Restricted Conformal Property of Compressive Sensing

TL;DR

This paper introduces the Restricted Conformal Property (RCP) in compressive sensing, which preserves signal directions via angle preservation, complementing the energy-focused Restricted Isometry Property (RIP).

Contribution

The paper proposes RCP as a novel property that captures signal direction information, extending the theoretical framework of compressive sensing beyond energy preservation.

Findings

RCP can accurately reflect the direction of signals.

Two proofs of RCP, RCP_JL and RCP_IP, are provided.

RCP enhances the understanding of signal properties in compressive sensing.

Abstract

Energy and direction are tow basic properties of a vector. A discrete signal is a vector in nature. RIP of compressive sensing can not show the direction information of a signal but show the energy information of a signal. Hence, RIP is not complete. Orthogonal matrices can preserve angles and lengths. Preservation of length can show energies of signals like RIP do; and preservation of angle can show directions of signals. Therefore, Restricted Conformal Property (RCP) is proposed according to preservation of angle. RCP can show the direction of a signal just as RIP shows the energy of a signal. RCP is important supplement and development of RIP. Tow different proofs of RCP are given, namely, RCP_JL and RCP_IP.