- Research
- Open Access
On segmentation model for vector valued images and fast iterative solvers
- Noor Badshah^{1}Email authorView ORCID ID profile,
- Fahim Ullah^{1} and
- Matiullah^{1}
https://doi.org/10.1186/s13662-018-1669-9
© The Author(s) 2018
- Received: 7 October 2017
- Accepted: 6 June 2018
- Published: 25 June 2018
Abstract
In this paper, we propose a new convex variational model for segmentation of vector valued images. The data term of the proposed model is based on the coefficient of variation, which works well in vector valued images having intensity inhomogeneity. Due to convexity of the model, it is independent of the placement of initial contour. Better performance of the proposed model can be seen from experimental results qualitatively and quantitatively. Images in practice are of large sizes, which makes numerical methods more important. In this paper, we also develop fast and stable numerical methods for solution of partial differential equation arisen from the minimization of the proposed model. We have developed a novel multigrid method based on a locally supported smoother. The proposed method is compared with the existing methods in terms of iterations and CPU time for vector valued images having large sizes.
Keywords
- Active contours
- Vector valued images
- Level sets
- Partial differential equations
- AOS method
- MG method
- Jacobi method
- Gauss Seidal method
1 Introduction
Segmentation of images refers to dividing an image into disjoint subdomains, which are homogeneous in some sense, i.e., of the same intensity, color, or texture. To detect objects of interest in an image is the basic objective of image segmentation. Many variational models, like edge-based [1], region-based [2], and active contour models [3–5], have already been developed in connection to image segmentation. Our main focus in this paper is on active contour models. The concept of active contours has been applied for detection of objects in a given image \(F_{0}\) by applying the techniques of curve evolution. In this approach, an initial curve C is evolved towards the edges of objects in a given image under some conditions/constraints.
Minimization of the above discussed model leads towards a highly nonlinear partial differential equation, whose solution is always challenging. Most of the methods for solution of these PDEs are based on explicit discretization, which is conditionally stable (see [10] and the references therein). This method requires very small time step, which in result increases the number of iterations and consequently causes the increase of computational cost. Another approach for solution of these PDEs is the semi-implicit (SI) scheme which is unconditionally stable. The SI method works well in 1D problems, while in higher dimension problems it becomes very slow in convergence and computational cost increases. Additive Operator Splitting (AOS) method was developed in [10] for diffusion problem and was implemented for segmentation models in [9, 11, 12]. The AOS method is fast in convergence as compared to the SI method. However, real images (medical images) are usually of large sizes and in such a case these methods are very slow in convergence. To overcome this problem, the multigrid method based on novel smoothers was proposed in [11] for a two-phase segmentation model (CV model) of gray valued images and in [12] for a multi-phase segmentation model of gray valued images. In this paper, we develop AOS and multigrid methods for solution of PDEs arisen from minimization of the proposed model for two-phase segmentation of vector valued images. The multigrid method is based on a new smoother which is supported locally by freezing the differential coefficients locally. Results of the proposed methods are compared with the existing methods (explicit and implicit), and our methods outperformed the existing methods.
Organization of the rest of the paper is as follows: In Sect. 2, related work is discussed. In Sect. 3, our proposed model is described in detail. In Sect. 4, details of the proposed numerical methods for the solution of partial differential equations are given. In Sect. 5, experimental results and comparison with the existing literature are discussed, and in the last section the conclusion of the paper is given.
2 Related work
In this section, we discuss image segmentation models for vector valued images.
2.1 Chan–Vese model for vector valued images (M1)
2.2 X. Cai joint model for image restoration and segmentation (M2)
3 Proposed model
For segmentation of vector valued images having intensity inhomogeneity, we propose a novel model based on the coefficient of variation. To discuss the proposed model in detail, we first define coefficient of variation (CoV). Data terms based on CoV are used for segmentation of gray images [9, 15]. We first define CoV as follows.
Definition 1
(Coefficient of variation)
Convex formulation of the model
4 Numerical methods
In this section we discuss numerical methods for solution of partial differential equations (15) and (21). We describe semi-implicit and additive operator splitting methods for Eq. (15) and the same can be extended to Eq. (21).
4.1 Semi-implicit method
4.2 Additive Operator Splitting (AOS) method
4.3 Multigrid (MG) method
V-cycle of a multigrid algorithm
Choice of smoother
Here, the coefficients are first updated locally and are stored for relaxation use. In this way Eq. (37) becomes linear and easy to solve.
5 Experimental results and discussion
In this section, we give experimental comparison of the proposed model and method with the existing models and methods qualitatively and quantitatively.
Qualitative comparison
We first give qualitative comparison of the proposed model by testing it on different synthetic and real images having intensity inhomogeneity. It can be seen from the experimental results that the proposed model outperforms the existing models.
Quantitative comparison
Image | CV model M1 | Cai model M2 | Proposed Model | ||||||
---|---|---|---|---|---|---|---|---|---|
No. of Itr. | CPU | JSI | No. of Itr. | CPU | JSI | No. of Itr. | CPU | JSI | |
1 | 360 | 386 | 0.7883 | 180 | 94 | 0.7882 | 8 | 8 | 1 |
2 | 190 | 204 | 0.8842 | 180 | 93 | 0.6785 | 33 | 31 | 1 |
3 | 550 | 642 | 0.7682 | 180 | 95 | 0.6828 | 33 | 30 | 1 |
Image Size | SI Method | AOS Method | MG Method | ||||
---|---|---|---|---|---|---|---|
Itr | CPU | Itr | CPU | Cycle | CPU | ||
Real Image Fig. 4(c) | 128 × 128 | 138 | 51.28 | 115 | 15.13 | 2 | 6.38 |
256 × 256 | 150 | 408.42 | 125 | 39.02 | 2 | 13.18 | |
512 × 512 | 168 | 4016.75 | 135 | 160.18 | 2 | 24.04 | |
1024 × 1024 | 194 | 26,767.43 | 195 | 938.00 | 2 | 69.23 | |
2048 × 2048 | – | – | 500 | 9948.70 | 2 | 261.11 | |
4096 × 4096 | – | – | – | – | 2 | 1086.83 |
Furthermore, in Fig. 4, we have tested the proposed model on different types of synthetic and real images. These results show effectiveness of the proposed model in different types of synthetic and real images. Fig. 4(b), (d) are final segmented results of natural images, Fig. 4(f), (h) are segmented results of synthetic or artificial images, Fig. 4(j), (l) are segmented results of biological cell images, and Fig. 4(n), (p) are segmented results of medical MR images. We observe that our model segments images with intensity variation or inhomogeneity efficiently in the objects like in Fig. 4(e), (g), (p). It can also segment images having inhomogeneity in their background as in Fig. 4(k).
6 Conclusion
The proposed model is based on the coefficient of variation, which works well in images having intensity inhomogeneity. The model is then formulated in convex framework to make it independent of initial contour. The model is minimized through variation to get a partial differential equation which is solved by using AOS method for images of moderate sizes. For images of large sizes, we have proposed a multigrid method based on locally supported smoother. The proposed convex model and multigrid method are compared with the exiting models and methods, and it was found that the proposed model and method outperformed the exiting models and methods.
Declarations
Acknowledgements
The authors would like to express their gratitude to the editors and anonymous reviewers for their valuable suggestions, which substantially improved the standard of the paper.
Availability of data and materials
Not applicable.
Funding
The work is supported financially by the Higher Education Commission (HEC) Islamabad, Pakistan.
Authors’ contributions
The authors equally contributed in the paper. All authors have read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This 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.
Authors’ Affiliations
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