# Isometric Signal Processing under Information Geometric Framework

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Intrinsic Parameter Submanifold

**Definition 1**(Intrinsic parameter submanifold)

**.**

**Remark**

**1.**

## 3. Signal Processing on the Intrinsic Parameter Submanifold

#### 3.1. Geometric Structure Change by Signal Processing

**Proof.**

**Lemma**

**1.**

**Proof.**

**Corollary**

**1.**

**Proof.**

#### 3.2. Isometric Signal Processing

**Definition 2**(Isometry)

**.**

**Theorem**

**1.**

**Proof.**

**Corollary**

**2.**

**Proof.**

## 4. Linear Form of Signal Processing

#### 4.1. Model Formulation

#### 4.2. Fisher Information Loss of Linear Signal Processing

#### 4.2.1. White Gaussian Noise

#### 4.2.2. Colored Gaussian Noise

#### 4.3. The Construction of the Isometric Linear Form of Signal Processing

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Remark**

**2.**

#### Sample of the Construction

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Rao, C.R. Information and the Accuracy Attainable in the Estimation of Statistical Parameters. In Breakthroughs in Statistics: Foundations and Basic Theory; Kotz, S., Johnson, N.L., Eds.; Springer: New York, NY, USA, 1992; pp. 235–247. [Google Scholar]
- Chentsov, N.N. Statistical Decision Rules and Optimal Inference; Number v. 53 in Translations of Mathematical Monographs; American Mathematical Society: Providence, RI, USA, 1982. [Google Scholar]
- Efron, B. Defining the Curvature of a Statistical Problem (with Applications to Second Order Efficiency). Ann. Stat.
**1975**, 3, 1189–1242. [Google Scholar] [CrossRef] - Efron, B. The Geometry of Exponential Families. Ann. Stat.
**1978**, 6, 362–376. [Google Scholar] [CrossRef] - Amari, S.I. Information Geometry and Its Applications, 1st ed.; Springer Publishing Company, Incorporated: Berlin, Germany, 2016. [Google Scholar]
- Chern, S.S.; Chen, W.H.; Lam, K.S. Lectures on Differential Geometry. Ann. Inst. Henri Poincare-Phys. Theor.
**2014**, 40, 329–342. [Google Scholar] - Rong, Y.; Tang, M.; Zhou, J. Intrinsic Losses Based on Information Geometry and Their Applications. Entropy
**2017**, 19, 405. [Google Scholar] [CrossRef] - Cheng, Y.; Wang, X.; Caelli, T.; Moran, B. Tracking and Localizing Moving Targets in the Presence of Phase Measurement Ambiguities. IEEE Trans. Signal Process.
**2011**, 59, 3514–3525. [Google Scholar] [CrossRef] - Cheng, Y.; Wang, X.; Caelli, T.; Li, X.; Moran, B. Optimal Nonlinear Estimation for Localization of Wireless Sensor Networks. IEEE Trans. Signal Process.
**2011**, 59, 5674–5685. [Google Scholar] [CrossRef] - Cheng, Y.; Wang, X.; Moran, B. Optimal Nonlinear Estimation in Statistical Manifolds with Application to Sensor Network Localization. Entropy
**2017**, 19, 308. [Google Scholar] [CrossRef] - Smith, S.T. Covariance, subspace, and intrinsic Cramér-Rao bounds. IEEE Trans. Signal Process.
**2005**, 53, 1610–1630. [Google Scholar] [CrossRef] - Wang, L.; Wong, K.K.; Wang, H.; Qin, Y. MIMO radar adaptive waveform design for extended target recognition. Int. J. Distrib. Sens. Netw.
**2016**, 2015, 84. [Google Scholar] [CrossRef] - Manton, J.H. Optimization algorithms exploiting unitary constraints. IEEE Trans. Signal Process.
**2002**, 50, 635–650. [Google Scholar] [CrossRef] - Abrudan, T.E.; Eriksson, J.; Koivunen, V. Steepest Descent Algorithms for Optimization Under Unitary Matrix Constraint. IEEE Trans. Signal Process.
**2008**, 56, 1134–1147. [Google Scholar] [CrossRef] - Abrudan, T.; Eriksson, J.; Koivunen, V. Conjugate gradient algorithm for optimization under unitary matrix constraint. Signal Process.
**2009**, 89, 1704–1714. [Google Scholar] [CrossRef] - Barbaresco, F. Innovative Tools for Radar Signal Processing Based on Cartan’s Geometry of SPD Matrices and Information Geometry. In Proceedings of the Radar Conference, Rome, Italy, 26–30 May 2008; pp. 1–6. [Google Scholar]
- Barbaresco, F. Robust statistical Radar Processing in Fréchet metric space: OS-HDR-CFAR and OS-STAP Processing in Siegel homogeneous bounded domains. In Proceedings of the International Radar Symposium, Leipzig, Germany, 7–9 September 2011; pp. 639–644. [Google Scholar]
- Wu, H.; Cheng, Y.; Hua, X.; Wang, H. Vector Bundle Model of Complex Electromagnetic Space and Change Detection. Entropy
**2018**, 21, 10. [Google Scholar] [CrossRef] - Hua, X.; Cheng, Y.; Wang, H.; Qin, Y.; Li, Y.; Zhang, W. Matrix CFAR detectors based on symmetrized Kullback-Leibler and total Kullback-Leibler divergences. Digit. Signal Process.
**2017**, 69, 106–116. [Google Scholar] [CrossRef] - Hua, X.; Fan, H.; Cheng, Y.; Wang, H.; Qin, Y. Information Geometry for Radar Target Detection with Total Jensen-Bregman Divergence. Entropy
**2018**, 20, 256. [Google Scholar] [CrossRef] - Dong, G.; Kuang, G.; Wang, N.; Wang, W. Classification via Sparse Representation of Steerable Wavelet Frames on Grassmann Manifold: Application to Target Recognition in SAR Image. IEEE Trans. Image Process.
**2017**, 26, 2892–2904. [Google Scholar] [CrossRef] [PubMed] - Zegers, P. Fisher Information Properties. Entropy
**2015**, 17, 4918–4939. [Google Scholar] [CrossRef] - Zamir, R. A proof of the Fisher information inequality via a data processing argument. IEEE Trans. Inf. Theory
**1998**, 44, 1246–1250. [Google Scholar] [CrossRef] - Blackwell, D. Conditional Expectation and Unbiased Sequential Estimation. Ann. Math. Stat.
**1947**, 18, 105–110. [Google Scholar] [CrossRef] - Lehmann, E.L.; ScheffÉ, H. Completeness, Similar Regions, and Unbiased Estimation-Part I. In Selected Works of E. L. Lehmann; Rojo, J., Ed.; Springer US: Boston, MA, USA, 2012; pp. 233–268. [Google Scholar]
- Lehmann, E.L.; ScheffÉ, H. Completeness, Similar Regions, and Unbiased Estimation—Part II. In Selected Works of E. L. Lehmann; Rojo, J., Ed.; Springer US: Boston, MA, USA, 2012; pp. 269–286. [Google Scholar]
- Kay, S.M. Fundamentals of statistical signal processing: Estimation theory. Control Eng. Pract.
**1994**, 37, 465–466. [Google Scholar]

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wu, H.; Cheng, Y.; Wang, H.
Isometric Signal Processing under Information Geometric Framework. *Entropy* **2019**, *21*, 332.
https://doi.org/10.3390/e21040332

**AMA Style**

Wu H, Cheng Y, Wang H.
Isometric Signal Processing under Information Geometric Framework. *Entropy*. 2019; 21(4):332.
https://doi.org/10.3390/e21040332

**Chicago/Turabian Style**

Wu, Hao, Yongqiang Cheng, and Hongqiang Wang.
2019. "Isometric Signal Processing under Information Geometric Framework" *Entropy* 21, no. 4: 332.
https://doi.org/10.3390/e21040332