Mathematical Problems in Engineering

Volume 2015, Article ID 268295, 10 pages

http://dx.doi.org/10.1155/2015/268295

## Approximate Image Reconstruction in Landscape Reflection Imaging

^{1}Laboratoire Equipes de Traitement de l’Information et Systèmes (ETIS), ENSEA/Université de Cergy-Pontoise/CNRS UMR 8051, 95302 Cergy-Pontoise, France^{2}Institute of Applied Mathematics, University of Saarland, 66041 Saarbrücken, Germany

Received 8 February 2015; Accepted 24 June 2015

Academic Editor: Franklin A. Mendivil

Copyright © 2015 Rémi Régnier et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Simple reflection imaging of landscape (scenery or extended objects) poses the inverse problem of reconstructing the landscape reflectivity function from its integrals on some particular family of spheres. Such data acquisition is encoded in the framework of a Radon transform on this family of spheres. In spite of the existence of an exact inversion formula, the numerical landscape reflectivity function reconstitution is best obtained with an approximate but judiciously chosen reconstruction kernel. We describe the working of this reflection imaging modality and its theoretical handling, introduce an efficient and stable image reconstruction algorithm, and present simulation results to prove the validity of this choice as well as to demonstrate the feasibility of this imaging process.

#### 1. Introduction

Imaging science is a rapidly developing field in all areas of human activity ranging from medical diagnostics to industrial nondestructive evaluation. In the last several decades, it has expanded vigorously in environmental/navigational surveillance, national security monitoring, weather forecast, hazard assessment, and so forth. It plays an essential role in remote sensing by providing information about objects or areas from a distance, typically from aircraft or satellites. By collecting data across a wide range of the electromagnetic spectrum at small spectral resolution (5–15 nm) and high spatial resolution (1–5 m), it allows detailed spectral signatures to be identified for different imaged materials.

So the aim is to obtain rapidly accurate images of large areas (or landscapes/sceneries) of the earth surface. Two main technologies have been conceived to this end: aerial or satellite photography (including television imaging) [1] and radar imaging, in particular the so-called Synthetic Aperture Radar Imaging (SAR) [2], which has the advantage of being weather independent.

The aim of this paper is not to focus on neither aerial photography nor SAR imaging and discuss their specific functioning problems. Its object is to single out an imaging concept based on the phenomena of wave reflection on more or less opaque objects and the registering of reflected wave energy by a single detector. It turns out that the appropriate mathematical description for this imaging modality is an integral transform, which is a generalization of the Radon transform, popular in medicine and industrial control. The crucial point is to show that imaging with this principle is viable and exploitable in practice.

The paper is organized as follows. Section 2 is devoted to defining active reflection imaging. It discusses the way information is recorded and used to produce images. Section 3 reviews the main mathematical tool which supports this active reflection imaging: the Radon transform on spheres centered on a plane. The next point is the derivation of an approximate reconstruction formula for the reflectivity function in Section 4. Numerical simulations and comparison comments on the results obtained by the exact and approximate reconstruction formulas are given in Section 5. A conclusion closes the papers with perspectives on possible future research directions.

#### 2. Reflection Imaging

Reflection imaging is the simplest way to acquire an image of an extended object or a landscape of macroscopic dimensions. Viewing an object under the illumination of a light is the simplest example of reflection imaging. More generally if a signal (or wave pulse) is sent through space and gets reflected by an opaque surface before being recorded by an apparatus, information on the presence of this surface may be obtained provided that the signal propagation properties are known. A scanning of the object by a large number of such signals and their detection may allow us to obtain some image of this object. This is the principle of reflection imaging.

One important mode of reflection imaging known to everyone is human vision. Natural light reflected on the surface of objects passes through the eye aperture and gets projected on the retina. This way of producing an image is also used in photographic and television cameras. However there are limitations to this modality of reflection imaging when large objects or sceneries are to be imaged due to weather conditions (precipitation, fog, and clouds), variations of radiation intensity, and distance related blurring [3]. Therefore it may be useful to seek an alternative reflection imaging principle. In this paper, we discuss a simple way of acquiring reflection data to obtain the image of an object.

Concretely, we will be concerned here in particular by large objects such as a landscape. We will consider first the case of flat scenery or landscape before going to the case of a hilly or structural landscape when no aperture (with retina or photographic film) is used. The first case is meant to facilitate the understanding of the mechanism of reflection imaging but is not a topic of main interest in itself.

##### 2.1. Flat Scenery or Flat Landscape

Let us consider a source emitting isotropically and uniformly bursts of signals (wave-packets). This source is at first motionless and is maintained at a constant height ; see Figure 1. Below is a planar landscape represented by a reflectivity function , which gives the percentage of signal energy sent back by reflection at point of the plane. Let us assume that signals travel at a constant velocity along all rectilinear trajectories in air and that can simultaneously register returning signal energies, and this is advocated, for example, in [4], and also called monostatic mode in radar technology. An emitted burst of signals emerges from at time and will expand spherically around at a distance from at time . It is clear that at time , the signal burst hits the floor plane at site and the reflected signal will be detected along the same propagation path at at time . At time , the return signal at is made of all the reflected energies at points situated on a circle centered at of radius . If is the emitted energy flux density at , then the received reflected signal from the two-dimensional landscape is the integral of on the circle . To keep the discussion simple, we have neglected signal spreading and attenuation along the propagation direction. So such circle integral of the reflectivity function is for the moment only a function of one variable . Collecting all such integrals will not be sufficient to find , because it is a function of two variables . To overcome this problem of insufficient data, we can move the point source along a given trajectory (or curve) which will introduce a second variable: the curvilinear abscissa of on its trajectory. One may take the simplest trajectory possible: a straight line parallel to the landscape plane at height . In this situation and with all the stated assumptions, the reflected signal flux density is given by the integral of on the circle , whose center is at abscissa on the orthogonal projection of the line trajectory of on the plane and radius Here is the integration measure of . The totality of for the unknown is what is called the Radon transform of on the family of circles centered on a straight line parallel to the landscape plane. This integral functional transform has been studied by many authors who have worked out the inverse transform; see, for example, [5]. In the inversion of this Radon transform on circles centered on a line, only -even part of can be reconstructed and a special scanning mode is required to obtain the full .