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Recently, an innovative matrix CFAR detection scheme based on informationgeometry, also referred to as the geometric detector, has been developedspeedily and exhibits distinct advantages in several practical applications.These advantages benefit from the geometry of the Toeplitz Hermitian positivedefinite (HPD) manifold $\mathcal{M}_{\mathcal{T}H_{++}}$, but thesophisticated geometry also results in some challenges for geometric detectors,such as the implementation of the enhanced detector to improve the SCR(signal-to-clutter ratio) and the analysis of the detection performance. Tomeet these challenges, this paper develops the dual power spectrum manifold$\mathcal{M}_{\text{P}}$ as the dual space of$\mathcal{M}_{\mathcal{T}H_{++}}$. For each affine invariant geometric measureon $\mathcal{M}_{\mathcal{T}H_{++}}$, we show that there exists an equivalentfunction named induced potential function on $\mathcal{M}_{\text{P}}$. By theinduced potential function, the measurements of the dissimilarity between twomatrices can be implemented on $\mathcal{M}_{\text{P}}$, and the geometricdetectors can be reformulated as the form related to the power spectrum. Theinduced potential function leads to two contributions: 1) The enhancement ofthe geometric detector, which is formulated as an optimization problemconcerning $\mathcal{M}_{\mathcal{T}H_{++}}$, is transformed to an equivalentand simpler optimization on $\mathcal{M}_{\text{P}}$. In the presented exampleof the enhancement, the closed-form solution, instead of the gradient descentmethod, is provided through the equivalent optimization. 2) The detectionperformance is analyzed based on $\mathcal{M}_{\text{P}}$, and the advantageouscharacteristics, which benefit the detection performance, can be deduced byanalyzing the corresponding power spectrum to the maximal point of the inducedpotential function.

Dual Power Spectrum Manifold and Toeplitz HPD Manifold: Enhancement and Analysis for Matrix CFAR Detection