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l(p)-Multiresolution analysis: how to reduce ringing and sparsify the error.

Arrate Muñoz1, Thierry Blu, Michael Unser

  • 1Biomed. Imaging Group, Swiss Fed. Inst. of Technol. Lausanne, Switzerland. arrate.munoz@epfl.ch

IEEE Transactions on Image Processing : a Publication of the IEEE Signal Processing Society
|February 5, 2008
PubMed
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This study introduces a new image pyramid reduction operator to minimize approximation errors using L(p)-sense, applicable for non-integer p values. Results show reduced ringing and sparser detail images for p near 1 compared to standard methods.

Area of Science:

  • Image processing and computer vision
  • Mathematical analysis and optimization

Background:

  • Image pyramids are crucial for multi-scale image analysis.
  • Traditional methods often use L(2)-sense minimization, which can lead to artifacts like ringing.
  • Non-integer L(p)-norms offer potential for improved image representation and processing.

Purpose of the Study:

  • To design an image pyramid reduction operator that minimizes approximation error in the L(p)-sense for non-integer p.
  • To explore the impact of varying p on image reconstruction quality and artifact reduction.
  • To develop an efficient and convergent algorithm for this optimization problem.

Main Methods:

  • Formulating the image pyramid reduction as an L(p)-norm minimization problem.
  • Utilizing shift-invariant basis functions (e.g., B-splines) for the image model.

Related Experiment Videos

  • Developing an iterative optimization algorithm incorporating digital filtering and derivative acceleration.

    • Main Results:

    • The proposed method provides a well-defined solution through an iterative optimization algorithm.
    • Convergence of the algorithm is accelerated using first and second-order derivatives.
    • For L(p)-norms with p close to 1, significant reduction in ringing artifacts was observed.
    • Detail images generated with p near 1 exhibited increased sparsity compared to the standard L(2) case.

    Conclusions:

    • The designed reduction operator effectively minimizes approximation errors in the L(p)-sense for non-integer p.
    • The L(p)-sense minimization, particularly for p near 1, offers advantages in reducing visual artifacts and enhancing image sparsity.
    • The iterative optimization algorithm provides an efficient method for implementing this approach in image processing applications.