Optimal image alignment with random projections of manifolds: algorithm and geometric analysis

Area of Science:

• Computer Vision
• Image Processing
• Optimization

Background:

• Image alignment is crucial for tasks like image stitching and object recognition.
• Existing methods can be computationally intensive or sensitive to noise.
• Compressed image representations pose unique alignment challenges.

Purpose of the Study:

• To develop an efficient image alignment method using random linear measurements.
• To formulate image alignment as a manifold distance minimization problem.
• To leverage difference of convex (DC) programming for robust transformation estimation.

Main Methods:

• Representing image transformations on a manifold defined by sparse dictionary representations.
• Decomposing the objective function into a difference of convex functions (DC) for Gaussian dictionaries.
• Solving the resulting DC program globally using a cutting plane method.

Main Results:

• The proposed method effectively estimates relative transformations (shift, rotation, scaling).
• The objective function's DC property enables global optimization.
• Bounded curvature of the transformation manifold ensures optimal solution quality.

Conclusions:

• Random measurements combined with DC programming offer an effective approach to image alignment.
• The method provides a robust and computationally feasible solution for compressed image data.
• This work advances the field of image registration with a theoretically sound optimization framework.