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Total Variation Based Magnetic Field Mapping

Total Variation Based Magnetic Field Mapping. Zhuo Zheng Joint work with C. Anand and T. Terlaky Advanced Optimization Lab. Outline. Motivation. Phase shift in the ideal case. Field mapping methods. Phase shift in the presence of noise. Total variation. Conclusions. Motivation.

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Total Variation Based Magnetic Field Mapping

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  1. Total Variation Based Magnetic Field Mapping Zhuo Zheng Joint work with C. Anand and T. Terlaky Advanced Optimization Lab

  2. Outline • Motivation Phase shift in the ideal case • Field mapping methods Phase shift in the presence of noise Total variation • Conclusions

  3. Motivation • Magnetic fields determine the resonant frequency • The field inhomogeneity is inevitable • It causes geometric distortions and artifacts in images • SSFP imaging is quite sensitive to the resonant frequency

  4. Correction Methods Steps Involved By acquiring two images at different TEs Use of the field map to compensate in the reconstruction Measure spatial variation Field Compensation Reasonably good Images

  5. Previous Model Revisited • Steady-state model established

  6. The shift Theorem

  7. Search And Position (SAP) • Perform pixel-wise evaluation to locate a certain values • No noise in the ideal case • It results in a slightly different steady-state matrix S • We introduce a new P in the presence of the field variation

  8. Search And Position (SAP) Modeling in the rotating frame of reference

  9. Search And Position (SAP) Suppose we have a ,We need to know where it locates in the image

  10. Search And Position (SAP)

  11. Noise Corruption Case • Measurement becomes • A least square program has been formulated Subject to some constraints. Numerical results show that it works !

  12. Total Variation Approach • First proposed by Rudin, Osher and Fatemi • Total variation based noise removal algorithms • Denoising can be solved as a constrained optimization program Objective: TV norm of the denoised image Constraint: deviation from the observation

  13. Total Variation Approach Model Equation: Optimization Problem:

  14. Total Variation Approach However, we are concerned about the signal instead of the image Therefore, our model should be:

  15. Total Variation Approach However, we are concerned about the signal instead of the image Therefore, our model should be: We formulate the nonconstrained problem and derive the optimality condition.

  16. Total Variation Approach Therefore, our model should be: We formulate the nonconstrained problem and derive the optimality condition.

  17. Total Variation Approach Euler-Lagrange Equation: Substitute into the expression of field map term and tissue density, we try to solve a hard optimization program.

  18. Euler-Lagrange Equation: Substitute into the expression of field map term and tissue density, we try to solve a hard optimization program.

  19. Conclusions • We explore several methods to generate a field map • Total variation based method is promising in our case • Better formulation has to be called for

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