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Isocontour mapping is efficient for extracting meaningful information from a biomedical image in a topographic analysis. Isocontour extraction from real world medical images is difficult due to noise and other factors. As such, adaptive selection of contour generation parameters is needed. This paper proposes an algorithm for generating an adaptive contour map that is spatially adjusted. It is based on the modified active contour model, which imposes successive spatial constraints on the image domain. The adaptability of the proposed algorithm is governed by the energy term of the model. This work focuses on mammograms and the analysis of their intensity. Our algorithm employs the Mumford-Shah energy functional, which considers an image's intensity distribution. In mammograms, the brighter regions generally contain significant information. Our approach exploits this characteristic to address the initialization and local optimum problems of the active contour model. Our algorithm starts from the darkest region; therefore, local optima encountered during the evolution of contours are populated in less important regions, and the important brighter regions are reserved for later stages. For an unrestricted initial contour, our algorithm adopts an existing technique without re-initialization. To assess its effectiveness and robustness, the proposed algorithm was tested on a set of mammograms.

The extraction of meaningful information from images by means of digital image processing techniques is an important task in many application domains. To identify significant information in an image, one can exploit distinctive features of objects (e.g., shape, location, margin) and uncover significant regions or patterns by analyzing the topological and geometrical properties of the image. In the biomedical sector, medical imaging techniques, such as X-rays, magnetic resonance imaging (MRI), and tomography are used to visualize internal structures of the body. Medical images, such as mammograms, have inherently complex and variable features with blurred object boundaries, which make the use of explicit features of objects in image analysis difficult. Hence, image analysis methods based on isocontour mapping are better suited to complex medical images as mentioned below.

Meaningful information can be extracted efficiently from a digital image on an isocontour map. In biomedical image processing, isocontour mapping is used extensively to perform the topographic analysis of medical images. An isocontour map consists of a set of curves of equal value (e.g., height or intensity). Analyses based on isocontour maps can provide the association between image inclusions. They detect a region of interest (ROI) by analyzing the correlation (or the enclosure relationship) between objects. ROI analysis that highlights suspicious regions in medical images is an essential step in computer-aided diagnosis systems. Thus, isocontour maps can provide a robust topographic representation of medical images for ROI analysis [

To our current knowledge there is no deterministic way to find parameters like the number of quantization levels (contour interval and the difference in elevation between successive contours) that yield the best results for isocontour generation. However, the determination of contour generation parameters is an open question and adjustable. Hong and Sohn [

A number of active contour models have been developed. Kass

Geodesic active contour (GAC) models in [

The classical active contour model [

The proposed algorithm for adaptive contour mapping is based on two previous active contour models: active contours without edges (ACWE) and level set evolution without re-initialization (LSEWR). From the ACWE concept, we used force image terms to get regional information, whereas in LSEWR we selected penalizing terms to eliminate re-initialization. The ACWE model proposed by Chan and Vese [^{n} regions from the combination of each phase by level set functions. The curve evolution with the level set function requires costly re-initialization because the level set function deviates from a signed distance function (SDF) in each evolution. Li

Our algorithm is designed with a similar manner of isocontour mapping to detect an arbitrary number of contours for spatial adaptive isocontour mapping. The existing multi-phase method, which detects contours at multiple level sets, always produces 2^{n} regions. This indicates that many insignificant features might be included in the contour map, thereby influencing the image analysis results. Our approach divides a region into two sub-regions using the base contour. It then divides one of the segmented sub-regions into two sub-regions in successive iterations. The proposed algorithm detects sub-regions by minimizing the new energy model, restricting it to the characteristic function of a base sub-region. The iterative segmentation process automatically terminates when the stopping criterion is met. Note that only one of the two sub-regions is further segmented in successive iterations. This is associated with the characteristics of the mammographic image in addition to problems in initialization and local optimum of the active contour model.

In mammograms, bright regions contain information that is more significant (e.g., candidate masses). Our algorithm takes advantage of this mammographic characteristic to address the problems in initialization and local optimum of the active contour model. That is, the proposed algorithm starts with the initial contour found in the darkest (low intensity) region so that the local optima encountered during the contour evolution is placed in less important low intensity regions. In terms of initialization, this enables the our approach to start its operation with an initial contour in the darkest region of the image, and the LSEWR component allows free initial contour without satisfying SDF. Hence, the important brighter regions are kept for segmentation and refined analysis at later stages.

In this paper, we propose an adaptive contour mapping approach that can be used to analyze the topological and geometrical information on mammogram images. Mammograms have complex and variable structures with blurred object boundaries, which make the use of explicit object features that are less applicable to image analysis. Hence, this work utilizes an isocontour map to analyze mammographic features. It is relatively easy to create an isocontour map, but the parameters for generating active contours should be determined accordingly. As shown in

The rest of the paper is organized as follows: we discuss related work in Section 2. In Section 3, we discuss the proposed multipass active contour approach. In particular, our algorithm is designed to partition the image domain into an arbitrary number of sub-regions by applying a two-phase segmentation algorithm based on the recursive use of the Mumford-Shah energy functional. The recursive application of the two-phase segmentation algorithm on subsequently segmented sub-regions results in a tree structure of partitions for an adaptive contour map. In Section 4, we discuss denoising of the input image, numerical scheme, initialization of the level set function, and the parameters needed to implement the algorithm. We then describe the data set and the analytical results in Section 5. We finish our contribution with several conclusions.

Active contours are a core component of computer vision and medical imaging, and they can be divided into three areas: boundary driven, region driven, and hybrid contours that combine the boundary and region driven areas.

Malladi

In 2001, Chan and Vese [

Vese and Chan [

Our algorithm is performed in such a way that it partitions an image into two sub-regions. One of the sub-regions is then iteratively partitioned into two more sub-regions. The sub-regions are determined by the minimization of a new energy model restricted to a characteristic function of a sub-region, and no re-initialization is needed. Our method segments the image into any number of regions, and the process automatically terminates at the stationary solution. In this paper, the proposed approach is performed from the lowest intensity region to important high intensity regions, which are thus segmented in fine scale. Finally, the segmentation result provides an adaptive contour map of the image.

A curve ^{th} sub-region w_{k} ⊂ Ω into two sub-regions w_{k+1},w̅_{k+1} with,

The region _{k}_{0} as the input image, which is further segmented in two sub-regions _{1} and w̅_{1}, which are located inside and outside of the zero level set curve respectively. In our algorithm, the evolution of level set starts from outside the boundary of the given image and moves inwards. To move the level set inwards, we calculate the inner sub-region of the zero level curve using _{1},_{2},_{3},_{4} are calculated using

The isocontour of _{1} and _{1} is calculated from the region _{0}_{1} as shown in _{1} makes the next sub-regions _{2} and _{2}. Further contours make further sub-regions _{3},_{3},_{4} and _{4}.

The proposed segmentation algorithm can be expressed in the following iteration that the minimizer ^{k}_{1} and _{2} depending on _{k+}_{1} and _{k+}_{1}. If minimizer _{k}^{k}_{1}, λ_{2} > 0,

_{1} is energy of region;_{k+1}

_{2} is energy of region _{k+1}

_{g} is weighted arc length of curve

_{g} is weighted area of sub-region _{k+}_{1} by an edge indicator function;

The energy functional _{1},_{2} would drive the motion of the zero level curve _{g} would regularize the curve _{g} would accelerate the curve evolution when all regions of _{k}_{1},_{2} from the CV model [^{k}_{k}

The energy of sub-region _{k}^{k}^{k}_{g} with univariate Dirac Delta function _{g}. By the level set method, the weighted area functional _{g} is expressed as:

The small energy of _{g} accelerates the curve evolution. By definition, an SDF satisfies a desirable property |∇ _{g}_{g}_{g}

For the minimization, it is necessary to find the zero point of differentiation of the functional _{1}, _{2}, _{1}, _{2}, _{1} and _{2}, which partitions a region inside and outside of the zero curve _{g}(_{g}

A classic iterative process to minimize the functional _{0} is the initial level set function. The means _{1} and _{2} of regions _{k+1} and _{k+1} are calculated from the mean of intensity values on image _{k}

If one of the sub-regions _{k}_{+1} or _{k}_{+1} is empty, then the formulation degenerates, and therefore the algorithm automatically terminates. Finally, the principal steps of the algorithm are:

Initialize

Compute ^{k}_{0}

Solve the PDE for ^{k}^{+1}

Check whether the solution is stationary. If not,

In this paper, we note that the stationary problem obtained directly from the minimization problem could also be solved numerically using a similar finite difference scheme.

Medical images are affected by artifacts due to device noise and inhomogeneities in the body. The noise affects the segmentation process. The noise changes the mean of a region and behaves like a strong edge. We use an image de-noising technique to reduce the interference of noise. In Rudin, Osher, and Fatemi (ROF) [_{0}_{0}. We diffuse the _{0} by changing λ. The denoised image

In our work, the Dirac function _{ε}(_{ε}(

We use the regularized Dirac δ_{ε}(_{ε}_{x}_{y}_{x}_{y}_{d}_{x}_{y}

The Laplacian operator has been implemented in a similar manner.

In our experiments, we choose the following parameters: λ_{1}= λ_{2}=100, ^{2},

In outmoded level set methods, it is essential to initialize the level set function _{0}. If the initial level set function is expressively different from a signed distance function, then the re-initialization schemes are not able to re-initialize the function to a signed distance function. In our formulation, not only is the re-initialization procedure eliminated, but the level set function _{0}^{k}^{k}^{+1} value, that is, for every evolved level set function. Performing from the lowest intensity region, the initial level set function _{0}is defined as:
_{0} with ρ =4ε and κ = 1. By definition, this initial level set function _{0}

The proposed multi-phase segmentation algorithm has been applied to synthetic and real mammographic images from the mini-MIAS database [

In

This paper presents an adaptive contour map that captures topographic information in mammograms characterized by blurred object boundaries. The proposed multipass active contour algorithm for adaptive contour mapping is based on the two-phase piecewise constant segmentation model (ACWE) proposed by Chan and Vese [

Our algorithm produces an arbitrary number of regions, and it automatically terminates when its stopping condition is met. The proposed algorithm was tested using synthetic and real mammographic images that include masses varying in size and subtlety. The experimental results showed that our approach yields an accurate contour map of both distinctive and subtle masses in mammograms. It also successfully produced an adaptive contour map in synthetic images that have relatively clear edges. The experimental results show sensitive segmentation on the important region as well as intuitive segmentation structure.

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0025512).

Multiscale approach for isocontour maps [

Partition by the curve ^{th} sub-region _{k}

(_{0}

The mini-MIAS database of mammogram number 028. (

The mini-MIAS database of mammogram number 063 (

Results of a synthetic image with 5% uniform noise. (