Total Nuclear Variation and Jacobian Extensions of Total Variation for Vector Fields

Area of Science:

• Image processing
• Computer vision
• Applied mathematics

Background:

• Vectorial total variation (VTV) measures are used for image processing tasks.
• Existing VTV methods face challenges like color smearing and affine-coupling bias.

Purpose of the Study:

• To theoretically analyze VTV measures and their properties.
• To propose a new VTV regularizer, Total Nuclear Variation (TNV), for improved image denoising.
• To investigate the use of extended Jacobians for enhanced VTV performance.

Main Methods:

• Theoretical analysis of VTV measures and Jacobian rank minimization.
• Introduction of Total Nuclear Variation (TNV) using nuclear norm as a convex regularizer.
• Implementation of extended Jacobians for larger neighborhood analysis.
• Development of efficient VTV optimization algorithms.

Main Results:

• Color smearing and affine-coupling bias are fundamental to smoothing vector direction.
• Minimizing Jacobian rank is equivalent to encouraging shared gradient directions across vector channels.
• TNV demonstrated superior performance compared to other VTV regularizers in color image denoising.
• Extended Jacobians further improved TNV's denoising performance.
• TNV achieved competitive results against the non-local means method, often showing significant improvements in dB.

Conclusions:

• TNV is an optimal convex regularizer for enforcing shared gradient directions in vector fields.
• Extended Jacobians enhance the effectiveness of VTV-based image denoising.
• TNV offers a promising and efficient approach for color image denoising.