Image Processing for Remote Sensing - Chapter 16 (end)

16 Quality Assessment of Remote-Sensing Multi-Band Optical Images Bruno Aiazzi, Luciano Alparone, Stefano Baronti, and Massimo Selva CONTENTS 16.1 Introduction..................................................................................................................... 355 16.2 Information Theoretic Problem Statement ................................................................. 357 16.3 Information Assessment Procedure............................................................................. 358 16.3.1 Noise Modeling................................................................................................. 358 16.3.2 Estimation of Noise Variance and Correlation............................................ 359 16.3.3 Source Decorrelation via DPCM..................................................................... 362 16.3.4 Entropy Modeling............................................................................................. 363 16.3.5 Generalized Gaussian PDF.............................................................................. 364 16.3.6 Information Theoretic Assessment................................................................. 365 16.4 Experimental Results...................................................................................................... 366 16.4.1 AVIRIS Hyperspectral Data ............................................................................ 366 16.4.2 ASTER Superspectral Data.............................................................................. 370 16.5 Conclusions...................................................................................................................... 374 Acknowledgment....................................................................................................................... 374 References ................................................................................................................................... 374 16.1 Introduction Information theoretic assessment is a branch of image analysis aimed at defining and measuring the quality of digital images and is presently an open problem [1–3]. By resorting to Shannon’s information theory [4], the concept of quality can be related to the information conveyed to a user by an image or, in general, by multi-band data, that is, to the mutual information between the unknown noise-free digitized signal (either radi-ance or reflectance in the visible-near infrared (VNIR) and short-wave infrared (SWIR) wavelengths, or irradiance in the middle infrared (MIR), thermal infrared (TIR), and far infrared (FIR) bands) and the corresponding noise-affected observed digital samples. Accurate estimates of the entropy of an image source can only be obtained provided the data are uncorrelated. Hence, data decorrelation must be considered to suppress or largely reduce the correlation existing in natural images. Indeed, entropy is a measure 355 © 2008 by Taylor & Francis Group, LLC 356 Image Processing for Remote Sensing of statistical information, that is, of uncertainty of symbols emitted by a source. Hence, any observation noise introduced by the imaging sensor results in an increment in entropy, which is accompanied by a decrement of the information content useful in application contexts. Modeling and estimation of the noise must be preliminarily carried out [5] to quantify its contribution to the entropy of the observed source. Modeling of information sources is also important to assess the role played by the signal-to-noise ratio (SNR) in determining the extent to which an increment in radiometric resolution can increase the amount of information available to users. The models that are exploited are simple, yet adequate, for describing first-order statistics of memoryless information sources and autocorrelation functions of noise pro-cesses, typically encountered in digitized raster data. The mathematical tractability of models is fundamental for deriving an information theoretic closed-form solution yield-ing the entropy of the noise-free signal from the entropy of the observed noisy signal and the estimated noise model parameters. This work focuses on measuring the quality of multi-band remotely sensed digitized images. Lossless data compression is exploited to measure the information content of the data. To this purpose, extremely advanced lossless compression methods capable of attaining the ultimate compression ratio, regardless of any issues of computational complexity [6,7], are utilized. In fact, the bit rate achieved by a reversible compression process takes into account both the contribution of the ‘‘observation’’ noise (i.e., informa-tion regarded as statistical uncertainty, whose relevance is null to a user) and the intrinsic information of hypothetically noise-free samples. Once the parametric model of the noise, assumed to be possibly non-Gaussian and both spatially and spectrally autocorrelated, has been preliminarily estimated, the mutual information between noise-free signal and recorded noisy signal is calculated as the difference between the entropy of the noisy signal and the entropy derived from the parametric model of the noise. Afterward, the amount of information that the digitized samples would convey if they were ideally recorded without observation noise is estimated. To this purpose, an entropy model of the source is defined. The inversion of the model yields an estimate of the information content of the noise-free source starting from the code rate and the noise model. Thus, it is possible to establish the extent to which an increment in the radiometric resolution, or equivalently in the SNR, obtained due to technological improvements of the imaging sensor can increase the amount of information that is available to the users’ applications. This objective measurement of quality fits better the subjective concept of quality, that is, the cap-ability of achieving a desired objective as the number of spectral bands increases. Practically, (mutual) information, or equivalently SNR, is the sole quality index for hyperspectral imagery, generally used for detection and identification of materials and spectral anomalies, rather than for conventional multi-spectral classification tasks. The remainder of this chapter is organized as follows. The information theoretic fundamentals underlying the analysis procedure are reviewed in Section 16.2. Section 16.3 presents the information theoretic procedure step by step: noise model, estimation of noise parameters, source decorrelation by differential pulse code modulation (DPCM), and parametric entropy modeling of memoryless information sources via generalized Gaussian densities. Section 16.4 reports experimental results on a hyperspectral image acquired by the Airborne Visible InfraRed Imaging Spectrometer (AVIRIS) radiometer and on a test superspectral image acquired by the Advanced Spaceborne Thermal Emis-sion and Reflection Radiometer (ASTER) imaging radiometer. Concluding remarks are drawn in Section 16.5. © 2008 by Taylor & Francis Group, LLC Quality Assessment of Remote-Sensing Multi-Band Optical Images 357 16.2 Information Theoretic Problem Statement If we consider a discrete multi-dimensional signal as an information source S, its average information content is given by its entropy H(S)[8]. An acquisition procedure originates an observed digitized source S, whose information is the entropy H(S). H(S) is not adequate for measuring the amount of acquired information, since the observed source generally does not coincide with the digitized original source, mainly because of the observation noise. Furthermore, the source is not exactly band-limited by half of the sampling frequency; hence, the nonideal sampling is responsible for an additional amount of noise generated by the aliasing phenomenon. Therefore, only a fraction of the original source information is conveyed by the digitized noisy signal. The amount of source information that is not contained in the digitized samples is measured by the conditional entropy H(SjS) or equivocation, which is the residual uncertainty on the original source when the observed source is known. The contribution of the overall noise (i.e., aliasing and acquisition noise) to the entropy of the digitized source is measured by the conditional entropy H(SjS), which represents the uncertainty on the observed sourceS when the original source S is known. Therefore, the larger the acquisition noise, the larger H(S), even if the amount of information of the original source that is available from the observed (noisy) source is not increased, but diminished by the presence of the noise. A suitable measure of the information content of a recorded source is instead represented by the mutual information: I(S;S) ¼ H(S) ÿ H(SjS) ¼ H(S) ÿ H(SjS) ¼ H(S) þ H(S) ÿ H(S,S): (16:1) Figure 16.1 describes the relationship existing between the entropy of the original and the recorded source and mutual information and joint entropy H(S,S). In the following sections, the procedure reported in Figure 16.2 for estimating the mutual information I(S; S) and the entropy of the noise-free source H(S) is described later. The estimation relies on parametric noise and source modeling that is also capable of describing non-Gaussian sources usually encountered in a number of applica-tion contexts. H(S) H(S) H(S S) I(S;S) H(S S) FIGURE 16.1 Relationship between entropies H(S) and H(S), equivocation H(SjS), conditional entropy H(SjS), mutual information I(S; S), H(S,S) and joint entropy H(S,S). © 2008 by Taylor & Francis Group, LLC 358 Image Processing for Remote Sensing GG PDF PDF of Entropy of Noisy residuals of noisy residuals Noise-free residuals noise-free signal Parametric De convolution entropy Decorrelation estimation Noisy Signal FIGURE 16.2 Entropy of noisy signal H(S) Noise histogram Noise GG PDF estimation modelling Noise parameters (s , r) GG PFD of decorrelated noise + Noise – + entropy H(S S) Mutual information I(S;S) Flowchart of the information theoretic assessment procedure for a digital signal. 16.3 Information Assessment Procedure 16.3.1 Noise Modeling This section focuses on modeling the noise affecting digitized observed signal samples. Unlike coherent or systematic disturbances, which may occur in some kind of data, the noise is assumed to be due to a fully stochastic process. Let us assume an additive signal-independent non-Gaussian model for the noise g(i) ¼ f(i) þ n(i) (16:2) in which g(i) is the recorded noisy signal level and f(i) the noise-free signal at position (i). Both g(i) and f(i) are regarded as nonstationary non-Gaussian autocorrelated stochastic processes. The term n(i) is a zero-mean process, independent of f, stationary and auto-correlated. Let its variance sn and its correlation coefficient (CC) r be constant. Let us assume for the stationary zero-mean noise a first-order Markov model, uniquely defined by the r and the sn n(i) ¼ r n(i ÿ1) þ «n(i) (16:3) in which «n(i) is an uncorrelated random process having variance s2n ¼ s2 (1 ÿ r2) (16:4) The variance of Equation 16.2 can be easily calculated as s2(i) ¼ s2(i) þ s2 (16:5) due to the independence between signal and noise components and to the spatial statio-narity of the latter. From Equation 16.3, it stems that the autocorrelation, Rnn(m), of n(i) is an exponentially decaying function of the correlation coefficient r: © 2008 by Taylor & Francis Group, LLC Quality Assessment of Remote-Sensing Multi-Band Optical Images 359 Rnn(m)D E[n(i)n(i þm)] ¼ rjmjs2 (16:6) The zero-mean additive signal-independent correlated noise model (Equation 16.3) is relatively simple and mathematically tractable. Its accuracy has been validated for two-dimensional (2D) and three-dimensional (3D) signals produced by incoherent systems [3] by measuring the exponential decay of the autocorrelation function in Equation 16.6. The noise samples n(i) may be estimated on homogeneous signal segments, in which f(i) is constant, by taking the difference between g(i) and its average g(i) on a sliding window of length 2mþ1. Once the CC of the noise, r, and the most homogeneous image pixels have been found by means of robust bivariate regression procedures [3], as described in the next section, the noise samples are estimated in the following way. If Equation 16.3 and Equation 16.6 are utilized to calculate the correlation of the noise affecting g and g on a homogeneous window, the estimated noise sample at the ith position is written as vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n(i) ¼ t (2m þ 1) [g(i) ÿ g(i)] (16:7) (2mþ 1) ÿ 1 þ 2r 1ÿr The resulting set {n(i)} is made available to find the noise probability density function (PDF), either empirical (histogram) or parametric, via proper modeling. 16.3.2 Estimation of Noise Variance and Correlation To properly describe the estimation procedure, a 2D notation is adopted in this section, that is, (i, j) identifies the spatial position of the indexed entities. The standard deviation of the noisy observed band g(i, j) is stated in homogeneous areas as sg(i;j) ¼ sn (16:8) Therefore, Equation 16.8 yields an estimate of sn, namely ^n, as the y-intercept of the horizontal regression line drawn on the scatterplot of ^g versus g, in which the symbol ^ denotes estimated values, and is calculated only on pixels belonging to homogeneous areas. Although methods based on scatterplots have been devised more than one decade ago for speckle noise assessment [9], the crucial point is the reliable identification of homogeneous areas. To overcome the drawback of a user-supervised method, an auto-matic procedure was developed [10] on the basis of the fact that each homogeneous area originates a cluster of scatterpoints. All these clusters are aligned along a horizontal straight line having the y-intercept equal to sn. Instead, the presence of signal edges and textures originates scatterpoints spread throughout the plot. The scatterplot relative to the whole band may be regarded as the joint PDF of the estimated local standard deviation to the estimated local mean. In the absence of any signal textures, the image is made up by uniform noisy patches; by assuming that the noise is stationary, the PDF is given by the superposition of as many unimodal distributions as the patches. Because the noise is independent of the signal, the measured variance does not depend on the underlying mean. Thus, all the above expectations are aligned along a horizontal line. The presence of textured areas modifies the ‘‘flaps’’ of the PDF, which still exhibit aligned modes, or possibly a watershed. The idea is to threshold the PDF to identify a number of points belonging to the most homogeneous image areas, large enough to yield © 2008 by Taylor & Francis Group, LLC ... - tailieumienphi.vn