School of Mathematical Sciences

Image of Kewei Zhang

Kewei Zhang

Professor of Mathematical Analysis, Faculty of Science


Research Summary

Hausdorff stable Geometric methods for approximations, interpolations from general compact sets, scattered data and level set information, inpainting, image processing, singularity extraction from… read more

Recent Publications

UK Patent: Title: Image Processing,

Publication Number: GB2488294.

Proprietors: Kewei Zhang, Antonio Orlando and Elaine Crooks,

Inventors: Kewei Zhang, Antonio Orlando and Elaine Crooks,

Publication date: 28/10/2015.

KEA: Geometric Image & Data Analysis

Current Research

Hausdorff stable Geometric methods for approximations, interpolations from general compact sets, scattered data and level set information, inpainting, image processing, singularity extraction from images and geometric objects, multiscale medial axis and geometric interrogation.

Partial differential inclusions and applications to forward-backward diffusion equations in image processing and material microstructure and to the coercivity problem for linear elliptic systems of partial differential equations.

Past Research

Research Track Record

1. Quasimonotone mapping and elliptic systems. The notion of quasimonotone mappings for elliptic systems was introduced in [Z88a] which generalized the concept of quasiconvexity for functions due to Morrey and the existence of weak solutions under the strong-quasimonotonicity condition was established. The aim was to explore the possibility for solving elliptic systems which do not satisfy the usual monotonicity condition. Independently, M. Fuchs defined the same notion he called the `natural ellipticity condition' and established a partial regularity result. Quasimonotone mappings was later shown by Landes to be the necessary and sufficient condition for pseudomonotonicity for elliptic systems.

2. Counterexample for Garding's inequality. I gave an counterexample in [Z89b, Z89c] showing that under the Legendre-Hadamard strong ellipticity condition, Garding's inequality (or coercivity) may not hold for linear elliptic systems if the coefficients are in L. This was an open problem asked by M. Giaquinta. This indicates that for general elliptic systems, it is not possible to linearize the system at a non-smooth solution and apply the implicit function theorem. A later result [Z96] shows that the BMO-seminorm of the coefficients gives necessary conditions for coercivity.

3. Biting theorem for Jacobian and compensated compactness. I established the weak continuity of Jacobians in the sense of the biting lemma [Z90b] (also see [BZ90] for an alternative approach). This was a major new discovery since John Ball's work on weak continuity of Jacobian in 1977. The result led to the further discovery of higher integrability of Jacobian by Muller and marked the beginning of an intensive study on the connections between the theory of compensated compactness of Tartar-Murat and the Hardy spaces in harmonic analysis. These connections became the key for the regularity theory for harmonic maps obtained by Helein, Evans and many others.

4. Implicit function theorem and minimizer. An important discovery on connections between two different approaches to nonlinear elastostatics was made in [Z91a]. I showed that under reasonable conditions, the solution obtained by the traditional implicit function method and the minimizer of the variational method due to John Ball are the same. This has also been the only non-trivial regularity result for the minimizers for nonlinear elasticity models under the polyconvexity assumption up to now.

5. Zhang Lemma. A very useful mathematical tool, now known as Zhang's Lemma was established in [Z92c] showing that for any bounded W01,1 sequences whose gradients approach a compact set of matrices, the sequence can be truncated into a bounded sequence in W1,∞ which shares the same oscillation behaviour as the original one. This lemma has many applications in later studies of variational approach to material microstructure and the study of gradient Young measures.

6. Krein-Milman theorem and the equal hull property. The notion of quasiconvex extreme points was introduced in [Z98c] for compact sets of matrices and the well-known Krein-Milman theorem in convex analysis was established for quasiconvex sets. This notion has found interesting applications in searching of `optimal' existence theorems for partial differential inclusions by Dacorogna-Marcellini and Muller-Sychev's work.

Another surprising observation on quasiconvex hull was made in [Z98b] which is now known as the equal hull property. The result also has interesting consequences in the study of partial differential inclusions. Given a compact set K of N× n matrices, there are a number of useful notions of semiconvex hulls in the calculus of variations. They are the closed lamination convex hull, Lc(K), the rank-one convex hull R(K), the quasiconvex hull, Q(K), the polyconvex hull P(K) and the convex hull C(K), satisfying:

Lc(K)⊂ R(K)⊂ Q(K)⊂ P(K)⊂ C(K).

It was proved in [Z98b] that in general, Q(K)=C(K) if and only if Lc(K)=C(K). If N=n=2, P(K)=C(K) if and only if Lc(K)=C(K).

7. Harmonic analysis method for elliptic systems, calculus of variations, compensated compactness and microstructure. The maximal function method was developed by Acerbi and Fusco in the study of weak lower semicontinuity problems for multiple integrals in the calculus of variations. I have developed this tools to be applicable to many other problems [Z88a, Z90b, Z92b-c, Z97c, Z04a]. Basic theory of singular integral operators has been used in several estimates in my study of quasiconvexity [Z97b, Z98a, Z06b].

The connection of general quadratic forms in the theory of compensated compactness and Hardy spaces was established in the joint work [LMWZ97] with C. Li, J. Hogan, A. McIntosh and Z. Wu where paracommutators and Lojasiewicz Lemma were both used to remove a constant rank restriction in the earlier work of Coifman, Lions Meyer and Semmes. Further optimal results on global higher integrability of Jacobian [HLMZ00] were obtained by using Orlicz-Sobolev spaces.

8. The two well model - mountain pass point and separation of Young measure. The first and the simplest multiwell model concerned with the two-point set K={A, B} of matrices. Ball and James showed in 1987 that if a sequence of gradients approach the two-point set K with rank(A-B)>1, then the sequence of gradients can only approach either A almost everywhere or B almost everywhere in the domain. Kohn found the explicit formula for the quasiconvex envelope of W(X)=Qf(X) where f(X)=min{ |X-A|2, |X-B|2}. It was later showed in [Z01b] that the gradient DW(X) is a quasimonotone mapping. Furthermore, the variational integral I(u)=∫Ω W(Du)+g • u dx, under the homogeneous Neumann condition has at least three critical points when the dead-load perturbation g is small. The functional has a global minimizer, a local minimizer and a mountain pass point.

Another interesting result based on the two-point set K={A,B} led to a further understanding of how to control oscillations of sequences of gradients or the gradient Young measures. It was established in [Z03a] that if rank(A-B)>1, then one can give an estimate of the ε-neighbourhood Kε of K which consists of two balls Kε= B (A,ε) ∪ = B(B,ε) such that Kε separates gradient Young measures, that is if Duj approaches Kε, then it approaches either B(A,ε) a.e. or B(A,ε) a.e.

9. Topological restrictions on subspaces without rank-one matrices. The notion of subspaces of matrices without rank-one matrices can be viewed as the simplest example for sets which do not support any non-trivial gradient Young measures. Let Mm × n be the space of m× n real matrices. Let l(m,n) be the largest possible dimension of subspaces of matrices without rank-one matrices. It was established by E. Rees in 1996 that l(m,n)≥ (n-1)(m-1). However, for a given subspace E ⊂ Mm × n without rank-one matrices, a natural problem is to find universal lower bound of the dimension χ(m,n) of the largest subspace without rank-one matrices that contains E. Obviously, l(m,n)≥ χ(m,n). In [Z01c] I showed that χ(m,n)≥ (m-1)(n-1) and in the physically interesting case m=n=3, one has χ(3,3)=4 which is sharp. While in this case l(3,3)=5.

Let E⊂ Mm × n be any linear subspace. It was established [Z04b] that there exists non-trivial quasiconvex/rank-one convex functions in the form X → f(PE(X)) if and only if PE(V(m,n))≠ E, where PE(V(m,n)) is the closure of the image set of the orthogonal projection for the rank-one cone to E. In particular, I proved that dim(E)>n+m-1 implies PE(V(m,n))≠ E. This result is a consequence of Tarski-Seidenberg theorem in real algebraic geometry.

The modelling of microstructure using geometrically linear elasticity goes back to 1960's where the linear strain e(A)=(A+AT)/2=PE(A) (A ∈ M3× 3) was used to replace the deformation gradient, where PE is the orthogonal projection to the subspace E of 3× 3 symmetric matrices. The notion of compatible strain (when A is a rank-one matrix) is very useful for construction laminated microstructure. Naturally, one might ask: how large the subspaces of symmetric matrices without compatible strain would be. The answer was provided in [Z03b] showing that such subspaces can be at most one-dimensional.

10. Partial Differential Inclusions, forward-backward diffusion equations and the Perona-Malik model. Perona-Malik anisotropic diffusion edge-enhancement model (1990) is arguably the most influential and intensively studied evolutional model for edge enhancement by the image processing community. The model is a non-coercive forward-backward parabolic equation under the Neumann boundary condition. A main unsolved mathematical problem is on the well-posedness of the model, such as existence and uniqueness of solutions in the two-dimensional case. Most of the effort (in mathematics) so far has been on the one-dimensional version of the problem. The existence of a unique smooth solution was established by Kawohl and Kutev (1998) for initial values with small derivative.

By using the partial differential inclusion method, I established in [Z04d] the existence of infinitely many W1,∞-weak solutions for the one-dimensional version of the Perona-Malik model ut=(σ(ux))x, where σ(s)=s/(1+s2). The result was established for all non-constant smooth initial value u0 under the homogeneous Neumann condition. This result is not only the first nontrivial existence result for the above problem, but also an instability result concerning possible numerical schemes.

An important tool for solving partial differential inclusion systems such as the Peronal-Malik equation is the so-called `controlled L convergence implies almost everywhere convergence for gradient' principle for Lipschiz mappings developed by Muller-Sychev, and Kirchheim. I found a natural generalization of this principle to a be applicable for mappings in BV∩ L in order to solve partial differential inclusion problems for unbounded sets [Z05c].

11. Compensated convexity and its applications. In [Z07b], I introduced the notions of lower and upper quadratic compensated convex transforms Cl2,λ(f) and Cu2(f) respectively and the mixed transforms by composition of these transforms for a given function f:RnR and for possibly large λ>0. I studied general properties of such transforms, including the so-called `tight' approximation of Cl2(f) to f as λ→ +∞ and compare our transforms with the well-known Moreau-Yosida regularization (Moreau envelope) and the Lasry-Lions regularization. I also studied analytic and geometric properties for both the quadratic lower transform Cl2( dist2(x,K)) of the squared-distance function to a compact set K and the quadratic upper transform Cu2(f) for any convex function f of at most quadratic growth. I showed that both Cl2( dist2(x,K)) and Cu2(f) are C1,1 approximations of the original functions for large λ>0 and Cu2(f) remains convex. Explicitly calculated examples of quadratic transforms were given, including the the lower transform of squared distance function to a finite set and upper transform for some non-smooth convex functions in mathematical programming.

Future Research

Further development of geometric methods for image processing and geometric shape analysis. Applications to engineering and medical imaging and geosciences.

Partial differential equation methods for image processing.

School of Mathematical Sciences

The University of Nottingham
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Nottingham, NG7 2RD

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