On the Spatial Resolution Enhancement of Microwave Radiometer ...

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On the Spatial Resolution Enhancement of Microwave Radiometer Data in Banach Spaces Flavia Lenti, Student Member, IEEE, Ferdinando Nunziata, Member, IEEE, Claudio Estatico, and Maurizio Migliaccio, Senior Member, IEEE

measurements and by the sampling pattern. Spatial resolution enhancement is obtained at the expense of noise amplification, therefore, in real cases, a compromise between these two factors must be considered [3]. The problem which describes the relationship between the coarse, but partially correlated, radiometer measurements, T A , and the temperature to be retrieved on the finer resolution grid, TAp , belongs to the class of Fredholm integral equations of the first kind, whose kernel depends on the antenna gain and on the scanning configuration. In this paper, atmospheric effects are neglected, hence TAp is replaced by the scene brightness Index Terms— Banach spaces, microwave radiometer, resolu- temperature, TB . A Fredholm integral equation of the first kind with a smooth kernel leads to an ill-posed problem, hence tion enhancement. the solution does not depend continuously on the data [4]. To reconstruct the surface brightness temperature at enhanced I. I NTRODUCTION resolution, an ill-posed inverse problem is to be solved, and ARTH-orbiting microwave radiometers are a key remote regularization methods must be accounted for. In the literature, several methods to enhance the spatial sensing tool to provide valuable and effective largescale information for oceanic and atmospheric applications. resolution of radiometer measurements have been proposed. However, there is a growing interest in other applications The Backus–Gilbert algorithm has been extensively used that require finer spatial resolution [1], [2]. The resolution [5]–[8]. Backus–Gilbert is a method for obtaining a solution of radiometer data can be enhanced by using either image- of the ill-posed problem introducing a minimization constraint processing techniques or special reconstruction algorithms [3]. on the noise amplification; resolution and noise are both These latter do not enhance the resolution of end-products, as minimized simultaneously. In [3] and [9], the iterative scatdone by ad hoc image-processing procedures; rather, once a terometer image reconstruction (SIR) algorithm is extended to low resolution measure of geophysical parameters is provided, the radiometer case and compared to the classical Backus– they attempt to reconstruct the geophysical parameters on a Gilbert approach. Experiments undertaken on both simufiner grid [3]. To this aim, a linear inversion problem, which lated and actual radiometer measurements show that the two can be physically considered as the analog of an antenna- approaches have similar resolution enhancement capabilities pattern deconvolution, has to be formulated and solved [4]. but the SIR algorithm is better in terms of processing time. In It is important to point out that although one may think to [1], the spatial resolution of soil moisture and ocean salinity consider an arbitrarily fine grid, the actual resolution enhance- (SMOS) radiometer data is enhanced by using six different ment capabilities are physically limited by the overlap of the deconvolution algorithms. Results indicate that the algorithms allow enhancing the spatial resolution of SMOS measurements Manuscript received July 28, 2012; revised January 8, 2013 and March 11, by a factor of 1.75. In [10], a stable and effective approach 2013; accepted March 21, 2013. based on the Tikhonov regularization procedure is proposed F. Lenti is with the Dipartimento di Scienze e Alte Tecnologie, Unito enhance the spatial resolution of 1-D simulated radiometer versità degli Studi dell’Insubria, Como 22100, Italy (e-mail: flavia.lenti@ uninsubria.it) measurements. The procedure is shown to achieve a significant F. Nunziata and M. Migliaccio are with the Dipartimento per le Tecnologie, spatial resolution improvement. In [11], a truncated singular Università degli Studi di Napoli Parthenope, Napoli 80143, Italy (e-mail: [email protected]; [email protected]). value decomposition (TSVD) approach is proposed to enhance C. Estatico is with the Dipartimento di Matematica, Università degli Studi the spatial resolution of 1-D radiometer measurements. The di Genova, Genova 16146, Italy (e-mail: [email protected]). rationale that lies at the basis of the TSVD approach consists Color versions of one or more of the figures in this paper are available in truncating the SVD solution to discard the components online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TGRS.2013.2255614 dominated by noise. The performance of the TSVD approach Abstract— A reconstruction technique, mathematically based on a generalization of the gradient method in Banach spaces, is first proposed to enhance the spatial resolution of radiometer earth observation measurements. This approach allows reducing the over-smoothing effects and the oscillations that are often present in standard Hilbert-spaces procedures without any drawback on the numerical complexity. Experiments undertaken on a data set consisting of both simulated and actual 2-D special sensor microwave imager radiometer measurements show the accuracy and the effectiveness of the proposed technique. A typical radiometer scene is processed in few minutes by a standard PC processor. Furthermore, since the proposed approach is iterative, the processing time increases slowly with the problem’s size.

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is compared to the Backus–Gilbert one using both simulated and actual 1-D radiometer measurements. In [12], the TSVD is first extended to the 2-D case and then applied to radiometer data. In this paper, a reconstruction technique, based on the gradient method in Banach spaces [13], [14], is proposed to enhance the spatial resolution of 2-D radiometer data. At the best of our knowledge, it is the first time that inversion procedures in Banach spaces are used to enhance the spatial resolution of radiometer products. Banach spaces are complete vector spaces endowed with a norm that only allows to measure “length” and “distance” between its elements without any scalar product, that is, without measuring any “angle” between them. Only few seminal applicative studies, related to nonlinear inverse scattering, have been accomplished by using the gradient method in Banach spaces [15], [16] and no study concerns resolution enhancement of EO products. The gradient method in Banach spaces is first exploited to enhance spatial resolution of both simulated and actual radiometer data. Since the method is based on the Banach spaces, it is expected to overcome the drawbacks of classical Hilbert space approaches: over-smoothing and oscillations due to Gibbs phenomenon. Experiments undertaken on both simulated and actual 2-D data show: 1) the accuracy of the proposed technique in reconstructing edges and abrupt intensity changes (reduction of the over-smoothness) typically obtained using L 2 ; 2) the reduction of the oscillations due to Gibbs phenomenon; and 3) the effectiveness of the approach in terms of processing time. The remainder of this paper is organized as follows. In Section II, the background theory is reviewed, in Section III experiments undertaken on both simulated and actual SSM/I radiometer data are presented and discussed, while conclusions are drawn in Section IV. II. T HEORETICAL BACKGROUND In this section, the theoretical aspects of the inversion procedure proposed to enhance the spatial resolution of the coarse but partially correlated 2-D radiometer measurements are described. The tth measured brightness temperature can be expressed as [11] ! ! G t (!)TB (!)d! (1) T At = G¯ −1 t !

where G¯ t =

! !

G t (!)d!.

(2)

!

! = (ϑ, φ) where ϑ and φ are the azimuth and the elevation angle in the reconstructed domain, G t (·) is a smooth function, which depends on the projection of the antenna pattern on the surface. Equation (1) represents a Fredholm integral equation of the first kind with a smooth kernel. Therefore, to retrieve the brightness field TB at a finer grid, an ill-posed inverse problem is to be solved [4]. The unknown brightness field TB and the observable field T A are represented by n 1 × n 2 and m 1 × m 2 matrices, with

n 1 > m 1 and n 2 > m 2 , termed as X and B, respectively. Hence, (1) can be cast in the form L(X) = B

(3)

where L(·) is a linear operator. By vectorizing X and B, (3) can be rewritten in the standard matricial form Ax = b

(4)

where A is a m 1 m 2 ×n 1 n 2 matrix, x = vec(X) and b = vec(B) where vec(·) is the operator that transforms a matrix into a lexicographically ordered vector. To retrieve TB , the linear system (4) has to be solved. The problem is underdetermined, hence, the system has infinite solutions. The solution sought is the one with minimum energy. A. Gradient Method in Hilbert Space Several regularization methods have been proposed in the context of Hilbert spaces to solve (4) [18]. They consist in minimizing the two-norm of the residual Ax − b, that is, in minimizing the least square functional !2 defined as M

!2 (x) =

1 1" # Ax − b #22 = (Ax − b)2i 2 2

(5)

i=1

where M = m 1 m 2 and the subscript i means that the i th component of the vector is considered. Hilbert spaces are the direct generalization of Euclidean spaces. Indeed, the norm and the scalar product of any Hilbert space have the same geometrical meaning and properties of the classical Euclidean distance and the Euclidean scalar product, so that the “distance” and the “angle” between its elements can be measured. Different methods have been developed to minimize (5). These methods can be divided into two categories: direct (Backus–Gilbert and TSVD method) and iterative methods. The iterative methods are preferable to direct methods when the matrix A is large [4]. The simplest iterative method to minimize the least square functional !2 is the gradient one (also known as Landweber method), defined as follows: starting from an arbitrary initial guess x0 , the k-iteration xk is given by xk = xk−1 − λ∇!2 (xk−1 )

(6)

where xk−1 is the k-1-iteration solution and λ > 0 is the step size. Since ∇!2 (x) = A∗ (Ax − b), where A∗ denotes the adjoint operator of A, (6) results xk = xk−1 − λA∗ (Axk−1 − b).

(7)

Unfortunately, regularized solutions obtained by minimizing the classical least square functional (5) in Hilbert spaces often show over-smoothness and the oscillations due to Gibbs phenomenon, which usually does not allow to obtain a good location of the edges [4].

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that is

B. Gradient Method in Banach Spaces In this paper, the inversion of (4) is revisited in Banach spaces [13]. In fact, in this manner, the L 2 effects are expected to be mitigated [15]. Accordingly, the method is based on the minimization of the functional ! p , which measures the p-norm of the residual, for 1 < p < +∞ M 1 1" p p ! p (x) = # Ax − b # p = (Ax − b)i . p p

(8)

i=1

The p-norm of (8) is the discrete version of the norm of the space L p , which is the space of the p-power integrable functions on the 2-D domain. Unlike the L 2 space, the L p space, for p '= 2, is not a Hilbert space, but is only a Banach space. A Banach space is a complete normed linear space, which is not equipped with any scalar product. This means that at variance of Hilbert spaces where the scalar product is defined, in Banach spaces the “angle” between two elements does not need to be defined and hence it has no meaning. Basically, for p '= 2, the functional ! p , which measures the residual in the L p Banach space, is intrinsically different from the functional !2 , which measures the residual in the classical L 2 Hilbert space. However, since for 1 < p < ∞ the functional ! p is always convex, a good technique to minimize (8) is the gradient method [13]. Although the gradient method can be implemented in straightforward way in Hilbert space, leading to the simple and well know iteration (7), its extension in Banach spaces is not straightforward, since in Banach spaces (7) is not welldefined. Let us illustrate this point. In this case, we recall that the operator A∗ :L q → L q is the dual operator of A:L p → L p , where (L p )∗ = L q is the dual space of L p and q is the Holder conjugate of p (i.e., 1p + q1 = 1) [19]. Hence, in the L p Banach space with p '= 2, A∗ (Ax−b) of (7) is not well defined, since Ax − b belongs to L p , while A∗ is defined on L q '= L p . As a matter of fact, in Banach spaces, the operator A∗ cannot be applied to the residual Ax−b because they belong to different spaces. Moreover, even the sum in (7) is not well defined in Banach spaces, since xk−1 belongs to L p , while A∗ (Ax − b) belongs to L q , so that their sum cannot be done. To overcome this key problem and generalize (7) to the L p Banach space, the so-called duality map J p [20] must be invoked. Basically, the duality map J p :L p → L q is a function associating an element of the Banach space L p with an element of its dual space, and it is explicitly defined as follows: J p (x)i = |x i | p−1 sign(x i ),

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i = 1, . . . , N

(9)

where again the subscript i means that the i th component of the vector is considered, and the function sign: R → R is defined as  s>0  1, 0, s=0 sign(s) = (10)  −1, s < 0.

By using the duality map J p , the (sub)gradient of ! p (x) can be rewritten as ∇! p (x) = (A∗ )J p (Ax − b)

(11)

∇! p (x)i =

M " (A∗ )i, j |(Ax − b) j | p−1 sign((Ax − b) j ), j =1

i = 1, . . . , N

(12)

which is now the generalization to L p spaces of A∗ (Ax − b) of (7). Accordingly, by replacing xk−1 in (7) with J p (xk−1 ), the sum J p (xk−1 ) − λA∗ J p (Ax − b) becomes well-defined, since both the addenda belong to L q . Finally, by using the adjoint duality map Jq , the sum J p (xk−1 ) − λA∗ J p (Ax − b) comes back to the L p space. Therefore, in the L p Banach space, the k-iteration of the gradient method is defined as follows [13]: xk = Jq (J p (xk−1 ) − λA∗ J p (Axk−1 − b)).

(13)

To initialize the method, the zeroth iteration x0 ∈ L p has to be chosen, which can be obtained by means of some a priori information of the expected solution x. In this paper, no information is required for the initial guess x0 , hence the null element is simply adopted. The convergence of the gradient method in Banach spaces is proved in [13]. In [13], it is also shown that for noisy-data, the scheme (13) is an iterative regularization algorithm, where the regularization parameter is represented by k, i.e., the number of iterations. To determine k, the discrepancy principle is used [18], which consists in choosing the smallest iteration index k such that (14) # Axk − b #≤ c where c is an upper bound related to the noise. A key issue in the proposed regularization method is the choice of the p parameter, which specifies a particular L p Banach space and, hence, the norm (8). In this paper, the p value is chosen according to the following mathematical rationale. The gradient method minimizes ! p (x), that is, the p-norm of the residual Ax − b. Given a generic vector v, for 1 < p < 2, its p-norm results in & |v i | p > |v i |2 for |v i | < 1 (15) |v i | p < |v i |2 for |v i | > 1.

This means that the small components of v, (i.e., |v i | < 1) are emphasized, whereas the large components (i.e., |v i | > 1) are reduced when compared to the gradient method in Hilbert space. In simple terms, a lesser penalty is put on elements with large but few components and a higher penalty on elements with many small components. Accordingly, the gradient method in L p space for 1 < p < 2 helps for sparsity and reduce over-smoothness. These reasonings suggest a p-value ranging between 1 and 2. However, p-values too close to 1 make the regularization method unstable [15] while values approaching 2 make the solution of the regularization method affected by the Gibbs phenomenon. Hence, to best reconstruct abrupt discontinuities is advisable to choose a p-value closer to 1 than to 2. Hence, several p-values were tested, and p = 1.2 was chosen since it provides the best tradeoff between the algorithm stability and the reconstruction performance.



TABLE I M AIN C HARACTERISTICS OF THE SSM/I R ADIOMETER Channel Frequency (GHz)

Pol.

19.35 19.35 22.235 37.0 37.0 85.5 85.5

V H V V H H V

3 dB Effective Field of View on Earth Surface (km) Along-Track Cross-Track 69 69 50 37 37 15 15

43 43 40 28 28 13 13

Spacing (km) Along-Track 25 25 25 25 25 12.5 12.5

Although the proposed inversion method does not call for the factorization of the projection of the antenna pattern on the surface, for sake of simplicity and since this paper aims at demonstrating the suitability of the Banach spaces approach; a sub-optimal but effective approach is considered, which consists in assuming all the projections of the antenna pattern on the surface separable in the same coordinate system G t (ϑ, φ) → G t 1 (ϑ)G t 2 (φ)

(16)

and, therefore, the following matricial form is obtained for (3) [12]: A1 X A2T = B (17) where the tth line of A1 and of A2 is the discretization of G t 1 (ϑ) and G t 2 (φ), respectively. Equation (17) is used to evaluate ! p (xk ) in a efficient way. It must be noted that (16) is rigorously true only when ϑ is close to 0 of a scanning radiometer. In fact, its oblong spatial response functions are not all separable in the same coordinate system. On the numerical complexity, the gradient method in Banach spaces is similar to iterative methods, such as the SIR [9].

Fig. 1. (a) Reference brightness field related to an area of 1400 × 700 km where some discontinuities of different sizes and brightness temperature values are present. (b) Simulated noisy measurements obtained by considering SSM/I parameters and 1.06-K additive noise.

III. E XPERIMENTAL R ESULTS In this section, the performance of the proposed gradient approach in L 1.2 is analyzed against both simulated and actual special sensor microwave imager (SSM/I) radiometer measurements. SSM/I is a conical scanning seven-channel, fourfrequency radiometer. The seven channels are given by the 19.35-GHz, the 37.0-GHz, and the 85.5-GHz measurements made at both vertical (V) and horizontal (H) polarization and by the 22.235-GHz measurements made at vertical polarization, see Table I. The radiometer orbital altitude is 833 km with a nominal swath width of about 1400 km. Along with the along-scan direction, it performs 128 measurements at the two 85.5-GHz channels, and 64 measurements, at the remaining five channels. The intrinsic spatial resolution depends on the antenna-beam pattern and ranges between 15 × 13 km and 69 × 43 km (along-track × cross-track), see Table I. In this paper, the 19-GHz channel measurements that are characterized by the coarsest spatial resolution, i.e., 69 × 43 km, are considered. The along-track dimension consists of 28 scans with a 25-km pixel spacing.

In all the subsequent experiments, the SSM/I integrated antenna pattern is approximated by a Gaussian function with a maximum gain of 43 dB since it has been shown to be a good approximation of realistic gain functions [23]. To obtain a sparse solution, the background brightness temperature is subtracted from the measurements and then added back to the solution. To perform experiments on simulated data, first of all, the 1400 ×700 km reference brightness field, whose pixel spacing is 13 pixel/deg, is generated. Then, the matrices A1 and A2 (of dimensions 64 × 1400 and 28 × 700, respectively) are simulated and the radiometer measurements are obtained through (17) on a uniform grid. To simulate realistic measurements noise is accounted for. In this paper, a zero mean additive Gaussian noise, whose standard deviation is equal to 1.06 K is used [9]. Finally, the 2-D gradient method in L 1.2 space is applied to reconstruct the brightness field. It must be explicitly pointed out that the spatial resolution of the reconstructed field will be always coarser than the pixel resolution [3].


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Fig. 3. Along-scan transect (see Fig. 1) related to the reference field (sky blue dotted line), the reconstructed field at enhanced resolution obtained by using gradient method in L 2 (blue line) and in L 1.2 (magenta circle) and the ones at non-enhanced resolution (red star).

Fig. 2. Reconstructed field at enhanced resolution obtained by using the gradient method in (a) L 1.2 and (b) L 2 spaces.

Experiments undertaken on real data are accomplished using the SSM/I data collected by the F08 satellite in 1988. The 2-D gradient method in L 1.2 is applied to obtain the reconstructed brightness field at enhanced resolution. To subtract the background, the image histogram is used and a constant value is subtracted. The first experiment is related to the 1400 × 700 km simulated reference field of Fig. 1(a). The reference field includes nine different hot spots, labeled H1–H9 in Fig. 1, characterized by different brightness temperatures and by different dimensions. H2 and H3 (H6 and H7) are very close to each other, about 20 km, in the along-scan (along-track) direction. Further, the reference field includes several lines that are characterized by different widths and by a brightness temperature 20 K smaller than the background one. Hence, this reference field is a very challenging scenario to test the proposed approach. The simulated measurements are shown in Fig. 1(b). Due to the coarse radiometer spatial resolution, all the edges are very blurred: H2, H3 and H6, H7 are not resolved, H1, H4, H5, and H9 are not visible as well as the narrowest part of the lower temperature lines. The reconstructed field at enhanced resolution obtained using the gradient method in L 1.2 space is shown in Fig. 2(a). It can be noted that all the edges are better retrieved in the reconstructed field. H2, H3 and H6, H7 are resolved and there are some hints related to the smaller hot spots H1, H4, H5, and H9. This result confirms the capabilities of the proposed technique in retrieving high frequency components attenuates (but not nulled out) by the low-pass system function. To analyze the performance of the proposed inversion technique against conventional L 2 ones, the gradient method

in L 2 (7) is used to obtain the reconstructed field shown in Fig. 2(b). It can be noted that, even in this case, the edges are better visible in the reconstructed field and the hot spots can be resolved. However, the L 2 reconstructed field exhibits the typical artifacts and oscillations due to Gibbs phenomenon. Hence, it is noisier than the L 1.2 reconstructed field. Moreover, the abrupt discontinuities due to both the lines and the hot spots are better reconstructed by the L 1.2 technique. To discuss the reconstruction performance in a more quantitative way, the reconstructed fields at enhanced resolution in L 1.2 and in L 2 are compared with the one at non-enhanced resolution, i.e., obtained by applying the inversion procedure to A+ x = b where A+ is m 1 m 2 square matrix. An along-scan transect [see Fig. 1 (a)], related to the reference field, the enhanced reconstructed fields and the non-enhanced one are shown in Fig. 3. It can be noted that the reconstructed fields fit well the reference profile and allow distinguishing all the hot spots. The reconstructed field at enhanced resolution with p = 1.2 does not show the oscillations which affect the L 2 solution and is less over-smoothed. In fact, the shape of the hot spots is better retrieved and the abrupt discontinuities are better reconstructed. To provide a further objective analysis of the quality related to the L 1.2 reconstructed field, two norms are introduced. The first one is the root mean square error (RMSE) which, according to the Frobenius norm, is defined by [21] RMSE(X k , X ref ) =

# X k − X ref # F # X ref # F

(18)

where the reference (X ref ) and reconstructed (X k ) fields are considered. The second norm consists in evaluating the correlation coefficient between the reference and the reconstructed field cov(X k , X ref ) . (19) r (X k , X ref ) = std(X k )std(X ref ) The two norms are evaluated for the reconstructed field at enhanced resolution using the gradient method with p = 1.2 and with p = 2 and the one at non-enhanced resolution (to obtain the same size of the reference field, the approach proposed in [9] is used: the value of the nearest measurement is assigned to each pixel), see Table II. It can be noted that both the fields at enhanced resolution result in a larger correlation and a lower RMSE if compared with the non-enhanced one



TABLE II RMSE AND r VALUES R ELATED TO THE F IRST E XPERIMENT Method

Gradient method with p = 1.2 Gradient method with p = 2

Resolution

Correlation

RMSE

non-enh.

0.6552

7.540 10−2

enh.

0.8071

5.956 10−2

enh.

0.7246

6.3420 10−2

Fig. 5. Reconstructed field at enhanced resolution obtained by using gradient method in L 1.2 .

Fig. 6. Along-track transect (see Fig. 4) related to the reference field (sky blue dotted line), the reconstructed field at enhanced resolution obtained by using gradient method in L 1.2 (blue line) and the ones at non-enhanced resolution (magenta circle).

Fig. 4. (a) Reference brightness field related to an area of 1400 × 700 km where some vertical and horizontal lines and some hot spots are present. (b) Simulated noisy measurements obtained by considering SSM/I parameters and 1.06 K additive noise.

but the p = 1.2 method results in the largest correlation and the lowest RMSE. The second experiment is related to the 1400 × 700 km simulated reference field of Fig. 4(a). In this experiment, the performance of the proposed L 1.2 approach is analyzed following the rationale suggested in [1]: vertical and horizontal bars are created with different widths and with different separations. The reference field includes several lines characterized by a width of about 40 km and by a separation ranging from about 30 to 60 km in along-scan direction and characterized by a separation ranging from about 10 to 40 km in the along-track direction. Moreover, the simulated scene includes several hot spots characterized by different dimensions ranging from about 10 × 10 to 40 × 40 km, and by a constant separation of 20 km. The simulated noisy measurements are shown in Fig. 4(b), where it can be noted that both the lines and the hot spots are not resolved. The reconstructed field at enhanced resolution is shown in Fig. 5. It can be noted that the bars can be correctly resolved once their separation is at least 20 and 15 km in the along-scan and along-track direction, respectively. The hot spots can be resolved once their dimension is larger than 10 × 10 km. To provide a quantitative analysis, an alongtrack transect [see Fig. 4(a)], related to reference field, the reconstructed field at enhanced resolution and the one at nonenhanced resolution, is shown in Fig. 6. The reconstructed

field fits best the reference field. Moreover, it can be noted that although bars separated by about 10 km are correctly located, the method fails in reconstructing the exact brightness level. A similar comment applies with respect to the hot spots. This result can be physically explained by noting that the product between the spatial resolution and the radiometric sensitivity is constant [1]. The third experiment is related to the 19-GHz V-polarized SSM/I actual radiometer data collected in 1988 (day 276, F08 satellite). A 1400 × 700 km area whose top left and bottom right corner coordinates (lat, lon) are −0.38°, −52.53° and 0.24°, −48.71°, respectively, is considered. The brightness field obtained by 19-GHz radiometer measurements is shown as a gray-tones image in Fig. 7(a). The Amazon River Delta is visible. This region was selected because it contains a relatively homogenous vegetation, a river, and the sea; hence it presents some abrupt discontinuities. The reconstructed field at enhanced resolution obtained using the gradient method in L 1.2 is shown in Fig. 7(b). It can be noted that the land/river and land/sea abrupt discontinuities are better visible in the reconstructed field. Moreover, the edges and the river’s profile are better reconstructed. It can also be noted that there are some artifacts along the river and the coastline and in the upper-side of the image. To provide a qualitative analysis, as suggest in [9], the enhanced resolution field is compared to radiometer data collected by 37-GHz channel, see Fig. 7(c). It can be noted that, due to the finer spatial resolution (37 × 28 km, see Table I), the 37-GHz radiometer image is characterized by better defined edges. The visual comparison of Fig. 7(b) and (c) confirms the


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Fig. 8. Along-scan transect [see Fig. 7(a)] related to the 19-GHz enhanced obtained by using gradient method in L 1.2 (red line), in L 2 (blue circle), and the non-enhanced field (sky blue dotted line) field.

Fig. 7. (a) 19-GHz V-polarized brightness temperature field. (b) 19-GHz V-polarized reconstructed field at enhanced resolution using L 1.2 gradient method. (c) 37-GHz V-polarized brightness field. (d) 19-GHz V-polarized brightness field at enhanced resolution using L 2 gradient method.

accuracy of the reconstructed field and the presence of artifacts in the reconstructed image. To analyze the performance of the proposed inversion technique against conventional L 2 ones, the gradient method in L 2 (7) is used to obtain the reconstructed field shown in Fig. 7(d). It can be noted that, even in this case, the edges are better visible in the reconstructed field. However, the L 2 reconstructed field exhibits stronger ribbs than the L 1.2 reconstructed field and these ribbs are not present in the 37-GHz field. Hence, the L 2 reconstructed field is much noisy than the L 1.2 reconstructed field. To provide a quantitative analysis of the reconstructed accuracy, an along-scan transect [see Fig. 7(a)] related to the 19-GHz brightness field at enhanced resolution obtained by using the gradient method in L 1.2 and in L 2 and the ones at nonenhanced resolution is shown in Fig. 8. It can be noted that in both the reconstructed fields, the abrupt discontinuity is more

Fig. 9. (a) 19-GHz V-polarized brightness temperature field. (b) 19-GHz V-polarized reconstructed field at enhanced resolution. (c) 37-GHz V-polarized brightness field.

pronounced but both the reconstructed fields present artifacts along the abrupt discontinuity and near the left edge of the image. The L 2 reconstructed field exhibits stronger ribbs than the L 1.2 reconstructed field. The forth experiment is related to the 19-GHz V-polarized SSM/I actual radiometer data collected in 1988 (day 278, F08 satellite). A 1400 × 700 km area whose top left and bottom right corner coordinates (lat, lon) are 58.01°, −14.86° and 47.01°, 7.81°, respectively, is considered. The brightness reference field obtained by radiometer measurements is shown as a gray-tones image in Fig. 9(a). Part of United Kingdom, Ireland, France, and Benelux are visible. This region was selected because it contains a small hotspot the Isle of Man.



ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their useful suggestions. R EFERENCES

Fig. 10. Along-scan transect [see Fig. 9 (a)] related to the 19-GHz enhanced (blue dotted line), the 19-GHz non-enhanced (violet line), and the 37-GHz (red star) field.

The edges are very blurred and the coastlines are not well defined. The brightness field at enhanced resolution obtained using the gradient method in L 1.2 space is shown in gray-tones in Fig. 9(b). It can be noted that the edges and the abrupt intensity changes are better visible and the coastlines are well defined. Moreover, some hints related to the small Isle of Man (22×52 km), whose center coordinates are 54.09°, −4.29°, are visible in Fig. 9(b) (see mangenta circle). It must be pointed out that no feature clearly associated to this island is visible in the non-enhanced field. The enhanced resolution field is compared to radiometer data collected by 37-GHz channel, see Fig. 9(c) where some hints related to the Isle of Man are visible. The reconstructed field presents some oscillations not presented in the 37-GHz field. These oscillations can be attributed to artifacts due to the reconstruction. To provide a quantitative analysis of the reconstructed accuracy, an alongscan transect [see Fig. 9(a)] related to the 37-GHz brightness field, the 19-GHz brightness field at enhanced resolution, and the one at non-enhanced resolution is shown in Fig. 10. It can be noted that the 19-GHz reconstructed field at enhanced resolution fits best the 37-GHz profile and the reconstructed field at enhanced resolution presents small features related to the Isle of Man, which are not visible in the non-enhanced field. These results have been confirmed by several other experiments on SSM/I radiometer data that are not shown to save space. IV. C ONCLUSION A robust and accurate approach to enhance the spatial resolution of single-pass radiometer products is proposed. The approach is mathematically based on the theory of regularization in Banach spaces; hence, it allows obtaining solutions characterized by an over-smoothness significantly lower than the classic Hilbert-space approaches and by minor oscillations due to Gibbs phenomenon. Further, the numerical complexity of the method does not change when moving from Hilbert to Banach spaces. Experiments undertaken on both simulated and actual SSM/I radiometer data confirm the theoretical rationale and show that the L 1.2 method performs better than the L 2 one, resulting in a more accurate reconstruction of abrupt discontinuities and a significant reduction of ribbs and oscillations.

[1] M. Piles, A. Camps, M. Vall-llossera, and M. Talone, “Spatial-resolution enhancement of SMOS data: A deconvolution-based approach,” IEEE Trans. Geosci. Remote Sens., vol. 47, no. 7, pp. 2182–2192, Jul. 2009. [2] M. A. Goodberlet, C. T. Swift, and J. C. Wilkerson, “Ocean surface wind speed measurements of the Special Sensor Microwave/Imager (SSM/I),” IEEE Trans. Geosci. Remote Sens., vol. 28, no. 5, pp. 823–828, Sep. 1990. [3] D. G. Long, “Reconstruction and resolution enhancement techniques for microwave sensors,” in Frontiers of Remote Sensing Information Processing, C. H. Chen, Ed. Singapore: World Scientific, 2003, ch. 11. [4] P.C. Hansen, Rank-Deficient and Discrete Ill-Posed Problem: Numerical Aspects of Linear Inversion. Philadelphia, PA, USA: SIAM, 1987, chs. 6–7, pp. 132–208. [5] G. E. Backus and J. F. Gilbert, “Numerical applications of a formalism for geophysical inverse problem,” Geophys. J. R. Astron. Soc., vol. 13, nos. 1–3, pp. 247–276, 1967. [6] A. Stogryn, “Estimates of brightness temperatures from scanning radiometer data,” IEEE Trans. Antennas Propag., vol. 26, no. 5, pp. 720–726, Sep. 1978. [7] M. R. Farrar and E. A. Smith, “Spatial resolution enhancement of terrestrial features using deconvolved SSM/I microwave brightness temperatures,” IEEE Trans. Geosci. Remote Sens., vol. 30, no. 2, pp. 349–355, Mar. 1992. [8] W. D. Robinson, C. Kummerow, and W. S. Olson, “A technique for enhancing and matching the resolution of microwave measurements from the SSM/I instrument,” IEEE Trans. Geosci. Remote Sens., vol. 30, no. 3, pp. 419–429, May 1992. [9] D. G. Long and D. L. Daum, “Spatial resolution enhancement of SSM/I data,” IEEE Trans. Geosci. Remote Sens., vol. 36, no. 2, pp. 407–417, Mar. 1998. [10] A. Gambardella and M. Migliaccio, “On the superresolution of microwave scanning radiometer measurements,” IEEE Geosci. Remote Sens. Lett., vol. 5, no. 4, pp. 796–800, Oct. 2008. [11] M. Migliaccio and A. Gambardella, “Microwave radiometer spatial resolution enhancement,” IEEE Trans. Geosci. Remote Sens., vol. 4, no. 5, pp. 1159–1169, May 2005. [12] F. Lenti, M. Migliaccio, F. Nunziata, and G. Rodriguez, “Twodimensional TSVD resolution enhacement for EO applications,” Atti Della Fondazione Ronchi, vol. 1, pp. 81–92, Feb. 2012. [13] F. Schopfer, A. K. Louis, and T. Schuster, “Nonlinear iterative methods for linear ill-posed problems in Banach spaces,” Inverse Problems, vol. 22, no. 1, pp. 311–329, 2006. [14] T. Hein and K. S. Kazimierski, “Accelerated Landweber iteration in Banach spaces,” Inverse Problems, vol. 26, no. 5, p. 055002, 2010. [15] C. Estatico, M. Pastorino, and A. Randazzo, “A novel microwave imaging approach based on regularization in L p Banach spaces,” IEEE Trans. Antennas Propag., vol. 60, no. 7, pp. 3373–3381, Jul. 2012. [16] C. Estatico, M. Pastorino, and A. Randazzo, “A Banach space regularization approach for microwave imaging,” in Proc. IEEE Antenna Propagat., Mar. 2012, pp. 1382–1386. [17] F. J. Wentz, “User’s manual SSM/I antenna temperature tapes,” Remote Sensing System, Santa Rosa, CA, USA, Tech. Rep. 032588, 1991. [18] M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging. Bristol, England: Institute of Physics Publication, 1998, chs. 4–5, pp. 73–120. [19] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations. New York, NY, USA: Springer-Verlag, 2010, chs. 4–5, pp. 86–157. [20] M.D. Contreras and R. Paya, “On upper semicontinuity of duality mapping,” Amer. Math. Soc., vol. 121, no. 2, pp. 451–459, 1994. [21] A. Bouhamidi, K. Jbilou, L. Reichel, and H. Sadok, “An extrapolated TSVD method for linear discrete ill-posed problems with Kronecker structure,” Linear Algebra Appl., vol. 434, no. 7, pp. 1677–1688, 2010. [22] D. S. Early and D. G. Long, “Image reconstruction and enhanced resolution imaging from irregular samples,” IEEE Trans. Geosci. Remote Sens., vol. 39, no. 2, pp. 291–302, Feb. 2001. [23] J. P. Hollinger, “DMSP special sensor microwave/imager user’s guide,” Naval Res. Lab., Washigton, DC, USA, Sep. 1987.


Flavia Lenti (S’11) received the B.Sc. and M.Sc. degrees (summa cum laude) in applied mathematics from Università "La Sapienza" of Rome, Rome, Italy, in 2006 and 2009, respectively, and the Master’s degree in space remote sensing technology from the Italian Space Agency in 2011. She is currently pursuing the Ph.D. degree in applied mathematics with Università of Insubria, Como, Italy. She has been cooperating with Remote Sensing Lab, Università "Parthenope" of Napoli, Napoli, Italy, since 2011, working on resolution enhancement of radiometer data. Her current research interests include numerical linear algebra and inverse problems applied to remote sensing data and image restoration.

Ferdinando Nunziata (S’03–M’12) was born in Italy in 1982. He received the B.Sc. and M.Sc. degrees (summa cum laude) in telecommunications engineering and the Ph.D. degree in curriculum electromagnetic fields from the Università degli Studi di Napoli "Parthenope," Napoli, Italy, in 2003, 2005, and 2008, respectively. He has been a Researcher in electromagnetic fields with the Faculty of Engineering "G. Latmiral," Università degli Studi di Napoli "Parthenope", Napoli, since 2010. He has been a Lecturer with the National Oceanography Centre, Southampton, U.K., the Université Catholique de Louvain, Louvain-la-Neuve, Belgium, the Helsinki University of Technology, Espoo, Finland, the City College of New York, New York, NY, USA, and Beihang University, Beijing, China. He has authored or co-authored more than 90 papers published in peer-reviewed journals and refereed conferences. His current research interests include electromagnetic modeling, single- and multipolarization sea surface scattering, radar polarimetry, synthetic aperture radar sea oil slick and metallic target monitoring and spatial resolution enhancement techniques. Dr. Nunziata was the recipient of the 2003 IEEE GRS South Italy Chapter Best Remote Sensing Thesis Award, the 2009 Sebetia-Ter International Award for his research activities in remote sensing, and the 2012 Latmiral Prize, provided by Italian society of electromagnetics. He was the Organizing Committee of the 2008, 2010, and 2012 IEEE Graduate of Last Decade (GOLD) Remote Sensing Conferences at the European Space Agency, Frascati, Italy, the Italian Naval Academy, Livorno, Italy, and the National Research Council, Rome, Italy. He served as a Session Chair at 2008 and 2012 IEEE International Geoscience and Remote Sensing Symposia, held in Boston, MA, USA, and in Munich, Germany, respectively. He is the Chairman of the Università degli Studi di Napoli "Parthenope" IEEE Student Branch and he is a GOLD representative to the GRSS (Geoscience and Remote Sensing Society) AdCom.

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Claudio Estatico received the Laurea degree in mathematics from the University of Genoa, Genoa, Italy, and the Ph.D. degree in computational mathematics and operation research from the University of Milan, Milan, Italy, in 1995 and 2002, respectively. He is currently an Associate Professor of numerical analysis with the Department of Mathematics, University of Genoa, Genoa, Italy. His current research interests include numerical linear algebra and inverse problems. They include structured matrices and regularization preconditioning for linear and nonlinear ill-posed problems, boundary conditions and reblurring in image restoration, inverse scattering and remote sensing. He was a participant in the semester program on inverse problems with the Institute for Pure and Applied Mathematics, University of California, Los Angeles, CA, USA, in 2003.

Maurizio Migliaccio (M’91–SM’00) was born in Naples, Italy, in 1962. He received the Laurea degree (Hons.) in electronic engineering from the Università degli Studi di Napoli “Federico II,” Naples, Italy, in 1987. He was a Visiting Scientist with the German Aerospace Center (DLR), Oberpfaffenhofen, Germany. He is currently a Full Professor of electromagnetics with the Università degli Studi di Napoli “Parthenope,” Naples. He is also a Lecturer in the U.S., Spain, Germany, and Italy. He has authored or co-authored more than 250 papers on applied electromagnetics. Dr. Migliaccio was the recipient of the IEEE Geoscience and Remote Sensing Society (GRS-S) Chapter Excellence Award in 2007 and the IEEE GRS-S Letters Prize Paper Award in 2009. He was also the Chapter Chair of IEEE GRS-S. He was the European Union Secretary of the COST 243 Action. He was the IEEE Italy Section Graduate of Last Decade (GOLD) delegate. He was also the General Chairman of the 2008, 2010, and 2012 IEEE Graduate of Last Decade (GOLD) Remote Sensing Conferences at the European Space Agency (ESA), Frascati, Italy, at the Italian Naval Academy, Livorno, Italy, and at the National Research Council, Rome, Italy, respectively. He was a member of the Italian Space Agency Scientific Board. He is national delegate of the SMOS-MODE Cost Action. He is a member of the ESA PBEO, the Scientific Board of the Indian Journal of Radio and Space Physics and SpaceMag.