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The intensity or gray-level derivatives have been widely used in image segmentation and enhancement. Conventional derivative filters often suffer from an undesired merging of adjacent objects because of their intrinsic usage of an inappropriately broad Gaussian kernel; as a result, neighboring structures cannot be properly resolved. To avoid this problem, we propose to replace the low-level Gaussian kernel with a bi-Gaussian function, which allows independent selection of scales in the foreground and background. By selecting a narrow neighborhood for the background with regard to the foreground, the proposed method will reduce interference from adjacent objects simultaneously preserving the ability of intraregion smoothing. Our idea is inspired by a comparative analysis of existing line filters, in which several traditional methods, including the vesselness, gradient flux, and medialness models, are integrated into a uniform framework. The comparison subsequently aids in understanding the principles of different filtering kernels, which is also a contribution of this paper. Based on some axiomatic scale-space assumptions, the full representation of our bi-Gaussian kernel is deduced. The popular γ-normalization scheme for multiscale integration is extended to the bi-Gaussian operators. Finally, combined with a parameter-free shape estimation scheme, a derivative filter is developed for the typical applications of curvilinear structure detection and vasculature image enhancement. It is verified in experiments using synthetic and real data that the proposed method outperforms several conventional filters in separating closely located objects and being robust to noise.