A polynomial piecewise constant approximation method based on dual constraint relaxation for segmenting images with intensity inhomogeneity

•A novel piecewise constant approximation method is proposed to segment images with intensity inhomogeneity.•The objective function is defined to transform original image into piecewise constant image.•The polynomial piecewise constant approximation realizes dual constraint relaxation for objective function.•The proposed method is not limited by the requirement that the intensity inhomogeneity of image is slowly varying.

In the past decade, several local region-based level set models have been proposed to segment images with intensity inhomogeneity. In general, these models are designed based on one assumption, i.e. intensity inhomogeneity is slowly varying in image domain. However, this assumption is not valid for those images with serious intensity inhomogeneity, which inevitably leads to poor segmentation performance of existing models. In this paper, we propose a novel level set method named as polynomial piecewise constant approximation (PPCA) to well segment images with serious intensity inhomogeneity. The basic idea of the PPCA method is to transform the original image to piecewise constant image so as to make piecewise constant segmentation criterion become applicable. Specially, we firstly define an initial objective function with some constraint conditions to transform the original image. Then, in order to obtain desirable piecewise constant image and highlight the anti-noise performance, the PPCA method is used to realize the dual constraint relaxation for objective function. The dual constraint relaxation reflects in two parts: on one hand, the objective function based on local region is exploited to replace the pointwise approximation method; on the other hand, considering the variance of local intensity distribution and the reliability of polynomial approximation, we utilize a Gaussian pyramid convolution strategy to devise polynomial fitting. The PPCA method relaxes the constraint condition of objective function so that the piecewise constant image is easily approximated. According to piecewise constant segmentation criterion, we obtain the partial differential equation based on polynomial piecewise constant approximation. Finally, we utilize level set method to construct the energy functional. The visual and quantitative experimental results demonstrate that the proposed PPCA method can yield better results than existing classical local models for segmenting images with serious intensity inhomogeneity.