Generalized reaction–diffusion equations

This Letter proposes generalized reaction–diffusion equations for treating noisy magnetic resonance images. An edge-enhancing functional is introduced for image enhancement. A number of super-diffusion operators are introduced for fast and effective smoothing. Statistical information is utilized for robust edge-stopping and diffusion-rate estimation. A quasi-interpolating wavelet algorithm is utilized for numerical computations. Computer experiments indicate that the present algorithm is efficient for edge detecting and noise removing. The generalized reaction–diffusion equations have potential applications in modeling boundary-restricted diffusion, multiphase chemical reactions and reaction–diffusion in porous media. q 1999 Elsevier Science B.V. All rights reserved. It is well known that the evolution of an image Ž . under certain partial differential equation PDE opw x erators is an image processing operation 1–3 . Correspondingly, an image processing operation, under appropriate restrictions, can be regarded as action of w x a PDE operator 4 . This link between PDEs and image processing has stimulated much interest in Ž . both fields. The use of linear PDE operators in image processing began with the diffusion operator and has its roots in the theory of multiscale analysis and Gaussian smoothing initiated by Rosenfeld and w x w x Thurston 1 and others 2,3 . It can be proved that the heat equation satisfies the maximum and minimum principles, which means the maximum and minimum can only be attained for the initial image w x data or at the boundary of an image 5 . Hence, the ) Corresponding author. Fax: q65 774 6756; e-mail: [email protected] edges of an image may lose definition under the action of the diffusion operator. w x Perona and Malik 6 addressed the edge loss Ž . problem by introducing a non-linear anisotropic w x diffusion operator 6 . Such a formalism has stimuw x lated a great deal of interest 7–20 . It is commonly believed that the Perona–Malik algorithm provides a potential means for noise removing, edge detection, image segmentation and enhancement. The basic idea behind the Perona–Malik algorithm is to regard Ž . an original image I r as evolving under an edgestopping diffusion operator Eu r ,t Ž . < < s=P d =u r ,t =u r ,t , Ž . Ž . Ž . Et u r ,0 s I r , 1 Ž . Ž . Ž . Ž < <. where d =u is a generalized diffusion coefficient which is so designed that it limits the diffusion near edges while allowing lots of diffusion in regions where the noise is to be suppressed. The edges of an 0009-2614r99r$ see front matter q 1999 Elsevier Science B.V. All rights reserved. Ž . PII: S0009-2614 99 00270-5 ( ) G.W. WeirChemical Physics Letters 303 1999 531–536 532 image are automatically detected by using a diffusion rate based on approximating the norm of the gradient across edges with the norm of the full image gradient. Examples of such coefficients, as suggested originally by Perona and Malik, include 2 < < =u < < d =u sexp y , 2 Ž . Ž . 2 2h