Fast image deconvolution using closed-form thresholding formulas of regularization

Wenfei Cao, Jian Sun, Zongben Xu

Abstract: In this paper, we focus on the research of fast deconvolution algorithm based on the non-convex Lq(q=12,23)<math><mrow is="true"><msub is="true"><mrow is="true"><mi is="true">L</mi></mrow><mrow is="true"><mi is="true">q</mi></mrow></msub><mo stretchy="false" is="true">(</mo><mi is="true">q</mi><mo is="true">=</mo><mfrac is="true"><mrow is="true"><mn is="true">1</mn></mrow><mrow is="true"><mn is="true">2</mn></mrow></mfrac><mtext is="true">,</mtext><mfrac is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mn is="true">3</mn></mrow></mfrac><mo stretchy="false" is="true">)</mo></mrow></math> sparse regularization. Recently, we have deduced the closed-form thresholding formula for L12<math><mrow is="true"><msub is="true"><mrow is="true"><mi is="true">L</mi></mrow><mrow is="true"><mfrac is="true"><mrow is="true"><mn is="true">1</mn></mrow><mrow is="true"><mn is="true">2</mn></mrow></mfrac></mrow></msub></mrow></math> regularization model (Xu (2010) [1]). In this work, we further deduce the closed-form thresholding formula for the L23<math><mrow is="true"><msub is="true"><mrow is="true"><mi is="true">L</mi></mrow><mrow is="true"><mfrac is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mn is="true">3</mn></mrow></mfrac></mrow></msub></mrow></math> non-convex regularization problem. Based on the closed-form formulas for Lq(q=12,23)<math><mrow is="true"><msub is="true"><mrow is="true"><mi is="true">L</mi></mrow><mrow is="true"><mi is="true">q</mi></mrow></msub><mo stretchy="false" is="true">(</mo><mi is="true">q</mi><mo is="true">=</mo><mfrac is="true"><mrow is="true"><mn is="true">1</mn></mrow><mrow is="true"><mn is="true">2</mn></mrow></mfrac><mtext is="true">,</mtext><mfrac is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mn is="true">3</mn></mrow></mfrac><mo stretchy="false" is="true">)</mo></mrow></math> regularization, we propose a fast algorithm to solve the image deconvolution problem using half-quadratic splitting method. Extensive experiments for image deconvolution demonstrate that our algorithm has a significant acceleration over Krishnan et al.’s algorithm (Krishnan et al. (2009) [3]). Moreover, the simulated experiments further indicate that L23<math><mrow is="true"><msub is="true"><mrow is="true"><mi is="true">L</mi></mrow><mrow is="true"><mfrac is="true"><mrow is="true"><mn is="true">2</mn></mrow><mrow is="true"><mn is="true">3</mn></mrow></mfrac></mrow></msub></mrow></math> regularization is more effective than L0,L12<math><mrow is="true"><msub is="true"><mrow is="true"><mi is="true">L</mi></mrow><mrow is="true"><mn is="true">0</mn></mrow></msub><mtext is="true">,</mtext><msub is="true"><mrow is="true"><mi is="true">L</mi></mrow><mrow is="true"><mfrac is="true"><mrow is="true"><mn is="true">1</mn></mrow><mrow is="true"><mn is="true">2</mn></mrow></mfrac></mrow></msub></mrow></math> or L1<math><mrow is="true"><msub is="true"><mrow is="true"><mi is="true">L</mi></mrow><mrow is="true"><mn is="true">1</mn></mrow></msub></mrow></math> regularization in image deconvolution, andL12<math><mrow is="true"><msub is="true"><mrow is="true"><mi is="true">L</mi></mrow><mrow is="true"><mfrac is="true"><mrow is="true"><mn is="true">1</mn></mrow><mrow is="true"><mn is="true">2</mn></mrow></mfrac></mrow></msub></mrow></math> regularization is competitive to L1<math><mrow is="true"><msub is="true"><mrow is="true"><mi is="true">L</mi></mrow><mrow is="true"><mn is="true">1</mn></mrow></msub></mrow></math> regularization and better than L0<math><mrow is="true"><msub is="true"><mrow is="true"><mi is="true">L</mi></mrow><mrow is="true"><mn is="true">0</mn></mrow></msub></mrow></math> regularization.