Wenjie He

A universal noise removal algorithm with an impulse detector

We introduce a local image statistic for identifying noise pixels in images corrupted with impulse noise of random values. The statistical values quantify how different in intensity the particular pixels are from their most similar neighbors. We continue to demonstrate how this statistic may be incorporated into a filter designed to remove additive Gaussian noise. The result is a new filter capable of reducing both Gaussian and impulse noises from noisy images effectively, which performs remarkably well, both in terms of quantitative measures of signal restoration and qualitative judgements of image quality. Our approach is extended to automatically remove any mix of Gaussian and impulse noise.

Examples of bivariate nonseparable compactly supported orthonormal continuous wavelets

We give many examples of bivariate nonseparable compactly supported orthonormal wavelets whose scaling functions are supported over [0,3]x[0,3]. The Holder continuity properties of these wavelets are studied.

Compactly Supported Tight Affine Frames with Integer Dilations and Maximum Vanishing Moments

When a cardinal B-spline of order greater than 1 is used as the scaling function to generate a multiresolution approximation of L2=L2(R) with dilation integer factor M≥2, the standard “matrix extension” approach for constructing compactly supported tight frames has the limitation that at least one of the tight frame generators does not annihilate any polynomial except the constant. The notion of vanishing moment recovery (VMR) was introduced in our earlier work (and independently by Daubechies et al.) for dilation M=2 to increase the order of vanishing moments. This present paper extends the tight frame results in the above mentioned papers from dilation M=2 to arbitrary integer M≥2 for any compactly supported M-dilation scaling functions. It is shown, in particular, that M compactly supported tight frame generators suffice, but not M−1 in general. A complete characterization of the M-dilation polynomial symbol is derived for the existence of M−1 such frame generators. Linear spline examples are given for M=3,4 to demonstrate our constructive approach.

Interactive sketch generation

In this paper, we propose an interactive system for generating artistic sketches from images, based on the stylized multiresolution B-spline curve model and the livewire contour tracing paradigm. Our multiresolution B-spline stroke model allows interactive and continuous control of style and shape of the stroke at any level of details. Especially, we introduce a novel mathematical paradigm called the wavelet frame to provide essential properties for multiresolution stroke editing, such as feature point preservation, locality, time-efficiency, good approximation, etc. The livewire stroke map construction leads the user-guided stroke to automatically lock on to the target contour, allowing fast and accurate sketch drawing. We classify the target contours as outlines and interior flow, and develop two respective livewire techniques based on extended graph formulation and vector flow field. Experimental results show that the proposed system facilitates quick and easy generation of artistic sketches of various styles.

Tight frames of compactly supported multivariate multi-wavelets

This paper is devoted to the study and construction of compactly supported tight frames of multivariate multi-wavelets. In particular, a necessary condition for their existence is derived to provide some useful guide for constructing such MRA tight frames, by reducing the factorization task of the associated polyphase matrix-valued Laurent polynomial to that of certain scalar-valued non-negative ones. We illustrate our construction method with examples of both multivariate scalar- and vector-valued subdivision schemes. Since our constructions for C^1 and C^2 piecewise cubic schemes are quite involved, we also include the corresponding Matlab code in the Appendix.

Construction of trivariate compactly supported biorthogonal box spline wavelets

We give a formula for the duals of the masks associated with trivariate box spline functions. We show how to construct trivariate nonseparable compactly supported biorthogonal wavelets associated with box spline functions. The biorthogonal wavelets may have arbitrarily high regularities.

Bivariate box spline wavelets in Sobolev spaces

We use bivariate boxspline functions to construct nonseparable wavelets in Sobolev spaces.

Digital filters associated with bivariate box spline wavelets

Battle-Lemaries wavelet has a nice generalization in a bivariate setting. This generalization is called bivariate box spline wavelets. The magnitude of the filters associated with the bivariate box spline wavelets is shown to converge to an ideal high-pass filter when the degree of the bivariate box spline functions increases to ∞. The passing and stopping bands of the ideal filter are dependent on the structure of the box spline function. Several possible ideal filters are shown. While these filters work for rectangularly sampled images, hexagonal box spline wavelets and filters are constructed to process hexagonally sampled images. The magnitude of the hexagonal filters converges to an ideal filter. Both convergences are shown to be exponentially fast. Finally, the computation and approximation of these filters are discussed.