Elsevier

Pattern Recognition

Volume 44, Issue 9, September 2011, Pages 1842-1851
Pattern Recognition

Structure-preserving smoothing of biomedical images

https://doi.org/10.1016/j.patcog.2010.08.003Get rights and content

Abstract

Smoothing of biomedical images should preserve gray-level transitions between adjacent tissues, while restoring contours consistent with anatomical structures. Anisotropic diffusion operators are based on image appearance discontinuities (either local or contextual) and might fail at weak inter-tissue transitions. Meanwhile, the output of block-wise and morphological operations is prone to present a block structure due to the shape and size of the considered pixel neighborhood.

In this contribution, we use differential geometry concepts to define a diffusion operator that restricts to image consistent level-sets. In this manner, the final state is a non-uniform intensity image presenting homogeneous inter-tissue transitions along anatomical structures, while smoothing intra-structure texture. Experiments on different types of medical images (magnetic resonance, computerized tomography) illustrate its benefit on a further process (such as segmentation) of images.

Introduction

Medical imaging scanners have been improving the quality of images over the years. Given that scanners capture either physical or chemical properties of tissue, the appearance of anatomical structures in images should be uniform. However, the presence of radiological noise (among other artifacts) disturbs structures homogeneity. It follows that images should be smoothed before any segmentation of anatomical structures. Medical imaging smoothing should homogenize the intensity inside anatomical structures, while preserving intensity changes at their boundaries without altering their shape. Existing smoothing methods for preserving image features (edges and corners) might be grouped into block-wise and differential operators.

Block-wise operators (like median, morphological [1], mean shift [2], or Kuwahara inspired [3]) replace the pixel intensity by a function (usually statistical [2], [3]) of neighboring values. Since they can be related to image level-sets evolution (rather than image intensity evolution) they naturally preserve contrast changes. The counterpart is that evolution of image contours alters their shape. Contours in filtered images deform according to the shape of the structure element defining the pixel neighborhood. In many cases [1], [3], even the smoothed image might present a block-wise appearance congruent with the shape of such structure element.

Differential operators use parabolic partial differential equations (PDE) to (iteratively) smooth an initial image [1]. Although higher order methods exist [4], many methods are given by second order elliptic operators admitting a formulation in general divergence form. Such methods have a physical interpretation in terms of heat diffusion and are backed by a solid mathematical theory [5].

For divergence-based methods, the redistribution of image intensity along time (iterations), as well as, final states are determined by the diffusion tensor. A main property is that evolution converges to a constant image unless the tensor cancels on some curves [5]. This is exploited by edge enhancement diffusions [6], which diffuse backwards (i.e. the tensor becomes negative) at pixels with an image gradient above a given threshold and like a standard heat equation otherwise. In this manner, they sharpen image contrast at edges while performing like a Gaussian filter at areas where contrast change is not significant. Although widely used by the medical imaging community, the choice of the threshold is a delicate issue [7], [8], [9] for preventing enhancement of noise and image artifacts.

In order to ensure smooth images, anisotropic diffusions [1] design positive tensors slowing down diffusion across structures and features of interest. Such features are determined by measures of image appearance discontinuity. Common trends are either the norm of image derivatives (first order for edges [10] and second order for ridges [11]) or global contextual discontinuities [12]. In order to ensure stability of the diffusion process, heat diffuses on the whole image plane, which implies convergence to a uniform intensity image [5]. This fact forces relying on a given number of iterations (termination problem) or adding close-to-data constraints to ensure preservation of the image most relevant features.

Close-to-data diffusions [13], [14] add a zero order fidelity term quantifying the difference between the original and the (current) diffused image to the anisotropic formulation. The iterative scheme is the minimization of a (total variation) functional searching for a compromise between smoothing and similarity to original data. The weight that controls the tradeoff between the regularity and fidelity terms is related to the degree of filtering of the solution of the minimization problem. A main concern is that smaller details, such as texture and some structures are destroyed if such parameter is too small [4], [15].

Finally, trace-based methods [16] remove first order terms from the divergence equation to make solutions consistent with the image level-sets geometry described by the diffusion tensor. Solutions are related to line integral convolution [17] along the diffusion tensor eigen-directions. Although sharp edges and curved structures are better preserved along the diffusion process, in the general case, a number of iterations must still be given in order to prevent convergence to constant final states. Besides, since the numeric scheme bases online integration of single vector fields [18] it cannot be easily generalized to diffusion on surfaces in volumetric data.

In this paper we introduce a differential operator, the structure-preserving diffusion, SPD, which restricts diffusion to a smooth approximation of image contours. Differential geometry arguments [19] ensure stability of the diffusion process. A main contribution is that SPD homogenizes gray-level along regular image contours without altering their shape. In this manner, SPD converges (i.e. the iterative scheme stabilizes) towards a non-uniform image presenting a uniform gray-level inside anatomical structures, while preserving transitions across tissue layers.

The contents of this paper are structured as follows. Our SPD approach is detailed in Section 2 and its mathematical issues are included in Appendix A. Validating experiments comparing with existing works are reported in Section 3 and an application illustrating SPD benefits is given in Section 4. Finally, concluding remarks are exposed in Section 5.

Section snippets

Implicit restricted diffusions

Smoothing methods provide a set of smoothed versions of a given (2D or 3D) function u0(x1,…,xn)=u0(x). For PDE-based methods, the set of smoothed versions is a time (scale) dependant family, u(x,t), that solves a PDE of parabolic type with initial condition u0(x). Many approaches for image smoothing are given by second order elliptic operators given in pure divergence formut=div(Ju)withu(x,0)=u0(x)for u=(ux1,,uxn) the function gradient and ut the derivative with respect to time. The symbol

Experiments

The goal of our experiments is to show the improvement in quality of SPD images (compared to other filtering approaches) for a further identification of anatomical structures. In order to illustrate SPD benefits regardless of the image modality and anatomic geometry considered, two different data sets have been considered:

  • Cardiac magnetic resonance (MR) images. Images have been extracted from a data set of healthy volunteers provided by Creu Blanca Clinic. Images have been acquired using a

Application to extraction of plant's xylem network

The xylem of plants is a tissue consisting of a tubular network that provides the main pathway for long distance transport of water from roots to leaves [25]. Its properties determine how much water can be transported by plants, as well as the vulnerability to transport dysfunctions (formation and propagation of emboli) associated to stress factors, such as droughts and frost. In particular, vulnerability to embolism has been recognized as one of the main factors explaining plants’ resistance

Conclusions

Smoothing of biomedical images preserving weak tissue transitions is a crucial step for a proper segmentation of anatomical structures. Smoothing methods given by solutions to a second order elliptic PDE are backed up by a solid mathematical theory and admit an explicit iterative numeric scheme. The asymptotic behavior of solutions is determined by the second order term, which is usually described by means of a diffusion tensor positive defined. If the eigenvectors of strictly positive

Acknowledgments

We would like to thank Francesc Carreras and Xavier Alomar from the Radiology Department of Creu Blanca clinic for providing MR data. This work was supported by the Spanish projects PI071188, TIN2009-13618, CSD2007-00018 and Explora (CGL2007-28784-E). The first author has been supported by The Ramon y Cajal Program.

Debora Gil received her degree in Mathematics from Universitat de Barcelona in 1994 and her PhD from Universitat Autònoma de Barcelona (UAB, Bellaterra, Spain) in 2004. Since 1997 she has been working for the Computer Vision Center (CVC) and the Computer Science Department of Universitat Autònoma de Barcelona. She is in charge of the Modelling and Visualization group in CVC since 2007. Her main interests are mathematical tools applied to image processing (especially, but not exclusively,

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    Debora Gil received her degree in Mathematics from Universitat de Barcelona in 1994 and her PhD from Universitat Autònoma de Barcelona (UAB, Bellaterra, Spain) in 2004. Since 1997 she has been working for the Computer Vision Center (CVC) and the Computer Science Department of Universitat Autònoma de Barcelona. She is in charge of the Modelling and Visualization group in CVC since 2007. Her main interests are mathematical tools applied to image processing (especially, but not exclusively, biomedical imaging) and she has expertise in pde-based methods, segmentation, motion analysis and shape modelling.

    Aura Hernandez received her degree in Mathematics from Universitat Autònoma de Barcelona (UAB, Bellaterra, Spain) in 2002. In 2005 and 2009, respectively, she received her MSc and PhD in Computer Science from the UAB. She is a lecturer at UAB and collaborates with the Computer Vision Center (www.cvc.uab.es). Her research interests are focused on Image Processing Techniques for Biomedical Images, such as, segmentation, and dynamical parameters estimation, and mathematical models for Biomedical Image Processing, like contour geometry, geometric flows and variational models.

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