I Introduction
Over the past two decades, the U.S. Army has been investigating the capability of lowfrequency, ultrawideband (UWB) synthetic aperture radar (SAR) systems for the detection of buried and obscured targets in various applications, such as foliage penetration [1], ground penetration [2], and sensingthroughthewall [3]. These systems must operate in the lowfrequency spectrum spanning from UHF frequency band to L band to achieve both resolution and penetration capability. Although a lot of progress has been made over the years, one critical challenge that lowfrequency UWB SAR technology still faces is discrimination of targets of interest from other natural and manmade clutter objects in the scene. The key issue is that that the targets of interest are typically small compared to the wavelength of the radar signals in this frequency band and have very low radar cross sections (RCSs). Thus, it is very difficult to discriminate targets and clutter objects using lowresolution SAR imagery.
Ia SAR geometry and image formation overview
Figure 1 shows a typical sidelooking SAR geometry where a vehicle or an airbornebased radar transmits wideband signals to the imaging area and measures the backscatter signals as it moves along an aperture. Let be the received rangecompressed data received at aperture; the backprojection algorithm [4] computes the value for each SAR image pixel from the imaging area as follows:
(1) 
where is the total number of aperture records used for the coherent integration, is the weighting value at the aperture position, and is the shift index to the signal for pixel at the aperture position. For a typical SAR image with aspect angle, and correspond to the angle values of and to form a SAR image with an integration angle of . Note that for a true constant integration angle SAR image formation, and are computed for every pixel of the SAR image. However, for computational efficiency, a large image area is divided into smaller subimages. For each subimage, SAR image formation is computed using the and values derived from the geometry of the center of the subimage and the radar aperture. To exploit the aspect dependence information of target, instead of forming a single SAR image with aspect angle as described above with a single sector , we form multiple sectors to generate the corresponding SAR images at different aspect angles. For example, in our consecutive multilook experiment, three SAR images are formed: left side image that covers sector , broadside image that covers sector , and right side image at sector .
To achieve a constant crossrange resolution SAR image, a large image area is also divided into smaller subimages. Each subimage is formed using a different subset of aperture. For a constant integration angle, subimages at farther range would be integrated using a longer aperture than nearrange subimages. The subimages are then mosaicked together to form a single large image that covers the area of interest. In consecutive multilook SAR image processing mode, instead of generating a single SAR image, multiple SAR images are formed at different viewing angles to exploit the aspect angle dependent information from targets. Thus, each aperture for each subimage is further divided into smaller segments (either disjoint or overlapped) that are used to form consecutive multilook SAR images as illustrated in Figure 1. The aspect angle dependence features from targets have been exploited in past research before using different techniques [5, 6, 4].
IB Closely related works and motivation
UWB radar techniques have recently attracted increasing attention in the area of penetration and object detection, thanks to their usage in security applications and surveillance systems [7]. T. Sakamoto et al. [8] proposed fast methods for ultrawideband (UWB) radar imaging that can be applied to a moving target. The technology has been also applied to 3D imaging applications [9], human posture [10], human activity [11], vital sign [12], and liquid material [13]
classification problems. In these papers, due to high dimensionality and small signaltoratio (SNR), the signals need to be preprocessed, e.g., dimensionality reduction and background subtraction, before being used to train a classifier. It has been shown that support vector machines usually provide the best performance
[11, 13]. It is worth noting that in the aforementioned applications, objects are usually big (human) and captured from a relatively small distance. On the contrary, objects in our problem are relatively small and quite far from the radar.In this paper, we consider the problem of discriminating and classifying buried targets of interest (metal and plastic mines, 155mm unexploded ordinance [UXO], etc.) from other natural and manmade clutter objects (a soda can, rocks, etc.) in the presence of noisy responses from the rough ground surfaces for lowfrequency UWB 2D SAR images. For classification problems, sparse representationbased classification [14] (SRC) has been successfully demonstrated in other imagery domains such as medical image classification [15, 16, 17, 18], hyperspectral image classification [19, 20, 21], highresolution Xband SAR image classification [22]
, video anomaly detection
[23], and several others [24, 25, 26, 27, 28, 29, 30, 31]. However, in the lowfrequency RF UWB SAR domain, although we have studied the feasibility of using SRC for higherresolution 3D downlooking SAR imagery [32], the application of SRC to lowfrequency UWB 2D SAR imagery has not been studied to date due to the aforementioned lowresolution issue. In this paper, we generalize the traditional SRC to address target classification using either a single channel (radar polarization) or multiple channels of SAR imagery. Additionally, we further propose a novel discriminative tensor sparsity framework for multilook multichannel classification problem, which is naturally suitable for our problem. In sparse representations, many signals can be expressed by a linear combination of a few basic elements taken from a “dictionary”. Based on this theory, SRC [14]was originally developed for robust face recognition. The main idea in SRC is to represent a test sample as a linear combination of samples from the available training set. Sparsity manifests because most of the nonzero components correspond to basic elements with the same class as the test sample.
Multichannel SRC has been investigated before in medical images [16, 18]. In these papers, one dictionary for each channel is formed from training data with locations of all channels of one training point being the same in all dictionaries. Then intuitively, when sparsely encoding each channel of a new test point using these dictionaries, we obtain sparse codes whose active (nonzero) elements tend to happen at the same locations in all channels. In other words, active elements are simultaneously located at the same location across all channels. This intuition is formulated based on pseudonorm, which is solved using a modified version of simultaneous orthogonal matching pursuit (SOMP) [33]. The cost function is nonconvex, and hence, it is difficult to find the global solution. Furthermore, when more constraints involved, there is no straightforward way to extend the algorithm. In this paper, we proposed another way of formulating the simultaneity constraint based on the norm, which enforces the row sparsity of the code matrix (in tensor form, we call it tube sparsity). The newly convex optimization problem can be solved effectively using the fast iterative shrinkage thresholding algorithm (FISTA) [34]. We also propose other different tensor sparsity models for our multichannel classification problems.
It has been shown that learning a dictionary from the training samples instead of concatenating all of them as a dictionary can further enhance performance of sparsitybased methods. On one hand, the training set can be compacted into a smaller dictionary, reducing computational burden at the test time. On the other hand, by using dictionary learning, discriminative information of different classes can be trained via structured discriminative constraints on the dictionary as well as the sparse tensor code. A comprehensive study of discriminative dictionary learning methods with implementations is presented at [26, 35]. These dictionary learning methods, however, are all applied to singlechannel problem where samples are often represented in form of vectors. While multichannel signals can be converted to a long vector by concatenating all channels, this trivial modification not only leads to the curse of dimensionality of highdimensional space, but also possibly neglects crosschannel information, which might be crucial for classification. In this paper, we also propose a method named TensorDL, which is a natural extension of singlechannel dictionary learning frameworks to multichannel dictionary learning ones. Particularly, the crosschannel information will be captured using the aforementioned simultaneity constraint.
Naturally, when a radar carried by a vehicle or aircraft moves around an object of interest, it can capture multiple consecutive views of that object (see Fig. 1). Consequently, if the multilook information is exploited, the classification accuracy will be improved. While the multilook classification problem has been approached before by SRCrelated methods [22, 25], none of these works uses the relative continuity of different views. We propose a framework to intuitively exploit this important information. More importantly, the optimization problem corresponding to this structure can be converted to the simultaneous sparsity model by using an elegant trick that we call ShiftSRC. Essentially, a tensor dictionary is built by circularly shifting an appropriate amount of a singlechannel dictionary. When we sparsely code the multilook signals using this tensor dictionary, the tensor sparse code becomes tube sparsity.
IC Contributions
The main contributions of this paper are as follows:

A framework for simultaneously denoising and classifying 2D UWB SAR imagery^{1}^{1}1The preliminary version of this work was presented in IEEE Radar Conference, 2017 [36]. Subtle features from targets of interest are directly learned from their SAR imagery. The classification also exploits polarization diversity and consecutive aspect angle dependence information of targets.

A generalized tensor discriminative dictionary learning (TensorDL) is also proposed when more training data involved. These dictionary learning frameworks are shown to be robust even with high levels of noise.

A relative SRC framework (ShiftSRC) is proposed to deal with multilook data. Lowfrequency UWB SAR signals are often captured at different views of objects, depending on the movement of the radar carriers. These signals contain uniquely important information of consecutive views. With ShiftSRC, this information will be comprehensively exploited. Importantly, a solution to the ShiftSRC framework can be obtained by an elegant modification on the training dictionary, resulting in a tensor sparse coding problem, which is similar to a problem proposed in contribution 1).
The remainder of this paper is organized as follows. Section II presents different tensor sparsity frameworks and the discriminative tensor dictionary learning scheme for multichannel classification problems. The ShiftSRC for multiplerelativelook and solutions to all proposed frameworks are also presented in this section. Section III shows extensive experimental results on a simulated dataset for several scenarios. An experiment with a realistic dataset is also included. Section IV concludes the paper.
Ii Sparse representationbased classification
Iia Notation
Scalars are denoted by italic letters and may be either lower or uppercase, e.g., . Vectors and matrices are denoted by bold lowercase () and bold upper case (), respectively. In this paper, we also consider 3D tensors (tensors for short) whose dimensions are named row, column, and channel. A tensor with only one column will be denoted by a bold, lowercase, calligraphic letter (). Tensors with more than one column will be denoted by an bold, uppercase, calligraphic letters ().
For any tensor , let be its th channel. For convenience, given two tensors , the tensor multiplication is considered channelwise multiplication, i.e., . For a tensor , we also denote the sum of square of all elements by and the sum of absolute values of all elements by . Tensor addition/subtraction simply represents elementwise addition/subtraction. Each target sample is represented by a UWB SAR image formed using either a single (using copol) or multiple polarization (using both copol and crosspol) channels. Thus, one target sample is denoted by , where is the total number of image pixels and is the number of polarization channels. A collection of samples is denoted by .
Consider a general classification problem with different classes. Let be the collection of all training samples from class , be the collection of samples in the shared class, and be the total dictionary with the concatenation being done at the second dimension (column). In our problem, the shared class can be seen as the collection of ground images.
IiB Classification scheme
Using the definition of tensor multiplication, a sparse representation of using can be obtained by solving
(2) 
where is a positive regularization parameter and is a function that encourages to be sparse. Denote by the sparse coefficient of on . Then, the tensor can be divided into tensor parts .
After solving the sparse coding problem (2), shared features (grounds in our problem) are eliminated by taking . Then the identity of one sample can be determined by the dictionary that provides the minimum residual:
(3) 
Confuser detection: In practical problems, the set of confusers is not limited to the training set. A confuser can be anything that is not a target; it can be solely the ground or a capture of an unseen object. In the former case, the test signal can be well represented by using only the ground , while in the latter case, the sparse representation assumption is no longer valid. Therefore, one test signal is classified as a confuser if one of following three conditions is satisfied: i) it is not sparsely interpreted by the total dictionary ; ii) it has the most active elements in the sparse code locating at ; and iii) it is similar to known confusers.
IiC Generalized sparse representationbased classification
In SRC [14] where only one channel is considered, is simply a function that forces sparsity as  or minimization. minimization is often used since it leads to a convex optimization problem with tractable solutions. In the following, we present two natural extensions of SRC for multichannel cases. We also proposed two tensor sparse representation methods that can enhance classification accuracy by exploiting crosschannel information. These four generalized SRC methods are as follows.
a) Apply a sparsity constraint on each column of the coefficient tensor , no crosschannel constraint.
We can see that this is similar to solving separate sparse coding problems, each for a polarization channel. The classification rule is executed based on the sum of all the squares of the residuals. We refer to this framework as SRCcumulative residual or SRCCR (CR in the short version). (See Figure 2a with sparse tensor ). The sparse code corresponding to the channel, , is computed via the traditional minimization:
(4) 
which can be solved effectively using FISTA [34], ADMM [37], etc., algorithms or the SPAMS toolbox [38].
b) Concatenate all channels.
The most convenient way to convert a multichannel problem to a singlechannel problem is to concatenate all channels of one signal to obtain a long vector. By doing so, we have the original SRC framework with norm minimization. After solving the sparse coding problem, if we break the long vectors and rearrange them back to tensor forms, then the tensor sparse coefficients can be formed by replicating the onechannel sparse coefficient at all channel (see Figure 2b with sparse tensor ). From this tensor viewpoint, the code tensor will have few active “tubes”; moreover, all elements in a tube are the same. We refer to this framework as SRCconcatenation or SRCCC (CC in the short version).
The optimization problem in SRCCC and its solution are very straightforward. First, we stack all channel dictionaries into a long one: (symbol ‘;’ represents the concatenation in the first dimension). Then for every test signal, we also stack all of its channels to form a long vector: . The optimization problem (2) becomes the traditional regularization problem:
(5) 
then also can be solved by aforementioned methods.
c) Use a simultaneous model.
Similar to the SHIRC model proposed in [16, 18], we can impose one constraint on active elements of tensor as follows: also has few nonzero tubes as in SRCCC; however, elements in one active tube are not necessarily the same. In other words, the locations of nonzero coefficients of training samples in the linear combination exhibit a onetoone correspondence across channels. If the th training sample in has a nonzero contribution to , then for , also has a nonzero contribution from the th training sample in . We refer to this framework as SRCSimultaneous or SRCSM (SM in the short version). (See Figure 2c with sparse tensor ). To achieve this requirement, we can impose on the tensor (with one column and T channels) the minimization constraint, which is similar to the rowsparsity constraint applied on matrices in [22].
Remarks: While SHIRC uses minimization on and applies the modified SOMP [33], our proposed SRCSM exploits the flexibility of regularizer since it is convex, and easily modified when more constraints are present (e.g., nonnegativity). In addition, it is more efficient especially when dealing with problems of multiple samples at input.
Concretely, the optimization problem of SRCSM can be written in the form
(6) 
where denotes the tube of the tensor code and is the total column of the dictionary , and is the vectorization of the tube . This problem is similar to the joint sparse representation (JSRC) problem proposed in [22] except that SRCSM enforces tubesparsity instead of the rowsparsity. Details of the algorithm that solves are described in Section IIF.
d) Use a group tensor model.
Intuitively, since one object is ideally represented by a linear combination of the corresponding dictionary and the shared dictionary, it is highly likely that number of active (tensor) parts in is small, i.e., most of are zero tensors. This suggests us a group tensor sparsity framework as an extension of [39] that can improve the classification performance, which is referred to as SRCGT (GT in the short version). The visualization of this idea is shown in Figure 2d.
The optimization problem of SRCGT is similar to (6) with a slight difference in the grouping coefficients:
(7) 
where is the total number of classes (including the shared ground class), and is the vectorization of the group tensor . Solution to this problem will be discussed next.
The overall algorithm of generalized SRC applied to multichannel signals in the presence of a shared class is shown in Algorithm 1.
(8) 
IiD Dictionary learning for tensor sparsity
As a natural extension, we can extend the tensor sparsity models to dictionary learning ones. Most of dictionary learning methods focus on a singlechannel signal, which is not suitable for models with crosschannel information. In this work, we extend singlechannel dictionary learning methods to multichannel dictionary ones by applying aforementioned tensor sparsity constraints.
In singlechannel, most of discrimination dictionary learning methods, such as FDDL [40], DLSI [41], DFDL [17], LRSDL [26], etc., have a cost function that is of the form
(9) 
where is a function enforcing the sparsity of , and , which includes fidelity and discriminant terms, is a function of and depends on the training samples .
One straightforward extension of these singlechannel models to a multichannel case is to apply the first term to each channel and join all channels by a sparsity constraint represented by . Naturally, can be one of four presented crosschannel sparsity constraints. Concretely, the overall cost function would be in the form
(10) 
where and is one of {CR, CC, SM, GT} sparsity constraints.
In this paper, we particularly focus on extending FDDL [40] to the multichannel case. FDDL is a special case of LRSDL [26] without an explicit shared dictionary. FDDL is chosen rather than LRSDL since in our problem, the shared dictionary, e.g., grounds, is already separated out. We also adopt fast and efficient algorithms proposed in the dictionary learning toolbox DICTOL [35] to update each channel of the dictionary . Also, the sparse code tensor is updated using FISTA [34] algorithm, which is discussed in Section IIF.
The proposed crosschannel dictionary learning method is named TensorDL suffixed by CR, CC, SM, or GT when different sparsity constraints are applied on .
IiE Tensor sparsity with multiple relative looks
In realistic situations, objects might be captured at different angles instead of only one. Moreover, these are often consecutive angles that a plane or a vehicle can capture (see Figure 3a left) while moving around objects. Based on this observation, in the training phase, we collect data from different views of objects, as in Figure 3a right, and orderly arrange them into dictionaries for each object, as illustrated in Figure 3b. The test objects, which are often captured at different relative views , are then sparsely coded by the whole dictionary.
Intuitively, if is the sparse code of the first view with only few active elements, then the sparse code of the next view will be active at locations shifted by one. Similarly, active locations in of the third view will be shifted by two compared to , and so on. In this case, active coefficients form a “stair”, as illustrated in Figure 3b right.
The sparse coding problem with the “stair” sparsity is a novel problem and has not been addressed before. In this paper, we propose a method called ShiftSRC to convert this problem to a previously known problem. The central idea is that if we stack all views into a tensor and “circularly shift” the ordered dictionary by one to the left at each view, then the obtained tensor code will be a tensor with active elements forming a “tube” sparsity (see Figure 3c). The solution to this problem is similar to the solution of SRCSM as stated in this paper.
IiF Solution to optimization problems
Both optimization problems (6) and (7) have the form
(11) 
where

is sum of norm 2, then it is a continuous convex function and nonsmooth.

is a continuous convex function of the type , i.e., continuously differentiable with a Lipschitz continuous gradient :
We observe that, with these properties, the optimization problem
(11) can be solved by FISTA [34]. There
are three main tasks in FISTA:
1. Calculating , which can be easily computed as
where each channel of is the transpose of the corresponding channel of .
2. Calculating a Lipschitz constant of . For our function , we can choose
(12) 
where
is the maximum eigenvalue of a square matrix.
3. Solving a suboptimization problem of the form
(13) 
with .
For SRCSM, problem (13) has the form
(14) 
Each problem in (14) is a minimum norm minimization with solution being
(15) 
Similarly, for SRCGT, problem (13) can be written as
(16) 
with solution being:
(17) 
A step by step description of SRCSM and SRCGT algorithms are given in Algorithm 2.
Iii Experimental results
In this section, we apply the proposed models to the problem of classifying objects of interest. Extensive results are presented on simulated and reallife datasets. A MATLAB toolbox for the tensor sparsity methods presented in this paper is available at [42].
Separate targets  All targets  
VV  HH  VV+HH  HH+HV  ALL  VV  HH  VV+HH  HH+HV  ALL  
SVM  62.17  74.47  65.00  77.77  70.33  83.68  89.36  83.36  66.36  79.78  
SRCCR  76.32  83.43  86.26  85.93  82.95  86.22  91.64  87.14  91.28  90.64  without nonnegativity constraint 
SRCCC  76.32  83.43  79.47  86.21  79.93  86.22  91.64  87.96  92.98  81.80  
SRCSM  76.32  83.43  81.07  86.52  85.87  86.22  91.64  89.66  95.12  94.14  
SRCGT  69.79  78.79  77.83  85.12  81.75  81.90  88.40  87.28  91.86  90.46  
SRCCR  79.05  89.41  79.85  88.52  84.92  86.32  92.98  87.48  94.86  94.26  with nonnegativity constraint 
SRCCC  79.05  89.41  83.99  92.27  85.11  86.32  92.98  89.20  93.82  87.08  
SRCSM  79.05  89.41  85.62  90.85  89.36  86.32  92.98  90.00  96.82  96.00  
SRCGT  75.55  88.97  81.85  90.57  87.28  84.16  90.32  88.10  94.58  94.12 
Iiia Electromagnetic (EM) Simulation data
SRC is applied to a SAR database consisting of targets (metal and plastic mines, 155mm unexploded ordinance [UXO], etc.) and clutter objects (a soda can, rocks, etc.) buried under rough ground surfaces. The electromagnetic (EM) radar data is simulated based on the fullwave computational EM method known as the finitedifference, timedomain (FDTD) software [43], which was developed by the U.S. Army Research Laboratory (ARL). The software was validated for a wide variety of radar signature calculation scenarios [44, 45]. Our volumetric rough ground surface grid – with the embedded buried targets – was generated by using the surface root meansquare (rms) height and the correlation length parameters. The targets are flush buried at 23 cm depth. In our experiments, the easiest case of rough ground surface in our experiments, the surface rms is 5.6 mm and the correlation length is 15 cm. The SAR images of various targets and clutter objects are generated from EM data by coherently integrated individual radar return signals along over a range of aspect angles. The SAR images are formed using the backprojection image formation [46] with an integration angle of . Figure 4a shows the SAR images (using vertical transmitter, vertical receiver – VV – polarization) of some targets that are buried under a perfectly smooth ground surface. Each target is imaged at a random viewing aspect angle and an integration angle of . Figures 4b and 4c show the same targets as Figure 4a, except that they are buried under a rough ground surface (the easiest case corresponds to ground scale/noise level^{2}^{2}2Note that a higher ground scale means more noisy images. Henceforth, in the text as well as figures we simply refer to ground scale as noise level for ease of exposition. = 1 and harder case corresponds to noise level = 5). Similarly, Figures 4d, 4e, and 4f show the SAR images of some clutter objects buried under a smooth and rough surface, respectively. For training, the target and clutter object are buried under a smooth surface to generate high signaltoclutter ratio images. We include 12 SAR images that correspond to 12 different aspect angles () for each target type. For testing, the SAR images of targets and confusers are generated at random aspect angles and buried under rough ground surfaces. Various levels of ground surface roughness are simulated by selecting different ground surface scaling factors when embedding the test targets under the rough surfaces. Thus, the resulting test images are very noisy with a very low signaltoclutter ratio. Each image is a polarization signal of object which is formed by one of transmitterreceiver setups: verticalvertical (VV), horizontalhorizontal (HH), or horizontalvertical (HV). Each data sample of one object is represented by either i) one SAR image using data from one copol (VV, HH) channel or ii) two or more images using data from copol (VV, HH) and crosspol (HV) channels. For each target type, we tested 100 image samples measured at random aspect angles.
IiiB Denoised signal visualization
We construct the dictionary of all three polarizations, with being dictionaries of targets, confusers, and grounds, respectively. The subdictionary can be seen as the dictionary of the objects of interest. For a noisy signal , we first solve the following problem:
(18) 
where is the proposed SM constraint. The sparse tensor code is then decomposed into two parts, and . The latter can be considered coefficients corresponding to the ground dictionary . The original signal can be approximately decomposed into two parts: as separated noise, and as the denoised signal. Visualization of these signals are shown in Figure 5. We can see that the tensor framework successfully decomposes noisy signals into a clean part and a ground signal. These results show the potential of the proposed tensor sparsity frameworks for classification.
IiiC Overall classification accuracy
We apply four methods presented in IIC to different combinations of three polarizations: VV, HH, VV+HH, HH+HV, and VV+HH+HV (or ALL), and also compare these results with those obtained by a support vector machine (SVM) using the libsvm library [47]. SVM was also applied to classify UWB signals [13, 11]. The training set comprises all five target sets, four out of five confuser sets (each time we leave one confuser set out, which is meant to be unseen), and the ground set. While all confuser sets can be seen as one class – confuser class – there are two ways of organizing target sets. First, each of five targets is considered one class in case the identity of each target is crucial (we name this scenario separatetarget with five target classes and one confuser class). Second, if we only need to know whether an object of interest is a target or not, we can consider all five targets as one class and its corresponding name is alltarget with one target class and one confuser class. We also consider two families of the sparse coding problem, one with and one without the nonnegativity constraint on the sparse code in each of the tensor sparse coding methods. Each experiment is conducted 10 times and their results are reported as the average number. Parameters in each method are chosen by a 5fold crossvalidation procedure. In this experiment, all test samples are corrupted by small noise, i.e., the noise level is set to one. In our experiments, we use overall classification accuracy as the performance metric, which computes percentage of correctly classified samples over total test samples across all classes.
Overall classification accuracies of different methods on different polarization combinations are reported in Table I. From the table, a few inferences can be made:

SRCrelated methods with nonnegative sparse coefficients perform better than those without this constraint^{3}^{3}3Based on this observation, from now on, all other results are implicitly reported with the nonnegativity constraint.. In addition, SVM is outperformed by all other methods in all tasks.

SRCSM provides the best results in all combinations for both the separatetarget and alltarget scenarios. The best accuracies in both cases are significantly high with slightly over 90% in the sixclass classification problem and nearly 97% in the binary classification problem.

If only one polarization is used, SRCCR, SRCCC, and SRCSM have identical results, since all of them are basically reduced to traditional SRC in this case. Additionally, these three methods slightly outperform SRCGT in this scenario.

If only one polarization can be obtained, HH always outperforms VV and by a significant margin. Additionally, the HH+VV combination often worsens the results versus using HH alone.

If the same method is used on different combinations, the best results are mostly obtained by the combination of HH and HV polarizations in both the separatetarget and alltarget scenarios.
IiiD Effect of noise levels on overall accuracy
The results in the previous section are collected in the presence of small corruption (noise level is only 1). In real problems, objects of interest are, deliberately or not, buried under extremely rough surfaces in order to fool classifiers. In this section, we conduct an experiment to see how each method performs when the level of corruption increases in the separatetarget scenario.
Classification results of five different methods on different polarization combinations and different noise levels are shown in Figure 6. First of all, similar trends to small corruption can be observed in that SRCSM shows best performance in all cases with bigger gaps occurring at high levels of noise. In addition, of the four SRCrelated methods, SRCCC is beaten by all three others when more than one polarization involved. This can be explained by the fact that SRCCC suffers from the curse of dimensionality when each sample is represented by concatenating long vectors. It can also be seen that SRCGT performs better than SRCCR and SRCCC in the presence of multiple polarizations. Last but not least, the best classification results can be seen at HH+HV and ALL among the five different combinations considered.
IiiE Effect of tensor dictionary learning on overall accuracy
We report the classification results for the alltarget scenario with different noise levels. We also include the results of experiments with discriminative tensor dictionary learning methods. The results of HH+HV and ALL are reported, since they yield the best results, as observed in previous sections.
The results of nine different methods are depicted in Figure 7. These methods include SVM (the dotted line), the four SRCrelated methods (dashed lines), and their corresponding tensor dictionary learning counterparts (prefixed by TensorDL, solid lines). We can see that except for the CC case, tensor dictionary learning methods outperform their corresponding SRC with gaps widening as the noise level increases. This observation confirms that tensor discriminative dictionary learning methods indeed provide accuracy improvements. Methods with the SM constraint once again emerge the winner in all noise levels, and the best accuracy numbers are observed using the HH+HV combination.
IiiF Multiplerelativelook classification accuracy
We describe a key realworld scenario where multilook signals of an object of interest can be obtained. Figure 8a depicts the relative location of a radar carrier, a jet plane in this case, and an object of interest. The plane moves around the object at an angle corresponding to the blue arrow. One sample of object can be captured at this whole range, which can be seen as one of its looks. At the same time, the radar can also capture multiple looks of the object at smaller angles represented by green arrows. By continuing considering even smaller angles, more representatives of the object can be captured. These multiple views can provide complementary information of the object, highly likely resulting in better classification accuracy of the system.
For the training samples, for each object, we generate two sets of signals. Each set contains samples captured after each , and each set has total of 24 views. Both sets start at the same relative angle but the first is captured by an integration angle of , the angle in the second set is . For the test samples, each object is represented by three signals: one by an integration angle of and two others by an integration angle of , as depicted in Figure 8a. Similar to previous experiments, test samples are captured at random relative angles. Ground samples are also simulated in the same way. Based on three signals captured, we establish three different way of using this information in the classification process:

1look: for each object, we use signals at integration angle of only. If only one polarization is used, an object of interest can be identified by SRC (and implicitly, SVM). If more than one polarization is involved, one object will be determined by SRCSM, as this is the best method based on previous experiments (see Figure 8b).

3look: uses all three signals to represent an object. In this case, the relationship between the signal and the first signal can be modeled by the SRCSM, while the relationship between two signals can be formed by either JSRC or ShiftSRC (see Figure 8d).
For this experiment, we consider the alltarget scenario and two polarization combinations, HH and HH+HV. It is worth noting that for the HH+HV case, there will be four channels in 2look and six channels in 3look. The results of different methods are shown in Figure 9. We can see that SVM still performs well at the lowest noise level (1), but drastically degrades with a little more corruption. On the other hand, SRCbased methods obtain good results even if the noise level is large for the HH+HV combination. Of sparse representation methods, ShiftSRC outperforms the others with the gap becoming larger for highly corrupted signals. Interestingly, ShiftSRC at 2look provides even better results than JSRC does at 3look. These results confirm the advantages of the proposed relativelook sparse representation model.
IiiG Overall accuracy on measured UWB SAR data
In this section, the results of this technique are illustrated using the data from the ARL UWB lowfrequency SAR, which transmits radar signals occupying the frequency spectrum that span approximately from 50 to 1150 MHz [48]. Figure 10 shows a SAR image formed using data collected at Yuma Proving Grounds (YPG) [2]. The scene includes several rows of buried mines surrounded by clutter objects such as bushes, rocks, tire tracks, etc.
The set contains signals of 50 targets and 50 confusers. Each signal has resolution of and already includes noise from the ground. Visualization of six samples in each class are shown in Figure 11. We conduct the alltarget experiment and report results of different methods on different polarization combinations in Figure 12. For each combination, three competing methods are considered: SVM, SRC, and TensorDL (both with the SM constraint). Since grounds are fixed in this data set, we report the results based on size of the training set. For each training size (10, 20, 30, or 40), we randomly choose samples from each class for training; the rest samples are considered test samples.
The results are reported in Figure 12 as the average of 10 tests. In general, the tensor dictionary learning performs better than sparse representation in all combinations except for the HH case. SVM also provides good results but is outperformed by other competing methods.
Iv Conclusion
We have developed a novel discrimination and classification framework for lowfrequency UWB SAR imagery using sparse representationbased methods. The framework is applicable for either single channel or multiple channel (polarizations, multilook) SAR imagery. The techniques are tested and the discrimination/classification performance of targets of interest versus natural and manmade clutter in challenging scenarios is measured using both rigorous electromagnetic simulation data and real data from the ARL UWB radar. The classification results show encouraging potential of tensor sparsity methods, even when the test images are very noisy (buried under extremely rough ground surfaces), targets have small RCSs, and the targets’ responses have very little detailed structures, i.e., targets are small compared to the wavelengths of the radar signals. The SRCSM technique and its dictionary learning version consistently offers the best results with the combination of co and crosspol data, e.g., HH+HV or ALL. In addition, the nonnegativity constraints on sparse tensor codes enhance the performance of the systems. Furthermore, we also show the ShiftSRC model is particularly suitable for problems with multiplerelativelook signals.
References
 [1] L. H. Nguyen, R. Kapoor, and J. Sichina, “Detection algorithms for ultrawideband foliagepenetration radar,” Proceedings of the SPIE, 3066, pp. 165–176, 1997.
 [2] L. H. Nguyen, K. Kappra, D. Wong, R. Kapoor, and J. Sichina, “Mine field detection algorithm utilizing data from an ultrawideband widearea surveillance radar,” SPIE International Society of Optical Engineers, 3392, no. 627, 1998.
 [3] L. H. Nguyen, M. Ressler, and J. Sichina, “Sensing through the wall imaging using the Army Research Lab ultrawideband synchronous impulse reconstruction (UWB SIRE) radar,” Proceedings of SPIE, 6947, no. 69470B, 2008.
 [4] L. Nguyen, R. Kapoor, D. Wong, and J. Sichina, “Ultrawideband radar target discrimination utilizing an advanced feature set,” SPIE, Algorithms for Synthetic Aperture Radar Imagery V, 1998.

[5]
T. R. Damarla, L. Nguyen, and K. Ranney, “Detection of UXO contaminated land fields using hidden Markov models in the SAR images generated by airborne radar system,” in
Proceedings of SPIE, vol. 4374, 2001, p. 61.  [6] P. Runkle, L. H. Nguyen, J. H. McClellan, and L. Carin, “Multiaspect target detection for sar imagery using hidden markov models,” IEEE Trans. on Geos. and Remote Sensing, vol. 39, no. 1, pp. 46–55, 2001.
 [7] M. Anabuki, S. Okumura, T. Sato, T. Sakamoto, K. Saho, M. Yoshioka, K. Inoue, T. Fukuda, and H. Sakai, “Ultrawideband radar imaging using adaptive array and doppler separation,” IEEE Transactions on Aerospace and Electronic Systems, vol. 53, no. 1, pp. 190–200, 2017.
 [8] T. Sakamoto, T. Sato, P. Aubry, and A. Yarovoy, “Fast imaging method for security systems using ultrawideband radar,” IEEE Transactions on Aerospace and Electronic Systems, vol. 52, no. 2, pp. 658–670, 2016.
 [9] A. Yamaryo, T. Takatori, S. Kidera, and T. Kirimoto, “Rangepoint migrationbased image expansion method exploiting fully polarimetric data for UWB shortrange radar,” IEEE Transactions on Geoscience and Remote Sensing, vol. 56, no. 4, pp. 2170–2182, 2018.

[10]
M. A. Kiasari, S. Y. Na, and J. Y. Kim, “Classification of human postures using ultrawide band radar based on neural networks,” in
IEEE International Conference on IT Convergence and Security (ICITCS), 2014, pp. 1–4.  [11] J. Bryan, J. Kwon, N. Lee, and Y. Kim, “Application of ultrawide band radar for classification of human activities,” IET Radar, Sonar & Navigation, vol. 6, no. 3, pp. 172–179, 2012.
 [12] X. Liang, T. Lv, H. Zhang, Y. Gao, and G. Fang, “Throughwall human being detection using UWB impulse radar,” EURASIP Journal on Wireless Communications and Networking, vol. 2018, no. 1, p. 46, 2018.
 [13] D. Wang, L. Chen, D. Piscarreta, and K. W. Tam, “Classification and regression of ultra wide band signals,” in IEEE Chinese Automation Congress (CAC), 2015, pp. 1907–1912.
 [14] J. Wright, A. Yang, A. Ganesh, S. Sastry, and Y. Ma, “Robust face recognition via sparse representation,” IEEE Trans. on Pattern Analysis and Machine Intelligence, vol. 31, no. 2, pp. 210–227, Feb. 2009.
 [15] T. H. Vu, H. S. Mousavi, V. Monga, U. Rao, and G. Rao, “DFDL: Discriminative featureoriented dictionary learning for histopathological image classification,” IEEE Int. Symposium on Biomedical Imaging, pp. 990–994, 2015.
 [16] U. Srinivas, H. S. Mousavi, C. Jeon, V. Monga, A. Hattel, and B. Jayarao, “SHIRC: A simultaneous sparsity model for histopathological image representation and classification,” IEEE Int. Symposium on Biomedical Imaging, pp. 1118–1121, Apr. 2013.
 [17] T. H. Vu, H. S. Mousavi, V. Monga, U. Rao, and G. Rao, “Histopathological image classification using discriminative featureoriented dictionary learning,” IEEE Trans. on Medical Imaging, vol. 35, no. 3, pp. 738–751, March, 2016.
 [18] U. Srinivas, H. S. Mousavi, V. Monga, A. Hattel, and B. Jayarao, “Simultaneous sparsity model for histopathological image representation and classification,” IEEE Trans. on Medical Imaging, vol. 33, no. 5, pp. 1163–1179, May 2014.
 [19] X. Sun, N. M. Nasrabadi, and T. D. Tran, “Taskdriven dictionary learning for hyperspectral image classification with structured sparsity constraints,” IEEE Trans. on Geos. and Remote Sensing, vol. 53, no. 8, pp. 4457–4471, 2015.
 [20] X. Sun, Q. Qu, N. M. Nasrabadi, and T. D. Tran, “Structured priors for sparserepresentationbased hyperspectral image classification,” IEEE Geos. and Remote Sensing Letters, vol. 11, no. 7, pp. 1235–1239, 2014.
 [21] Y. Chen, N. M. Nasrabadi, and T. D. Tran, “Hyperspectral image classification via kernel sparse representation,” IEEE Trans. on Geos. and Remote Sensing, vol. 51, no. 1, pp. 217–231, 2013.
 [22] H. Zhang, N. M. Nasrabadi, Y. Zhang, and T. S. Huang, “Multiview automatic target recognition using joint sparse representation,” IEEE Trans. on Aerospace and Electronic Systems, vol. 48, no. 3, pp. 2481–2497, 2012.
 [23] X. Mo, V. Monga, R. Bala, and Z. Fan, “Adaptive sparse representations for video anomaly detection,” IEEE Trans. on Circuits and Systems for Video Technology, vol. 24, no. 4, pp. 631–645, 2014.
 [24] T. H. Vu and V. Monga, “Learning a lowrank shared dictionary for object classification,” IEEE Int. Conference on Image Processing, 2016.
 [25] H. S. Mousavi, U. Srinivas, V. Monga, Y. Suo, M. Dao, and T. Tran, “Multitask image classification via collaborative, hierarchical spikeandslab priors,” in IEEE Int. Conference on Image Processing, 2014, pp. 4236–4240.
 [26] T. H. Vu and V. Monga, “Fast lowrank shared dictionary learning for image classification,” IEEE Trans. on Image Processing, vol. 26, no. 11, pp. 5160–5175, Nov 2017.
 [27] U. Srinivas, Y. Suo, M. Dao, V. Monga, and T. D. Tran, “Structured sparse priors for image classification,” IEEE Trans. on Image Processing, vol. 24, no. 6, pp. 1763–1776, 2015.
 [28] H. Zhang, Y. Zhang, N. M. Nasrabadi, and T. S. Huang, “Jointstructuredsparsitybased classification for multiplemeasurement transient acoustic signals,” IEEE Trans. on Systems, Man, and Cybernetics, Part B: Cybernetics, vol. 42, no. 6, pp. 1586–1598, 2012.
 [29] M. Dao, Y. Suo, S. P. Chin, and T. D. Tran, “Structured sparse representation with lowrank interference,” in IEEE Asilomar Conference on Signals, Systems and Computers, 2014, pp. 106–110.
 [30] M. Dao, N. H. Nguyen, N. M. Nasrabadi, and T. D. Tran, “Collaborative multisensor classification via sparsitybased representation,” IEEE Trans. on Signal Processing, vol. 64, no. 9, pp. 2400–2415, 2016.
 [31] H. Van Nguyen, V. M. Patel, N. M. Nasrabadi, and R. Chellappa, “Design of nonlinear kernel dictionaries for object recognition,” IEEE Trans. on Image Processing, vol. 22, no. 12, pp. 5123–5135, 2013.
 [32] L. Nguyen and C. Le, “Sparsity driven target discrimination for ultrawideband (UWB) downlooking Ground penetration radar (GPR),” TriService Radar, 2013.
 [33] J. A. Tropp, A. C. Gilbert, and M. J. Strauss, “Algorithms for simultaneous sparse approximation. part i: Greedy pursuit,” Signal Processing, vol. 86, no. 3, pp. 572–588, 2006.
 [34] A. Beck and M. Teboulle, “A fast iterative shrinkagethresholding algorithm for linear inverse problems,” SIAM Journal on Imaging Sciences, vol. 2, no. 1, pp. 183–202, 2009.
 [35] T. H. Vu, “DICTOL – A dictionary learning toolbox,” https://github.com/tiepvupsu/DICTOL, 2017, [Online; accessed 15November2017].
 [36] T. H. Vu, L. Nguyen, C. Le, and V. Monga, “Tensor sparsity for classifying lowfrequency ultrawideband (UWB) SAR imagery,” 2017, pp. 0557–0562.

[37]
S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed
optimization and statistical learning via the alternating direction method of
multipliers,”
Foundations and Trends® in Machine Learning
, vol. 3, no. 1, pp. 1–122, 2011.  [38] “SPArse Modeling Software,” http://spamsdevel.gforge.inria.fr/, accessed: 20141105.
 [39] Y. Yu, J. Huang, S. Zhang, C. Restif, X. Huang, and D. Metaxas, “Group sparsity based classification for cervigram segmentation,” IEEE Int. Symposium on Biomedical Imaging, pp. 1425–1429, 2011.

[40]
M. Yang, L. Zhang, X. Feng, and D. Zhang, “Sparse representation based
Fisher discrimination dictionary learning for image classification,”
Int. Journal of Computer Vision
, vol. 109, no. 3, pp. 209–232, 2014. 
[41]
I. Ramirez, P. Sprechmann, and G. Sapiro, “Classification and clustering via
dictionary learning with structured incoherence and shared features,” in
IEEE Int. Conference Computer Vision Pattern Recognition
, 2010, pp. 3501–3508.  [42] T. H. Vu, “Tensor Sparsity Toolbox,” https://github.com/tiepvupsu/tensorsparsity, 2017, [Online; accessed 10November2017].
 [43] T. Dogaru, “AFDTD user’s manual,” ARL Technical Report, Adelphi, MD, ARLTR5145, March 2010.
 [44] T. Dogaru, L. Nguyen, and C. Le, “Computer models of the human body signature for sensing through the wall radar applications,” ARL, Adelphi, MD, ARLTR4290, September 2007.
 [45] D. Liao and T. Dogaru, “Fullwave characterization of rough terrain surface scattering for forwardlooking radar applications,” IEEE Trans. on Antenna and Propagation, vol. 60, pp. 3853–3866, August 2012.
 [46] J. McCorkle and L. Nguyen, “Focusing of dispersive targets using synthetic aperture radar,” U.S. Army Research Laboratory, Adelphi, MD, ARLTR305, August 1994.
 [47] C.C. Chang and C.J. Lin, “LIBSVM: A library for support vector machines,” ACM Trans. on Intelligent Systems and Technology, vol. 2, pp. 27:1–27:27, 2011, software available at http://www.csie.ntu.edu.tw/~cjlin/libsvm.
 [48] M. Ressler, L. Happ, L. Nguyen, T. Ton, and M. Bennett, “The army research laboratory ultrawide band testbed radars,” in IEEE International Radar Conference, 1995, pp. 686–691.
Comments
There are no comments yet.