Interactive GPU active contours for segmenting inhomogeneous objects
 1.4k Downloads
Abstract
We present a segmentation software package primarily targeting medical and biological applications, with a high level of visual feedback and several usability enhancements over existing packages. Specifically, we provide a substantially faster GPU implementation of the local Gaussian distribution fitting energy model, which can segment inhomogeneous objects with poorly defined boundaries as often encountered in biomedical images. We also provide interactive brushes to guide the segmentation process in a semiautomated framework. The speed of our implementation allows us to visualize the active surface in real time with a builtin ray tracer, where users may halt evolution at any time step to correct implausible segmentation by painting new blocking regions or new seeds. Quantitative and qualitative validation is presented, demonstrating the practical efficacy of our interactive elements for a variety of realworld datasets.
Keywords
Segmentation Active contours Level set methods GPU Medical applications Biological applications1 Introduction
 Biosciences:

Cellular, developmental and cancer biology.

Plant biology, including plant–pathogen interactions.

Animal biology, including virus–host interactions and bacterial infections.

Microbiology, including food safety.

Neuroscience, including connectome projects and developmental neuroscience.

 Medicine:

Automated differential diagnosis.
 Diagnostic measurements, shape, and volume, of:

Macular holes in retinal degeneration.

Aneurysms, clotting and infarction.

Tumors, neoplasia and dermatological moles.

MRI segmentation in dementia and Alzheimer’s.

 Computerassisted surgery:

Presurgical planning and surgery simulation.

Guided surgical navigation.


The oldest and most widely cited segmentation approaches are active contours [20]; these are variational frameworks which allow users to define an initial open or closed curve that deforms so as to minimize a energy functional, outlining or surrounding the object of interest. While active contours have been realized as fully automatic approaches without initial contours [25], their original foundation as an assisted approach is still important today as it allows users, such as clinicians, to extract precise measurements from specific objects of interest within a complex image. However, such interactivity relies on realtime visual feedback; therefore, they must also be computationally efficient.
Graphics processing units (GPUs) provide energyefficient parallel computing and enable realtime interactive segmentation for larger 2D or 3D datasets [10, 43], but existing GPU segmentation methods currently rely on simple segmentation criteria restricting their usage and applications. The popular local Gaussian distribution fitting (LGDF) energy model [47] is much more powerful and able to segment a wider variety of general objects. However, it requires several intermediate processing steps that must be implemented sequentially, making it challenging to efficiently implement on graphics hardware. The current implementation of the LGDF energy model can segment small 2D images (\(99\times 120\) in 27.37 s), but requires several hours of processing for larger 2D or 3D images [47]. For a 3D image of size \(256\times 256\times 160\), this would take 6.6 hours if the implementation were available for 3D, preventing usage in many practical applications.
1.1 Contributions
In our approach, we: (1) significantly increase the performance of the LGDF energy model through an optimized GPU implementation, handling much larger 2D images and even 3D images at interactive performance, (2) introduce a novel set of interactive brush functions that are integrated into the GPU kernels such as to modify and constrain the evolving level set in real time, (3) provide a ray tracer to view the segmentation results at each time step, and (4) expose a simpler and more intuitive parameter space to the user, with suggested values and ranges. The combination of these four enhancements greatly improves the practicality of what is already considered a stateoftheart level set method of particular relevance to the biomedical image processing communities. Our software is shown to be stable with respect to its input parameters and robust to noise through a large experiment on synthetic data and is further evaluated through segmenting a wide variety of realworld images, such as those shown in Fig. 1.
2 Related work
The field of active contours first gained mainstream adoption with the ‘active snakes’ model published by [20]. This seminal work proposes iterative evolution of an initial spline curve, with the evolution being governed by the minimization of an energy functional, the local minima of which correspond to curves that fit along prominent edges in the image. Level set methods (core theory explained in [30]) model contours implicitly as the zerocrossing of a scalar field. Originally they were proposed in [31] to model the evolution of interregion boundaries in physical simulations. Malladi et al. [26] applied level sets to active contours, with the evolution of the contour being governed by its local mean curvature and the intensity gradient magnitude of the image, in such a way that local curvature is reduced and the motion of the contour stops as it approaches an edge. In [5], the authors develop a level setbased active contour framework in which the energy functional is based on the Mumford–Shah model, rather than image edges, which in practice are often faint, blurred, or broken. The Mumford–Shah energy model [28] is minimized by an optimal partition of an image into piecewise smooth segments, and highquality implementations exist on the GPU [33]. The global optimum can be found using a primaldual algorithm [4] resulting in a cartoonlike rendering of the original image. Local solutions, such as with a trustregion approach [14], have applications in interactive segmentation, where local edits need to be made frequently.
Deep convolutional neural networks are the state of the art in image segmentation, where millions of parameters of deeply layered convolutions are learned using backpropagation [22]. These models are capable of learning abstract features in the data; however, their current reliance on such large datasets makes them unusable for a number of applications.
The influential public datasets with groundtruth segmentations (such as BSDS, MSRC, iCoseg, FlickrMFC, SegTrack) include RGB videos or 2D images such as cars, chairs, and people. Of these, the interactive approaches take as input a set of scribbles where objects follow similar color distributions [53]. Graph cut segmentation is popular in this field, where Grady [15] and Vineet and Narayanan [46] propose GPU implementations. For interactive segmentation in the biosciences, we find the main limitations being (1) the initialization of the foreground–background scribbles in 3D datasets such as networks and (2) the opaque intermediate steps of the cutting algorithm making it difficult to obtain a high level of visual feedback. While popular and easy to validate, these approaches address a different problem to grayscale 3D segmentation as with imaging modalities (such as CT, PET, SPECT, MRI, fMRI, ultrasound, optical imaging and microscopy) in the biosciences [10]. There is still a need for benchmark medical datasets with welldefined interactive performance evaluation [51].
Accelerating image segmentation with GPUs is a large research field with several comprehensive surveys [10, 34, 41, 43]. The survey by [10] covers a broad range of algorithms and different imaging modalities, whereas Smistad et al. [43] focuses more on GPU segmentation with a detailed discussion on the current GPU architecture.
The GPU level set methods in the literature focus on limiting the active computational domain to a small region near the zerocrossing of the level set function, such as the traditional narrow band algorithm [1]. More recent extensions classify the active region using simple operations on the spatial and temporal derivatives of the level set function [36] and then discard unimportant regions through parallel stream compaction. While limiting the active computational domain produces excellent performance with lower memory usage, the current implementations all use simple speed functions that attract the level set to make it grow and/or shrink within a fixed intensity range [18, 23, 36]. In contrast, the LGDF model proposed by [47] is able to segment much more challenging images, in which objects exhibit intensity inhomogeneity or even have the same mean intensity as their background, being distinguished only by intensity variance. However, to date the only existing implementation runs on the CPU, likely due to the sequential dependency of convolutions in the intermediate steps. Further, the LGDF model is derived from [5] who introduce \(C^\infty \) regularization of the Heaviside and Dirac functions which are nonzero everywhere, unlike the \(C^2\) regularized Heaviside (proposed in [52]) which is nonzero only in the vicinity of the contour. \(C^\infty \) regularization restrains the algorithm from converging on local minima, but precludes traditional narrow band or sparse field algorithms because it requires the level set to update at all points on each time step.
GPU active contour methods parallelize the calculation of the energy forces described in the original snakes paper [20]. Traditional methods rely on simple intensity gradients and are prone to converging on local minima; however, [49] introduced a diffusion of the gradient vectors called gradient vector flow (GVF) to address this problem. [16] were one of the first GPU active contour implementations using GVF, and more recent optimizations in OpenCL exploit cached texture memory which has spatial locality in multiple dimensions [42]. The active contour can also be approximated by a surface mesh, such as in [39] who use Laplacian smoothing on local neighborhoods in conjunction with driving mesh vertices with gradient and intensity forces. However, these approaches still rely on the image gradient being a reliable indication of object boundaries, which is not the case in many realworld images [5].
Ever since the original snakes paper, active contours have gained popularity through being able to interactively edit the contour, or set up constraints to guide its motion [20]. Regionbased active contour methods provide the option to initialize with a simple primitive shape, or sketch a starting region [7]. The more advanced approach by [27] introduces nonEuclidean radialbasis functions, which are weighted by the image features and blended to form an implicit function whose sign can be fixed at userdefined control points. The tool by [50] provides an interactive interface with geodesic active contours [3] and region competition [55]. Region competition favors a welldefined intensity range, whereas the geodesic approach is better suited for images with clear edges; by combining both approaches, [50] can segment a broad range of images, yet it requires significant tuning and can still fail in complex images with neither a welldefined intensity range nor clear edges.
There are several GPU approaches that produce segmentation without relying on initialization of a seed region [25]. Clustering methods join regions of a highdimensional feature space [13], and superpixel approaches [35] form clusters that are deliberately oversegmented into more manageable regions. These approaches are good at simplifying complex images, yet they do not capture specific objects. In contrast, active shape and appearance methods fit a model to the data based on prior knowledge; however, this inherently makes assumptions of the overall shape of the objects and fails when these assumptions are not met.
3 Method
The LGDF model, originally proposed in [47], builds on existing active contour literature by introducing a new energy functional based on the local Gaussian distributions of image intensity. This functional drives a variational level set approach which is able to segment objects whose intensity mean and variance are inhomogeneous. Rather than creating segments whose intensity is as uniform as possible, this algorithm allows slow changes in intensity across an object, penalizing only sudden changes within it, without relying on a gradient based edge detector [5].
Due to the smooth form of the \(C^{\infty }\) regularized Heaviside (Eq. 7), \(\delta (\phi ) = H'(\phi )\) is nonzero everywhere. This allows \(\phi \) some freedom to change at any point in the image, not just in a narrow band around the contour. This helps prevent convergence on local energy minima [5].
3.1 GPU implementation
The goal of the implementation is to iteratively solve Eq. 8 for \(\phi (\mathbf {x}, t)\) and visualize the results at each iteration. This is done by discretizing \(\phi \) with respect to time and applying numerical integration: starting with \(\phi (\mathbf {x}, t=0)\) (which is specified by the user), an update loop computes \(\phi (\mathbf {x}, t+\varDelta t)\) by computing \(\frac{\partial \phi }{\partial t}\) according to Eq. 8 and assuming this quantity stays constant during the short time step \(\varDelta t\). Existing GPU level set methods implement their update rule inside a single kernel function; however, \(E^{\text {LGDF}}\) is more challenging as relies on intermediate stages with neighborhood operations, such as convolutions and derivatives, whose sequential dependencies must be considered such as to avoid race conditions.
3.2 GPU architecture
The six required Gaussian convolutions require a large number of buffer reads. However, an ndimensional Gaussian filter can be separated into the matrix product of n vectors allowing us to convolve with n 1D filters instead of one very large ndimensional filter. This reduces \(l^2\) texture samples to 2l in 2D or \(l^3\) texture samples to 3l in 3D, for a truncated Gaussian kernel of length l. Therefore, our overall algorithmic complexity is \(O(n \cdot l)\) for an input of size n.
The buffer reads for the horizontal Gaussian pass are coalesced, but for the vertical and depth passes the reads are not coalesced and therefore very slow. This could be alleviated by transposing the image between convolutions, making the buffer reads coalesced for vertical and depth passes. However, transposing the image three times per convolution is slow, even when this is optimized by using local/shared memory. In our architecture, we instead make use of texture memory, which preserves spatial locality among neighboring pixels in all three dimensions, making access time for all three passes comparable to coalesced buffer reads. This allows us to skip the transpositions altogether and convolve up to four images at once in the available texture memory channels, yielding faster overall performance than local/shared memory approaches.
The three Gaussian convolutions of the image and Heaviside (GIH, GH, \(GI^2H\), Fig. 2) are the result of neighborhood operations, but are not dependent on each other. This is also the case with the three Gaussian convolutions \(GE_0\), \(GE_1\), \(GE_2\). We therefore create kernels shown in Fig. 3 to perform each set of three Gaussian convolutions simultaneously, and two more kernels to prepare for them (called ‘Prep Conv 1’ to compute H, IH, \(I^2H\), and ‘Prep Conv 2’ to compute \(E_0\), \(E_1\), \(E_2\)). The curvature field \(\kappa \) (Eq. 9) requires all three (two in 2D) gradient components to be first stored in texture memory in order to avoid race conditions, since all differential operations are computed by central finite differences, a neighborhood operation. This is why we compute \(\kappa \) early on and pass it through the Gaussian convolution kernels in the conveniently available w channel of the texture buffer; computing \(\kappa \) immediately before ‘Update \(\phi \)’ would require an extra texture buffer since there is only one unused channel at that point. After updating, we force the partial derivatives of \(\phi \) to be zero at their corresponding image boundaries (in the ‘Neumann/Copy’ kernel) to prevent numerical instability and copy the result back into buffer A for the next iteration.
3.3 Interactive brushes
There are many applications in the biosciences, computer vision, medical, and pattern recognition communities where guidance by human experts is required [7, 20, 27, 48, 50]. The current interactive GPU level set methods, such as [36], provide interfaces to (1) initialize \(\phi \) inside/outside the object, (2) dynamically adjust parameters, and in some cases (3) allow \(\phi \) to be edited (a union operator on new objects/regions, followed by rerunning of the algorithm); however, it is difficult to refine evolution such as to prevent contour leaking or constrain the evolution. The graphcuts and radialbasis function approaches [15, 27] allow users to sketch lines or define control points which are tagged to both the desired object and the undesired regions, but we find the process difficult to refine where the segmented boundary lies somewhere between the input locations, where there may not be discernible image intensity features (see Fig. 4 topleft and in the accompanying video).
To address these issues, we follow the strategies outlined in the survey [29] with similar functions to the modeling/graphics literature [12]; however, we closely integrate brush functions with our segmentation kernels with the goal of editing and constraining \(\phi \) during the iterative evolution process itself. Specifically, we provide functions to initialize, append, erase, and constrain (locally stop evolution of \(\phi \)) after each iteration of the update step (Eq. 8), and visualize the results after each iteration. Note that for simplicity we define our functions with circular (2D) or spherical (3D) regions, but there is nothing to prevent implementing more bespoke functions, such as surface pulling [12].
In Fig. 4, we illustrate two simple usecases of our interactive brushes. In the top row, the user paints using the ‘barrier’ brush to cover the full image region, shown in blue. This is followed by the ‘erase’ brush (Eq. 17), to cut a permissible region in which a new seed region is placed (Eq. 16), which evolves to segment the macular hole without leaking into the opening. (We show this in 3D in the accompanying video.) Similarly, in the lower row, the vessels are segmented without leaking into the heart (see also Table 5 2b–c).
3.4 Realtime rendering
To render the zerocrossing of the level set function \(\phi \) in 3D, we launch a render kernel after the Neumann/Copy step in the update loop (Fig. 3). We send a camera matrix to initialize each pixel with a ray origin \(\mathbf {o}\) and direction unit vector \(\hat{d}\). We parameterize the ray’s position by \(\mathbf {r} = \mathbf {o} + \hat{d} s\) and, assuming \(\phi \) to be the signed distance to the zerocrossing, advance the ray in steps by \(s_{i+1} = s_i + \phi (\mathbf {r})\). However, \(\phi \) is not a perfect signed distance function; therefore, we must divide our step size by the maximum derivative of \(\phi \); this value is not known precisely, but in practice we find we can obtain sufficiently small visual artifacts at good performance by choosing a constant step size \(\varDelta s = 0.3 \phi (\mathbf {r})\). Further, given that \(\phi \) is not defined outside of the image boundaries, we initially advance \(s_0\) to the start of the image axisaligned bounding box (where the \(s_0\) is calculated using an analytical raybox intersection function [21]). To increase visual quality, we implement 3D ambient occlusion and softshadows by marching the ray in the directional of the normal and light source once it has hit a surface [11].
4 Results and validation
In this section, we provide quantitative results validating our algorithm’s performance, parameter insensitivity, and robustness to noise. We also provide qualitative results to justify the utility of our interactive brushes and assess the segmentation of realworld images from various domains.
Comparing the Jaccard index for our GPU implementation with the CPU implementation
Image  Jaccard index 

Synthetic objects 2D  1 
Tumor (small) 2D  1 
Tumor (large) 2D  0.981 
Macular hole 3D  0.990 
Brain 3D  0.984 
Tumor 3D  0.993 
Segmentation without interactive brushes attained from a single circular seed region inside the object
These results show the GPU to be nearidentical to the CPU implementation; we find small discrepancies at the boundary at subvoxel precision caused by different implementations of lowlevel math library functions and different (mathematically equivalent) algebra in the intermediate steps (Eqs. 11 and 12).
4.1 Noise and parameter insensitivity
The results in Fig. 6 show that the method can segment severely noisy images, corrupted with a PSNR of about \(10^{1.05}\), under a constant parameter assignment. While the results in Fig. 6 show the method is more robust to Gaussian noise than speckle noise, it is important to understand that this is only within the parameters chosen; improvements can generally be made by adjusting the parameters for individual scenarios. In addition to Gaussian, salt and pepper, and speckle noise, we implemented a multifrequency ‘cloud’ noise at a target PSNR, which simulates intensity inhomogeneity. In Fig. 6, it appears that the cloud noise improves under a PSNR of \(10^{0.81}\); however, this is caused by the cloudlike objects inside the synthetic object being captured. In such cases, we can still segment the underlying object, but only through decreasing \(\sigma \) or using the interactive brushes.
Our proposed parameters for controlling the method. All images in this paper are generated using these three parameters within their suggested range and constants \(\varDelta t=0.1\) and \(\mu =1.0\)
Description  Symbol  Suggested range  Default 

Capture range  \(\sigma \)  \([1.01 , \; 10]\)  3 
Smoothing weight  \(\nu \)  \([10 , \; 90]\)  50 
Shrink or grow  \(\lambda \)  \([\,0.1 , \; 0.1]\)  0.05 
We call \(\sigma \) a ‘capture range’ parameter as it describes the range from which a pixel’s energy may be affected by the contour (see Eqs. 2–4) and therefore determines the capture range. The parameter \(\nu \) penalizes the length of the contour (Eqs. 6 and 8); a larger \(\nu \) value results in a smoother contour which is less likely to burst through small gaps or capture small/sharp features. Traditionally many active contour methods have been designed to grow or shrink until they reach the object boundary and then stop; the parameter \(\lambda \) optionally enables this behavior by weighting the image terms \(e_1\) and \(e_2\) by \(\lambda _1\) and \(\lambda _2\), respectively (Eq. 8), biasing the contour toward shrinking or growing. By adjusting these parameters in real time, inexperienced users quickly learn to intuitively manipulate them in combination with our interactive brushes. In most cases, we set \(\lambda =0.05\) to prefer contour growth and adjust only \(\sigma \) and \(\nu \).
Following challenging scenarios are quickly and easily segmented with our interactive brushes
4.2 Segmenting realworld images
Segmentation results of multiple objects displayed in different colors. 1a shows a segmented image of HaCaT human cell culture cells using confocal microscopy, 1b shows the interdigitation of segmented layers of eisosome proteins from cryoEM tomography data [19], 1c shows a malaria sporozoite [38]. Row 2 shows medical CT scans of the abdomen, body, and thorax [37]. 3a shows an MRI of a cerebral aneurysm and 3b an XA angiogram [37]. 3c shows the structure of the Sec13/31 COPII coat cage from cryoEM data [44]. Row 4 shows the herpes simplex virus capsid [6], phi procapsid [40], and the mumps virus [9], all from cryoEM data. Row 5 shows applications outside of biology and medicine: 5a is a CT scan of an engine block [2], 5b sintered alumina [38], and 5c shows a selection of objects from a CT scan of a backpack [2]
Many of the segmentations (Table 5 1a, 3ab, and 5bc) are not possible with the current GPU level set segmentation approaches, which use simple speed functions to attract and/or shrink the contour within a fixed intensity range [18, 23, 36]. For example, when painting an initial seed region inside a vessel network with intensity inhomogeneity, the active contour will not grow along the vessel. In contrast, the adopted LGDF energy model allows us to paint a simple initial sphere anywhere on the object which then spreads through the network of vessels. In cases where the contour evolution misses a vessel or oversegments part of the object, evolution is temporarily halted (\(\varDelta t=0\)), local amendments are made, and then evolution is resumed (\(\varDelta t=0.1\)). By making local adjustments with a high level of visual feedback, we can spot such issues and make amendments immediately.
4.3 Performance and memory usage
We also investigated other optimizations given that the Gaussian convolution is the primary bottleneck of our approach. We implemented Gaussian convolution in the Fourier domain using MATLAB GPU arrays. While Fourier convolution allows for a lower order of growth, the benefits are outweighed by the large constant factor due to the algorithm complexity; this takes 400ms per frame using a GTX TITAN X, which is off the scale in Fig. 8.
The mean time of 100 iterations with our C++ OpenCL implementation is evaluated across different hardware and compared to our GPU Fourier implementation and the original MATLAB version on the CPU (which is vectorized and calls code written in C for the Gaussian convolution). These results are shown in Fig. 9.
In Fig. 9, our algorithm substantially outperforms the original implementation in all images. Given that we process the entire dataset with compact kernels and separable convolutions, we can fully utilize highend GPU hardware to obtain a substantial speedup of up to three orders of magnitude from the original version, and 1–2 orders of magnitude from our GPU Fourier convolution version. This means that segmentations which previously took over an hour can now be achieved in a few seconds, without any trade in quality.
5 Discussion
The primary limitation of our implementation is that we require storing the full dataset at the original resolution in GPU texture memory, as the \(C^\infty \) Heaviside and Dirac functions are nonzero everywhere to reduce convergence on local minima [5]. This also limits the algorithm’s speed. In future work, we will investigate dynamically adjusting the resolution away from the zerocrossing of the \(C^\infty \) Heaviside, to reduce the memory requirements and improve performance, and evaluate the impact of this approach on segmentation quality.
While there are some excellent publicly available datasets for interactive segmentation of realworld 2D color images and videos [53], the problem of segmenting everyday objects in color photographs, e.g., with a graph cut approach on distributions of color information, is fundamentally different to segmenting a tissue or organ. In the latter case, the challenge is more often due to intensity inhomogeneity or poorly defined edges, rather than complex backgrounds or discontinuities within the object. As with [51], we would like to see benchmark 3D biological and medical datasets for evaluating interactive performance.
6 Conclusion
In conclusion, we have shown that sophisticated level set segmentation energy models, with sequential dependencies among intermediate processing steps, can be implemented efficiently on the GPU through careful structuring of the GPU kernels within the constraints of the GPU memory architecture. While active contours are used in unsupervised algorithms, they continue to benefit from interactive approaches that enable users to guide and constrain the contour to capture specific parts of more challenging objects. We have shown that the LGDF energy model proposed by [47] requires little parameter tuning, is robust against different types of noise, and can be generalized to a broad range of realworld 3D images from biology and medicine. Segmenting many of these images was not possible with existing GPU level set algorithms due to their simple energy functionals. We have greatly enhanced the LGDF model’s performance, making it practical in many more usecases than before (including 3D images). We also extended its functionality through interactive brush functions that give direct influence over the dynamic contour evolution. In the future, we believe GPU adaptations of advanced segmentation algorithms will continue to proliferate, using similar design processes to ours.
7 Availability
We release our C++/OpenCL software and source code under the GNU General Public License Version 3, alongside an optional MATLAB wrapper. The implementation is crossplatform using GLFW with few dependencies, where binaries for Linux and Windows are also available: https://github.com/cwkx/IGAC
Notes
Acknowledgements
We are grateful to NVIDIA for providing a GTX TITAN X for this research. Table 5 1a shows fixed HaCaT human cell culture cells stained with SiRActin (Spirochrome) RED, rat antitubulin antibody/secondary antirat Alexa488 antibody GREEN and DNA DAPI BLUE. The cells are imaged with a Zeiss 880 Airyscan LSM confocal microscope, prepared, and imaged by Miss Bethany Cole, Miss Joanne Robson & Dr Tim Hawkins. Durham Centre for Bioimaging Technology, Department of Biosciences, Durham University.
Supplementary material
Supplementary material 1 (mp4 85151 KB)
References
 1.Adalsteinsson, D., Sethian, J.A.: A fast level set method for propagating interfaces. J. Comput. Phys. 118(2), 269–277 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
 2.Bartz, D.: Volvis datasets. http://www.volvis.org (2005). Accessed 2016 Mar 30
 3.Caselles, V., Catté, F., Coll, T., Dibos, F.: A geometric model for active contours in image processing. Numer. Math. 66(1), 1–31 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
 4.Chambolle, A., Pock, T.: A firstorder primaldual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. 40(1), 120–145 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
 5.Chan, T.F., Vese, L., et al.: Active contours without edges. IEEE Trans. Image Process. 10(2), 266–277 (2001)CrossRefzbMATHGoogle Scholar
 6.Chang, J.T., Schmid, M.F., Rixon, F.J., Chiu, W.: Electron cryotomography reveals the portal in the herpesvirus capsid. J. Virol. 81(4), 2065–2068 (2007)CrossRefGoogle Scholar
 7.Chen, H.L.J., Samavati, F.F., Sousa, M.C., Mitchell, J.R.: Sketchbased volumetric seeded region growing. In: Proceedings of the Third Eurographics Conference on SketchBased Interfaces and Modeling, pp. 123–130 (2006)Google Scholar
 8.Cocosco, C.A., Kollokian, V., Kwan, R.K.S., Pike, G.B., Evans, A.C.: BrainWeb: Online interface to a 3D MRI simulated brain database. NeuroImage 5, 425 (1997)Google Scholar
 9.Cox, R., Pickar, A., Qiu, S., Tsao, J., Rodenburg, Cynthia, Dokland, T., Elson, A., He, B., Luo, M.: Structural studies on the authentic mumps virus nucleocapsid showing uncoiling by the phosphoprotein. Proc. Natl. Acad. Sci. U.S.A. 111(42), 15208–15213 (2014)CrossRefGoogle Scholar
 10.Eklund, A., Dufort, P., Forsberg, D., LaConte, S.M.: Medical image processing on the GPU past, present and future. Med. Image Anal. 17(8), 1073–1094 (2013)CrossRefGoogle Scholar
 11.Evans, A.: Fast approximations for global illumination on dynamic scenes. In: ACM SIGGRAPH 2006 Courses, pp. 153–171. ACM (2006)Google Scholar
 12.Eyiyurekli, M., Breen, D.: Interactive freeform levelset surfaceediting operators. Comput. Graph. 34(5), 621–638 (2010)CrossRefGoogle Scholar
 13.Fulkerson, B., Soatto, S.: Really quick shift: image segmentation on a GPU. In: Trends and Topics in Computer Vision, pp. 350–358 (2010)Google Scholar
 14.Gorelick, L., Schmidt, F.R., Boykov, Y.: Fast trust region for segmentation. In: 2013 IEEE Conference on Computer Vision and Pattern Recognition, pp. 1714–1721 (2013)Google Scholar
 15.Grady, L.: Random walks for image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 28(11), 1768–1783 (2006)CrossRefGoogle Scholar
 16.He, Z., Kuester, F.: GPUbased active contour segmentation using gradient vector flow. In: International Conference on Advances in Visual Computing, pp. 191–201 (2006)Google Scholar
 17.Jarrin, M., Young, L., Wu, W., Girkin, J.M., Quinlan, R.A.: Chapter twentyone—in vivo, ex vivo, and in vitro approaches to study intermediate filaments in the eye lens. In: Omary, M.B., Liem, R.K.H. (eds.) Intermediate Filament Proteins, volume 568 of Methods in Enzymology, pp. 581 – 611. Academic Press (2016)Google Scholar
 18.Jeong, W.K., Beyer, J., Hadwiger, M., Vazquez, A., Pfister, H., Whitaker, R.T.: Scalable and interactive segmentation and visualization of neural processes in em datasets. IEEE Trans. Vis. Comput. Graph. 15(6), 1505–1514 (2009)CrossRefGoogle Scholar
 19.Karotki, L., Huiskonen, J.T., Stefan, C.J., Ziółkowska, N.E., Roth, R., Surma, M.A., Krogan, N.J., Emr, S.D., Heuser, J., Grünewald, K., Walther, T.C.: Eisosome proteins assemble into a membrane scaffold. J. Cell Biol. 195(5), 889–902 (2011)CrossRefGoogle Scholar
 20.Kass, M., Witkin, A., Terzopoulos, D.: Snakes: active contour models. Int. J. Comput. Vis. 1(4), 321–331 (1988)CrossRefzbMATHGoogle Scholar
 21.Kay, T.L., Kajiya, J.T.: Ray tracing complex scenes. In: Conference on Computer Graphics and Interactive Techniques, SIGGRAPH, pp. 269–278. ACM (1986)Google Scholar
 22.LeCun, Y., Bengio, Y., Hinton, G.: Deep learning. Nature 521(7553), 436–444 (2015)CrossRefGoogle Scholar
 23.Lefohn, A.E., Kniss, J.M., Hansen, C.D., Whitaker, R.T.: A streaming narrowband algorithm: interactive computation and visualization of level sets. IEEE Trans. Vis. Comput. Graph. 10(4), 422–433 (2004)CrossRefGoogle Scholar
 24.Li, C., Xu, C., Gui, C., Fox, M.D.: Level set evolution without reinitialization: a new variational formulation. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition, vol. 1, pp. 430–436 (2005)Google Scholar
 25.Li, M., He, C., Zhan, Y.: Adaptive levelset evolution without initial contours for image segmentation. J. Electron. Imaging 20(2), 023004 (2011)CrossRefGoogle Scholar
 26.Malladi, R., Sethian, J.A., Vemuri, B.C.: Shape modeling with front propagation: a level set approach. IEEE Trans. Pattern Anal. Mach. Intell. 17(2), 158–175 (1995)CrossRefGoogle Scholar
 27.Mory, B.: Interactive Segmentation of 3D Medical Images with Implicit Surfaces. Ph.D. thesis, STI, Lausanne (2011)Google Scholar
 28.Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42(5), 577–685 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
 29.Olabarriaga, S.D., Smeulders, A.W.M.: Interaction in the segmentation of medical images: a survey. Med. Image Anal. 5(2), 127–142 (2001)CrossRefGoogle Scholar
 30.Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Applied Mathematical Sciences. Springer, New York (2002)zbMATHGoogle Scholar
 31.Osher, S., Sethian, J.A.: Fronts propagating with curvaturedependent speed: algorithms based on Hamilton–Jacobi formulations. J. Comput. Phys. 79(1), 12–49 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
 32.Peng, D., Merriman, B., Osher, S., Zhao, H., Kang, M.: A PDEbased fast local level set method. J. Comput. Phys. 155(2), 410–438 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
 33.Pock, T., Cremers, D., Bischof, H., Chambolle, A.: An algorithm for minimizing the Mumford–Shah functional. In: IEEE International Conference on Computer Vision, pp. 1133–1140 (2009)Google Scholar
 34.Pratx, G., Xing, L.: GPU computing in medical physics: a review. Med. Phys. 38, 2685 (2011)CrossRefGoogle Scholar
 35.Ren, C.Y., Reid, I.: gSLIC: a realtime implementation of SLIC superpixel segmentation. Technical report, University of Oxford, Department of Engineering, Technical Report (2011)Google Scholar
 36.Roberts, M., Packer, J., Sousa, M.C., Mitchell, J.R.: A workefficient GPU algorithm for level set segmentation. In: Proceedings of the Conference on High Performance Graphics, pp. 123–132. Eurographics Association (2010)Google Scholar
 37.Rosset, A., Spadola, L., Ratib, O.: Osirix: an opensource software for navigating in multidimensional dicom images. J. Digit. Imaging 17(3), 205–216 (2004)CrossRefGoogle Scholar
 38.Schindelin, J., ArgandaCarreras, I., Frise, E., Kaynig, V., Longair, M., Pietzsch, T., Preibisch, S., Rueden, C., Saalfeld, S., Schmid, B., Tinevez, J.Y., White, D.J., Hartenstein, V., Eliceiri, K., Tomancak, P., Cardona, A.: Fiji: an opensource platform for biologicalimage analysis. Nat. Methods 9(7), 676–682 (2012)CrossRefGoogle Scholar
 39.Schmid, J., IglesiasGuitián, J., Gobbetti, E., MagnenatThalmann, N.: A GPU framework for parallel segmentation of volumetric images using discrete deformable models. Vis. Comput. 27(2), 85–95 (2010)CrossRefGoogle Scholar
 40.Sen, A., Heymann, J.B., Cheng, N., Qiao, J., Mindich, L., Steven, A.C.: Initial location of the RNAdependent RNA polymerase in the bacteriophage Phi6 procapsid determined by cryoelectron microscopy. J. Biol. Chem. 283(18), 12227–12231 (2008)CrossRefGoogle Scholar
 41.Shi, L., Liu, W., Zhang, H., Xie, Y., Wang, D.: A survey of GPUbased medical image computing techniques. Quant. Imaging Med. Surg. 2(3), 2223–2292 (2012)Google Scholar
 42.Smistad, E., Elster, A.C., Lindseth, F.: Realtime gradient vector flow on GPUs using OpenCL. J. Real Time Image Process. 10(1), 67–74 (2012)CrossRefGoogle Scholar
 43.Smistad, E., Falch, T.L., Bozorgi, M., Elster, A.C., Lindseth, F.: Medical image segmentation on GPUs a comprehensive review. Med. Image Anal. 20(1), 1–18 (2015)CrossRefGoogle Scholar
 44.Stagg, S.M., Gürkan, C., Fowler, D.M., LaPointe, P., Foss, T.R., Potter, C.S., Carragher, B., Balch, W.E.: Structure of the Sec13/31 COPII coat cage. Nature 439(7073), 234–238 (2006)CrossRefGoogle Scholar
 45.Steel, D.H.W., Lotery, A.J.: Idiopathic vitreomacular traction and macular hole: a comprehensive review of pathophysiology, diagnosis, and treatment. Eye 27(1), 1–21 (2013)CrossRefGoogle Scholar
 46.Vineet, V., Narayanan, P.J.: CUDA cuts: fast graph cuts on the GPU. In: IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pp. 1–8 (2008)Google Scholar
 47.Wang, L., He, L., Mishra, A., Li, C.: Active contours driven by local Gaussian distribution fitting energy. Signal Process. 89(12), 2435–2447 (2009)CrossRefzbMATHGoogle Scholar
 48.Whitaker, R., Breen, D., Museth, K., Soni, N.: Segmentation of Biological Volume Datasets Using a LevelSet Framework, pp. 249–263. Springer, Vienna (2001)Google Scholar
 49.Xu, C., Prince, J.L.: Snakes, shapes, and gradient vector flow. IEEE Trans. Image Process. 7(3), 359–369 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
 50.Yushkevich, P.A., Piven, J., Hazlett, H.C., Smith, R.G., Ho, S., Gee, J.C., Gerig, G.: Userguided 3D active contour segmentation of anatomical structures: significantly improved efficiency and reliability. Neuroimage 31(3), 1116–1128 (2006)CrossRefGoogle Scholar
 51.Zhao, F., Xie, X.: An overview of interactive medical image segmentation. Ann. BMVA 2013(7), 1–22 (2013)Google Scholar
 52.Zhao, H.K., Chan, T., Merriman, B., Osher, S.: A variational level set approach to multiphase motion. J. Comput. Phys. 127(1), 179–195 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
 53.Zhu, H., Meng, F., Cai, J., Shijian, L.: Beyond pixels: a comprehensive survey from bottomup to semantic image segmentation and cosegmentation. J. Vis. Commun. Image Represent. 34, 12–27 (2016)CrossRefGoogle Scholar
 54.Zhu, L., Karasev, P., Kolesov, I., Sandhu, R., Tannenbaum, A.: Interactive Image Segmentation From A Feedback Control Perspective. ArXiv eprints (2016)Google Scholar
 55.Zhu, S.C., Yuille, A.: Region competition: Unifying snakes, region growing, and Bayes/MDL for multiband image segmentation. IEEE Trans. Pattern Anal. Mach. Intell. 18(9), 884–900 (1996)CrossRefGoogle Scholar
Copyright information
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.