Original contribution
Bias field reduction by localized Lloyd–Max quantization

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Abstract

Bias field reduction is a common problem in medical imaging. A bias field usually manifests itself as a smooth intensity variation across the image. The resulting image inhomogeneity is a severe problem for posterior image processing and analysis techniques such as registration or segmentation. In this article, we present a novel debiasing technique based on localized Lloyd–Max quantization (LMQ). The local bias is modeled as a multiplicative field and is assumed to be slowly varying. The method is based on the assumption that the global, undegraded histogram is characterized by a limited number of gray values. The goal is then to find the discrete intensity values such that spreading those values according to the local bias field reproduces the global histogram as good as possible. We show that our method is capable of efficiently reducing (even strong) bias fields in 3D volumes.

Introduction

Medical image data often suffer from intensity inhomogeneities (bias fields). In magnetic resonance imaging (MRI), this artifact may have several causes, such as a lack of uniform sensitivity of the RF-emitting and -receiving coils, static field (B0) inhomogeneities, gradient-induced eddy currents, magnetic susceptibility of tissue, interslice cross-talk, RF standing wave effects and attenuation of the RF signal inside the object [1], [2].

A bias field usually manifests itself as a smooth intensity variation across the image. Automated extraction of useful information from images demands an automated detection and correction of the bias field. Indeed, tasks like image quantization, segmentation [3] or tissue classification can be severely impeded by degraded intensity homogeneities of images. For example, Kohn et al. [3] observed that a bias field elongates the clusters that represent the brain and cerebrospinal fluid such that the segmentation of the gray and white matter is severely affected.

Various methods have been proposed in order to reduce an image bias field, with different degrees of success. A recent review on methods for correction of intensity inhomogeneities has been written by Vovk et al. [4]. According to their work, current bias field reduction methods can be subdivided as follows:

  • Filtering methods: Low-pass filtering methods assume that the intensity inhomogeneity corrupting the image is a low-frequency signal component that can be separated from the high-frequency information of the imaged anatomical structures. However, this assumption is only accurate if the imaged anatomical structures are relatively small and hence contain no low frequencies that might be mistakenly removed by low-pass filtering. For most of the anatomical structures imaged by an MR scanner this assumption does not hold, which results in overlap of anatomy and inhomogeneity frequency spectra, thereby limiting the feasibility of filtering methods. Homomorphic filtering and homomorphic unsharp masking have been proposed. Morphological filtering [5] and simple high-pass filtering belong to this group of methods but have not shown to be useful for MRI.

  • Surface fitting methods: These methods fit a parametric surface to a set of image features (intensity or gradient based) that contain information on intensity inhomogeneity. The resulting surface, which is usually polynomial or spline based, represents the multiplicative inhomogeneity field that is used to correct the input image [6], [7].

  • Segmentation-based methods: It is well known that images that need to be segmented in homogeneous regions require preprocessing to remove the bias field prior to segmentation. In turn, removal of the bias field in an image becomes trivial when the image is subdivided into classes that should be homogeneous. Methods were proposed that exploit this duality based on maximum likelihood, expectation maximization [8], [9] and fuzzy C-means clustering [10], [11], and often solved by means of functional minimization.

Based on the categorization of Vovk et al. [4], the method we will present is novel to the best of our knowledge and is likely to be situated in the category segmentation-based methods. However, the functional that we will consider for minimization is not entropy related [12] but a measure simply based on the mean squared error (MSE) [11].

Segmentation and tissue classification are problems that are related to the field of image quantization. Where segmentation tries to classify the image domain into distinct regions that meet some criterion (being an element in some threshold interval, in its simplest form), tissue classification tries to label each pixel of the image domain by one or more tissue indices. Image quantization (Lloyd–Max, C-means quantization) [13] on the other hand tries to find a reduced set of intensity values that allows to represent a large number of gray values in an optimal way, which is usually provided by MSE measures.

In this article, we show how a local Lloyd–Max quantization (LMQ) approach can be effectively used to debias medical images. No presmoothing scheme is required, as our method works still rather well in situations with heavy noise. We will also show that the proposed method is very well capable of significantly reducing strong bias fields in simulated as well as real 3D medical data sets.

The article is organized as follows: First we start with a general description of the inspiration for our bias field reduction approach. Next, a formulation of the image model is presented. In Subsection 2.2 to 2.3, the well-known LMQ procedure (without bias field) is briefly summarized and then extended to image models that contain bias fields, after which the algorithm schemes are introduced in sections. In Section 3, various experiments to test the debiasing performance of the proposed method are conducted and discussed. Finally, in Section 4, we draw the conclusions.

Section snippets

Inspiration

In this article, the model we employed for the undegraded image (i.e., no bias field or noise) assumes that the image can be subdivided into a fixed number of classes, where image intensities inside each class are distinct from those of the others. This implies that the undegraded image can be segmented into distinct parts, each of which can be classified with a characteristic gray level. For example, an MR image of the brain can be modeled to consist of four classes: background, cerebrospinal

Results and discussions

In this section, we present the results from various experiments where we applied our method on phantom geometric images, simulated brain images as well as real brain images.

According to Vovk et al. [4], the nonparametric nonuniformity normalization (N3) method, described in Ref. [15], has become the standard method against which other debiasing methods are compared. Hence, during the tests for simulated data and real data, we have compared our proposed LMQ algorithm to the N3 method, which is

Conclusion

In this article, we have proposed a general debiasing method based on LMQ. Our method assumes the imaged structures to consist of distinct components, which is largely justifiable in anatomical scans. Apart from that, our method does not assume any prior knowledge about the bias field and is practically robust against either phantom, simulated or real images. Both simulation image and real image results showed that the LMQ bias reduction is significantly better than the commonly used N3 method.

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    The first step in the processing pipeline was to segment the skin layer. Hereto, the MR images were debiased by Lloyd–Max quantization (Mai et al., 2011) and thresholded for grayscale intensity between 300 and 1800. Connected components labeling (Dillencourt et al., 1992) was used to separate the head from smaller noise artifacts and head fixation braces.

This work is financially supported by the Institute for the Promotion of Innovation through Science and Technology in Flanders (IWT-Vlaanderen), through the SBO grant 060819 “Quantiviam,” and by the IAP grant P6/38 of the Belgian Science Policy.

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