. 2016, 6, 114 20 of 22 As seen in row (d) of Figure 14, most methods had some problems dealing with the contradictory contours. The surface reconstructed by the BPA method lay on the outer red contour, which led to a bump on the reconstructed surface, while the original surface of the kidney should lie on the inner blue contour. There was a concave near the contradictory contours on the surface reconstructed by the Power Crust method, which indicated that it could not produce a correct surface from contradictory contours. The surface reconstructed by the VSR method lay between the outer read contour and the inner blue contour, which was a better result than the BPA and Power Crust methods. The surface reconstructed by the Poisson method lay on the inner blue contour correctly, indicating that it was the best method for dealing with contradictory contours. Because contradictory contours usually come from inconsistencies in the input data, they can be compensated for by an additional preprocessing step that solves such inconsistencies before the surface reconstruction. They can also be slightly compensated for when using approximation instead of interpolation. The results of Figure 14 may suggest a new way of combining the VSR and Poisson methods to solve the problem. We will investigate this in future work. Moreover, the VSR method was computationally expensive and needed considerable memory. Computation times were slow for a large number of constraints. This problem could be solved by reducing the constraints, using GPU-based parallel computing strategies, and using a fast computing algorithm such as the fast multipole method, which was suggested by Carr et al. [50]. 7. Conclusions Three experiments were conducted to compare the performance of the VSR method with three other methods, i.e., BPA, Power Crust, and Poisson, when reconstructing a surface from sparse freehand 3D ultrasound contours. The research results show that VSR performed the best of the tested methods. Compared with the three other methods, the VSR method showed the best visual resemblance and the least quantitative differences to the original surface, especially when the data were very sparse. The VSR method closely approximated the original surface even with two contours, whereas the other algorithms failed to do so. The reconstructed surface produced by VSR appeared smooth and natural. The experiments’ results show that the reproducibility of the VSR method is also the best of the tested methods. Acknowledgments: This work was supported by the Key Projects in the National Science & Technology Pillar Program under the grant 2013BAI01B01. 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