Charles K. Chui

Coherent line drawing

This paper presents a non-photorealistic rendering technique that automatically generates a line drawing from a photograph. We aim at extracting a set of coherent, smooth, and stylistic lines that effectively capture and convey important shapes in the image. We first develop a novel method for constructing a smooth direction field that preserves the flow of the salient image features. We then introduce the notion of flow-guided anisotropic filtering for detecting highly coherent lines while suppressing noise. Our method is simple and easy to implement. A variety of experimental results are presented to show the effectiveness of our method in producing self-contained, high-quality line illustrations.

Flow-Based Image Abstraction

We present a non-photorealistic rendering technique that automatically delivers a stylized abstraction of a photograph. Our approach is based on shape/color filtering guided by a vector field that describes the flow of salient features in the image. This flow-based filtering significantly improves the abstraction performance in terms of feature enhancement and stylization. Our method is simple, fast, and easy to implement. Experimental results demonstrate the effectiveness of our method in producing stylistic and feature-enhancing illustrations from photographs.

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.

Surface subdivision schemes generated by refinable bivariate spline function vectors

The objective of this paper is to introduce a direct approach for generating local averaging rules for both the 3 and 1-to-4 vector subdivision schemes for computer-aided design of smooth surfaces. Our innovation is to directly construct refinable bivariate spline function vectors with minimum supports and highest approximation orders on the six-directional mesh, and to compute their refinement masks which give rise to the matrix-valued coefficient stencils for the surface subdivision schemes. Both the C1-quadratic and C2-cubic spaces are studied in some detail. In particular, we show that our C2-cubic refinement mask for the 1-to-4 subdivision can be slightly modified to yield an adaptive version of Loops surface subdivision scheme.

A unified scheme for adaptive stroke-based rendering

This paper presents a comprehensive scheme for automatically generating a broad class of artistic illustrations from photographs. Using strokes as the major building blocks, our system optimizes the stroke attributes subject to the desired rendering style. The stroke attributes are computed adaptively to enable importance-based control of the abstraction level at each pixel. We propose a novel outline detection and refinement paradigm called edge painting to construct an outline map, and from which to derive the pixel-wise importance. We also introduce an adaptive bilateral filter to adaptively guide the curved stroke directions based on the importance map. Given the outline, importance, and direction maps, the system creates the illustration via selecting the representative colors, setting the style parameters, and optimizing the stroke attributes based on simulated annealing. The experimental results show that our scheme facilitates automatic production of artistic illustrations in a wide range of rendering styles.

Balanced multi-wavelets in ℝ^{}

The notion of K-balancing was introduced a few years ago as a condition for the construction of orthonormal scaling function vectors and multi-wavelets to ensure the property of preservation/annihilation of scalarvalued discrete polynomial data of order K (or degree K - 1), when decomposed by the corresponding matrix-valued low-pass/high-pass filters. While this condition is indeed precise, to the best of our knowledge only the proof for K = 1 is known. In addition, the formulation of the K-balancing condition for K > 2 is so prohibitively difficult to satisfy that only a very few examples for K = 2 and vector dimension 2 have been constructed in the open literature. The objective of this paper is to derive various characterizations of the K-balancing condition that include the polynomial preservation property as well as equivalent formulations that facilitate the development of methods for the construction purpose. These results are established in the general multivariate and bi-orthogonal settings for any K > 1.

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.

From extension of Loop's approximation scheme to interpolatory subdivisions

The minimum-supported bivariate C^2-cubic spline on a 6-directional mesh constructed in our previous work [Chui, C.K., Jiang, Q.T., 2003. Surface subdivision schemes generated by refinable bivariate spline function vectors. Appl. Comput. Harmonic Anal. 15, 147-162] can be used to extend Loops approximation subdivision scheme to introduce some parameter for controlling surface geometric shapes. This extension is achieved by considering matrix-valued subdivisions, resulting in subdivision templates of the same 1-ring template size as Loops scheme, but with 2-dimensional matrix-valued weights. Another feature accomplished by considering such an extension is that the two components of the refinable vector-valued spline function can be reformulated, by taking certain linear combinations, to convert the approximation scheme to an interpolatory scheme, but at the expense of an increase in template size for the edge vertices. To maintain the 1-ring template size with guarantee of C^2 smoothness for interpolatory surface subdivisions, a non-spline solution is needed, by applying some constructive scheme such as the procedure discussed in our recent work [Chui, C.K., Jiang, Q.T., 2005b. Matrix-valued symmetric templates for interpolatory surface subdivisions I. Regular vertices. Appl. Comput. Harmonic Anal. 19, 303-339]. The main objective of this paper is to develop the corresponding matrix-valued 1-ring templates for the extraordinary vertices of arbitrary valences, for all of the three schemes mentioned above: the extended Loop approximation scheme, its conversion to an interpolatory scheme, and the non-spline 1-ring interpolatory scheme. The discrete Fourier transform (DFT) is applied to analyze the spectral properties of the corresponding subdivision matrices, assuring that the eigenvalues of the subdivision matrices satisfy certain conditions for C^1 smoothness at the extraordinary vertices for all of the three considerations in this paper.

Matrix-valued subdivision schemes for generating surfaces with extraordinary vertices

Subdivision templates of numerical values are replaced by templates of matrices in this paper to allow the introduction of shape control parameters for the feasibility of achieving desirable geometric shapes at those points on the subdivision surfaces that correspond to extraordinary control vertices. Formulation of the matrix-valued subdivision surface is derived. Based on refinable bivariate spline function vectors for matrix-valued subdivisions, the notion of characteristic map introduced by Reif is extended from (scalar) surface subdivisions to matrix-valued subdivisions. The C^1- and C^k-continuity of Reif and Prautzsch for matrix-valued subdivisions are discussed. To illustrate the general theory, the smoothness of matrix-valued triangular subdivision schemes for extraordinary vertices with valences 3 and 4 is analyzed. The issue of effective choices of the shape control parameters will also be discussed in this paper.

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.