Correlation properties of the vector signal representation for speckle pattern

Area of Science:

• Signal processing
• Mathematical analysis
• Metrology

Background:

• The complex analytic signal, derived from a real-valued signal and its Hilbert transform, is crucial for one-dimensional (1D) signal analysis and statistical analysis.
• The Riesz transform, a generalization of the Hilbert transform, extends analytic signal concepts to high-dimensional signal processing by creating vector signal representations.

Purpose of the Study:

• To introduce vector correlations as novel mathematical tools for vector calculus.
• To establish a mathematical foundation for vector analysis in statistical speckle metrology.

Main Methods:

• Development of vector correlation as a mathematical tool for vector calculus.
• Application of vector correlations to real-valued speckle patterns for statistical analysis.
• Investigation of correlation properties associated with vector correlations.

Main Results:

• Introduction of vector correlations for high-dimensional signal processing.
• Demonstration of vector correlations' utility in statistical speckle analysis.
• Establishment of correlation properties for vector analysis in speckle metrology.

Conclusions:

• Vector correlations offer a new mathematical framework for high-dimensional signal analysis.
• The introduced vector correlations provide a foundational basis for vector analysis within speckle metrology.
• This work extends the traditional analytic signal concept to vector signal representations for advanced metrology applications.