Spectral Super-Resolution of Hyperspectral Imagery Using Reweighted l 1 Spatial Filtering

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1 1 Spectrl Super-Resolution of Hyperspectrl Imgery Using Reweighted l 1 Sptil Filtering Adm S. Chrles, Student Member, IEEE, nd Christopher J. Rozell, Senior Member, IEEE Abstrct Sprsity-bsed models hve enbled significnt dvnces in mny imge processing tsks. Hyperspectrl imgery (HSI) in prticulr hs benefited from these pproches due to the significnt low-dimensionl structure in both sptil nd spectrl dimensions. Specificlly, previous work hs shown tht sprsity models cn be used for spectrl super-resolution, where spectrl signtures with HSI-level resolution re recovered from mesurements with multispectrl-level resolution (i.e., n order of mgnitude fewer spectrl bnds). In this pper we expnd on those results by introducing new inference pproch known s reweighted l 1 sptil filtering (RWL1-SF). RWL1-SF incorportes more sophisticted signl model tht llows for vritions in the SNR t ech pixel s well s sptil dependencies between neighboring pixels. The results demonstrte tht the proposed pproch leverges signl structure beyond simple sprsity to chieve significnt improvements in spectrl super-resolution. Index Terms Hyperspectrl imgery, sprse pproximtion, reweighted l 1 I. INTRODUCTION Hyperspectrl Imgery (HSI) cptures detiled terrestril informtion with high resolution in both the sptil nd spectrl dimensions [1]. While this level of detil is importnt in mny remote sensing tsks such s object detection nd mteril discrimintion, cquiring HSI cn be more prohibitive thn cquiring multispectrl imgery (MSI). For exmple, sensor fbriction cost nd imge cquisition time (for comprble SNR per spectrl bin) increses s the sensor s bndwidth nrrows [1]. The finncil cost differentil leds to MSI being more ccessible thn HSI, both in terms of currently flown instruments nd rchived dt. In pplictions where high temporl resolution is required (e.g., due to high velocity of the imging pltform), the decresing SNR per bin due to shorter cquisition times might mke it more desirble to combine HSI bins to collect imgery with MSI-level spectrl resolution. 1 Recent results hve shown the potentil to use MSI to obtin HSI-level resolution imges by performing spectrl superresolution [2]. The tsk of super-resolution is to use prior knowledge of the signl sttistics in post-processing to infer the content of the signl t finer resolution thn the originl observtions. In photogrphic imges, signl models bsed on the notion of sprsity (i.e. imges cn be described by smll Mnuscript received September 6, 212. This work ws supported in prt by NSF grnt CCF * Corresponding uthor. The uthors re with the School of Electricl nd Computer Engineering, Georgi Institute of Technology, Atlnt, GA, USA (e-mil: {chrles6,crozell}@gtech.edu). The uthors re grteful to C. Bchmnn t the Nvl Reserch Lbortory for generously providing the Smith Islnd HSI dt set. 1 In the reminder of the pper we will genericlly refer to MSI dt, including the cse of MSI-level spectrl resolution from n HSI sensor. number of toms in potentilly lrge dictionry) hve been very successful in sptil super-resolution pplictions [3]. In previous work we hve demonstrted the pplicbility of sprsity-bsed methods in spectrl super-resolution for HSI [2]. Specificlly, by lerning dictionry of spectrl signtures tht sprsely decompose the spectrl response in ech pixel, we lern n pproximtion to the dt mnifold tht cptures rich higher-order sttisticl structure in HSI dt. This model cn then be used to perform spectrl superresolution from MSI-level dt to HSI-level resolution with very high ccurcy [2]. In this work we improve on these previous results by proposing reweighted l 1 sptil filtering lgorithm to incorporte sptil regulrity to improve spectrl super-resolution. This pproch closely follows recent work in dynmic filtering where temporl correltions hve been used to improve recovery of time-vrying signls in reweighted l 1 frmework [4]. The min contribution of this work is to show tht more dvnced recovery lgorithms cn produce significnt improvements in the spectrl super-resolution results for scenes with significnt sptil regulrity, with most of the improvement coming from pixels tht re not well-modeled by bsic sprsity model. II. SUPER-RESOLUTION VIA SPARSE CODING A. The Sprse Coding Model The sprse coding model for HSI represents the observed spectrum t ech pixel x i,j R M by smll number of terms in liner genertive model x i,j = φ k i,j,k + ɛ i,j = Φ i,j + ɛ i,j, where Φ R M N is mtrix of (non-negtive) dictionry elements, i,j R N is the (non-negtive) coefficient vector for pixel {i, j} nd ɛ i,j R M is noise term. The coefficient vlues cn be recovered for given pixel by solving n l 1 regulrized lest-squres optimiztion problem (termed Bsis Pursuit De-Noising (BPDN)) â i,j = rg min x i,j Φ γ 1, under positivity constrints ( i,j,k ), where γ is prmeter tht trdes off between the dt fidelity (lest-squres) term nd the sprsity bsed regulrizer (the l 1 norm) [5]. The BPDN optimiztion cn be interpreted s finding mximum -priori (MAP) estimte of the coefficients under ssumptions of Gussin noise nd n independent identiclly distributed (i.i.d.) Lplcin prior distribution on the coefficients [2]. This prior probbly distribution for the coefficients is chosen to encourge only smll frction of them to be non-zero

2 2 simultneously (e.g., the prior distribution is chosen to hve high kurtosis so it hs pek round zero). The l 1 regulriztion pproch hs previously been shown to be effective for un-mixing in HSI [6] [8]. Furthermore, we hve previously shown tht completely unsupervised pproch cn be used with this model to lern effective dictionry elements [2]. While these lerned dictionry elements re not gurnteed to be mteril spectr (nd so re not clled endmembers), previous work hs shown tht they re often highly correlted with known mterils in the scene. There is no ssumption tht the coefficients in ech pixel sum to one, mening tht the coefficients cn be thought of s reltive contributions of ech dictionry to the spectrl signture (but the totl sum of the coefficients will vry depending on the totl rdince t tht pixel). A more detiled exmintion of the dictionry elements lerned under this model cn be found in [2]. Note tht lthough the genertive model bove is liner, the sprsity constrint introduces nonliner component to the model tht cn cpture dt sttistics tht re not well represented in typicl liner mixture model (e.g., forming locl pproximtion to mnifold structures in the dt) [2]. B. Spectrl Super-Resolution The min pproch to spectrl super-resolution will be to lern dictionry for HSI using set of trining dt, nd then use this dictionry (which cptures higher order sttistics in the HSI dt) to infer HSI-level resolution from MSI dt. Since ech MSI bnd corresponds to weighted pooling of mesurements tken over severl HSI bnds, we concisely write the reltionship between the (unobserved) HSI spectrum nd the (observed) MSI spectrum using mtrix multipliction y i,j = Bx i,j = BΦ i,j + ɛ i,j, where y i,j R P is the MSI spectrum, ɛ i,j = Bɛ i,j is the MSI resolution mesurement error, nd B R P M is the blurring mtrix which pools neighboring bnds. We note here tht in some cses y i,j completely omits some spectrl bnds tht re in HSI but not mesurble with some MSI sensors, but these bnds must still be inferred. Figure 1 shows n exmple of simulted MSI bnds nd their reltionship to HSI rnges. Note tht while we will typiclly consider pooling opertions tht hve flt spectrl responses in bnds tht do not overlp for simplicity, other more relistic models tilored to specific sensor could be used s well. Using the sprse coding model, we infer sprse coefficients in the HSI-level spectrl signtures from the MSI dt by performing n inverse problem using BPDN â i,j = rg min y i,j BΦ γ 1. (1) The HSI spectrum cn be recovered from these coefficients by x i,j = Φâ i,j. In words, the pproch bove solves n optimiztion problem tht seeks the HSI coefficients tht re both consistent with the model (i.e., they re sprse) nd tht explin the mesured MSI spectrum. In previous work, the bove pproch is demonstrted with different choices for the mtrix B (representing MSI sensors nd HSI sensors run t MSI-level spectrl resolution), nd with scenes collected Simulted MSI Mesurements Wter Ab. HSI Only { Bnd 8 Bnd 7 Bnd 6 Bnd 5 Bnd 4 Bnd 3 Bnd 2 Bnd Wvelength (μm) Fig. 1. Simulted MSI responses re comprised of eight spectrl bnds tht pool HSI mesurements. The brs show the wvelengths included in ech bnd for the simulted MSI mesurements. The top row shows the wter bsorption bnds tht re not included in ny dt. The second row shows the spectrl rnges tht re present in the HSI dt but not included in the simulted MSI mesurements (i.e., these bnds must be inferred with no dt). t different times of the yer (i.e., contining differences in the sttistics due to vegettion chnges, etc.) [2]. The reconstruction errors between the true (orcle) HSI nd the inferred HSI spectr were very low, in the rnge of 2-3% reltive men squred error (rmse) depending on the cse. While this prior work is encourging, the results lso showed tht the reltively smll subset of pixels inconsistent with the sprsity model (e.g., mixtures of mny types of vegettion) cn be outliers with much worse reconstruction performnce. III. REWEIGHTED l 1 FOR SUPER-RESOLUTION A. Reweighted l 1 (RWL1) As first step to improving super-resolution performnce, we generlize the sprsity model to llow the SNR for ech coefficient to be n unknown prmeter tht is estimted s prt of the inference process. In BPDN (eqution (1)), the trdeoff prmeter γ depends on the SNR (the rtio of the vrince in the sprse coefficients to the noise vrince [2]) nd is the sme for ech coefficient. In contrst, the reweighted l 1 (RWL1) frmework [9], [1] llows ech coefficient i,j,k its own prmeter γ i,j,k, where nd γ re inferred concurrently. Specificlly, RWL1 is equivlent to using the itertive Expecttion-Mximiztion (EM) lgorithm to find joint estimte of nd γ ssuming tht γ hs n i.i.d. Gmm hyperprior distribution. While more technicl detils of the model nd lgorithm cn be found in [1], the RWL1 lgorithm pplied to the super-resolution problem cn be stted succinctly s lternting weighted BPDN optimiztion nd n nlytic updte to the weights until convergence: â n i,j = rg min x i,j BΦ γ γ n i,j,k = â n i,j,k α, + β k γ n 1 i,j,k k, where α, β nd γ re prmeters relted to the hyperprior on γ nd n is the itertion number. One wy to intuitively understnd the RWL1 lgorithm is to understnd the effect ech γ i,j,k hs on the weighted l 1

3 3 optimiztion problem. Lowering given γ i,j,k vlue mkes it esier for the corresponding coefficient to be ctivted in the next BPDN itertion. By itertively reclculting the weights, coefficients tht re ctivted in the initil optimiztion become more esily ctivted in future itertions (vi smller weights) nd unused coefficients re more difficult to ctivte in future itertions (vi higher weights). Additionl literture hs linked RWL1 to pproximting solutions to l p regulrized lest squres problems for p < 1 [11] nd symptotic theoreticl gurntees in other inverse problems (e.g., compressed sensing) [12]. B. Reweighted l 1 Sptil Filtering (RWL1-SF) While spectrl sttistics re informtive enough to perform super-resolution in mny cses, sptil regulrity cn often be leverged in some types of scenes to improve performnce (especilly when the sprsity model is not good fit for given pixel). Sptil regulrity ws lso used recently in the context of mteril clssifiction, indicting its utility in HSI [13]. Therefore, s second step to improving super-resolution performnce, we further generlize the RWL1 model to incorporte sptil informtion into the inference process. Specificlly, in our proposed reweighted l 1 sptil filtering (RWL1-SF), we updte the weights for given coefficient using combintion of informtion from the previous itertion on neighboring pixels (similr to the reweighted l 1 dynmic filtering lgorithm developed in [4]). In this wy, even wek evidence from individul pixels in locl neighborhood cn be ggregted to improve the inference in cses tht would be prticulrly difficult when just considering individul pixels independently. To be precise, consider the mtrix of ll coefficients for the k th dictionry element, [A k ] i,j = i,j,k. In ech itertion of RWL1-SF, the weight for the k th coefficient t the pixel in row i nd column j is set by weighted pooling of the previous estimtes for the k th coefficient t the neighboring pixels. While there re mny potentil wys to implement this sptil ggregtion nd weight updting, in this pper we use simple liner weighted verge: α γ i,j,k = [Ψ A k ] i,j + β where the term [Ψ A k ] i,j represents the {i, j} th term of the kernel Ψ R L P convolved with the sptil field of previous estimtes for the k th coefficient. Note tht while this sptil regulriztion cn ccumulte wek evidence spred over severl neighboring pixels to perform inference, the model does not force sptil homogeneity so tht single-pixel (or sub-pixel) objects re missed. In other words, rther thn low-pss filtering the estimtes of interest (the i,j,k vribles), the sptil verging is pplied to second order vrible (γ i,j,k ) tht simply bises sprse inference process. In fct, though n explicit test with single-pixel nomlies is beyond the scope of this letter, previous work using this pproch for dynmic filtering [4] showed tht this method of stochstic filtering is prticulrly robust to model mismtch. The kernel Ψ incorportes the knowledge tht dependencies should hve limited sptil extent nd will be modulted Pixel t Kernel Centered t Fig. 2. The kernel Ψ determines the influence from neighboring pixels on coefficient inference t given loction. When the L P kernel is centered on the {i, j} th pixel it describes the weighted summtion of neighboring coefficient estimtes tht influence the next coefficient estimte in tht pixel. depending on the distnce between the pixels, s depicted in Figure 2. The vlue in the {l, p} th entry of Ψ indictes the mount which the {i + l L/2, j + p P/2} th element of A k influences the {i, j} th element of A k in the next itertion of the inference. Typiclly, the center (,) vlue of Ψ should be unity nd the kernel vlues should tper off towrds the edges to represent the decying dependence with distnce. In this work we use the sme 5 5 pixel Gussin kernel shpe for ll prts of the estimtion, but in generl ech coefficient or pixel loction could hve different kernel if there ws dvnced knowledge of the sptil nd spectrl dependencies in the dt. Indeed, in scenes with very different sttistics thn the HSI used s n exmple here (e.g., urbn scenes), the sptil regulriztion process my benefit from specilized tretment of edges in the imge. IV. PERFORMANCE COMPARISONS We test the performnce of RWL1 nd RWL1-SF ginst previous results on segments of HSI from Smith Islnd, VA. These two HSI imges were tken by the PROBE2 sensor on October 18, 21 nd August 22, 21 nd hve 113 usble spectrl bnds spnning the µm rnge (fter removl of wter bsorption bnds nd pplying tmospheric correction to estimte reflectnce) nd sptil resolution of pproximtely 4.5m 2. We simulte MSI mesurements by creting mtrix B to represent response function tht entirely eliminted mesurements in higher wvelength regions nd pooled the remining HSI mesurements into eight spectrl bnds shown in Figure 1 (ech row of B hs ones over bnds included nd zeros otherwise). We lern 44-element dictionry Φ on the October 18, 21 imge s in [2], nd test recovery on both imges. Of prticulr note is tht the two imges were tken severl months prt, nd the sttisticl chnges with the sesonl vritions mde the recovery of the August imge the most chllenging test cse in prior work [2]. We estimte the originl 113 bnds from the 8 simulted MSI bnds for both imges vi BPDN, RWL1 nd RWL1-SF. For testing purposes we recover contiguous 68x288 pixel region (omitting 11 pixels with severe sensor errors) from the Smith Islnd dtset, shown in Figure 3. This region yielded prticulrly poor performnce when using BPDN for superresolution in prior work [2]. As shown in Tble I, the previous 2 More detils bout this dtset cn be found in [14].

4 4 TABLE I SUPER-RESOLUTION FROM SIMULATED MSI MEASUREMENTS IN TERMS OF RELATIVE MSE AND SPECTRAL ANGLE (SA). October 18 (Sme Dy) rmse SA (degrees) Men Medin Men Medin BPDN 2.33%.35% RWL1.85%.24% RWL1-SF.68%.23% August 11 (Different Dy) rmse SA (degrees) Men Medin Men Medin BPDN 6.25% 6.25% RWL1 3.34% 3.2% RWL1-SF 2.45% 1.89% men rmse ws 6.3% nd the medin rmse ws 3.3% for this region on the August imge, which is considerbly worse thn the performnce seen on sets of pixels rndomly selected throughout the entire imge (nerly triple the 2.456% men nd n order of mgnitude higher thn the.1219% medin rmse observed on the full dtset [2]). As stted in [2], BPDN super-resolution resulted in the highest error in portions of the scene tht re expected to hve more heterogeneous compositions, therefore mking the bsic sprsity model poorer fit thn it is in more homogeneous regions. To illustrte this, Figure 3 shows the distribution of BPDN reconstruction errors (mesured in spectrl ngle) for the sme dy dtset, highlighting the difference in performnce in distinct regions of heterogeneous mterils on the ground. Unsurprisingly, the higher errors re lso concentrted in the HSI spectrl bnds tht re not mesured in the MSI dt. Previous work [2] shows tht if the sme number of mesurements re tken over the whole HSI spectrl rnge (corresponding to n HSI sensor operting in lower spectrl resolution mode for higher temporl resolution), this mbiguity is reduced nd performnce increses significntly. Tble I provides men nd medin recovery results, illustrting significnt performnce improvements when using RWL1 insted of BPDN, nd further substntil improvements when using RWL1-SF. Figure 4 illustrtes two exmple pixels tht re representtive of the esiest nd most chllenging performnce for the October imge. For the best cse, the spectr re nerly indistinguishble from the true HSI. For the worst cse reconstruction we note tht the errors re clerly concentrted in the unmesured (high wvelength) spectrl rnges nd tht the proposed lgorithms mke substntil improvements in the recovery over the previous results using BPDN. Figure 5 illustrtes tht the overll sttistics of the dt in the August imge re lso better preserved when using RWL1-SF insted of BPDN, with first four principl components of the reconstructed dt (ccounting for 99.99% of the energy in the imge segment) much more closely pproximting the principl components of the HSI when using RWL1-SF. Reflectnce Reflectnce Actul Spectrum BPDN RWL1 RWL1 SF Wvelength (µm) Wvelength (µm) Fig. 4. Two exmple spectr super-resolved from MSI-level dt. Top plot is representtive of best-cse performnce nd bottom plot is representtive of worst-cse performnce for previous pproches [2]. Note tht errors re highly concentrted in unmesured bnds..1.5 Rw Dt BPDN RWL1 RWL1 SF.15 PCA Wvelength λ (µm) PCA 2.2 PCA 3.2 PCA 4 Fig. 5. The first four principl components the recovered HSI spectr compred to the principl components of the originl HSI dt. V. CONCLUSIONS Super-resolving MSI dt to HSI-level spectrl resolutions is technique tht is of prticulr importnce given the vlue of high resolution spectrl informtion. The proposed lgorithms leverge both more dvnced sprsity models in

5 5 RGB Imge BPDN Spectrl Angle 1 RWL1 Spectrl Angle RWL1 SF Spectrl Angle Number Below BPDN RWL1 RWL1 SF Spectrl Angle (degrees) Fig. 3. Left: The RGB imge of the October region being tested nd the het mps depicting the spectrl ngle errors throughout the region using BPDN, RWL1 nd RWL1-SF. The lrgest improvements over BPDN occur long the shoreline where the mteril mixture is very heterogeneous (e.g., wter, snd, vegettion) nd the sprsity model lone is insufficient. Right: The cumultive distribution function (CDF) of the spectrl ngle errors. Note tht the BPDN CDF hs hevy til, indicting mny pixels with poor performnce. RWL1 improves performnce significntly. RWL1-SF uses model of sptil dependence to further reduce the outliers nd improve performnce, with 9% of the pixels hving spectrl ngle errors less thn degrees. ech pixel, s well s sptil regulrity between pixels. This incresed model structure improves on previous super-resolution results significntly, especilly in the pixels tht were outliers in previous results due to their poor super-resolution performnce [2]. Specificlly, using dditionl intr-pixel structure in RWL1 yielded 35.62% nd 16.29% improvement in the men nd medin SA, respectively. Incorporting sptil dependencies in RWL1-SF boosted these results further, giving totl of 4.96% improvement in the men SA nd 19.66% improvement in medin SA. While the vlue of this superresolution technique will ultimtely need to be verified in terms of performnce in specific pplictions, we note tht 9% of the recovered pixels in the current dtset hd spectrl ngle error less thn 7 degrees. While future improvements my continue to be mde, this level of error is well within the clss spectrl width of some clssifiers currently in use (e.g., 7 degrees to 3 degree in [15]). Agin, we note tht the presented dt includes some of the most chllenging problem spects from the previous work (i.e., difficult pixels nd MSI mesurements with no dt from some HSI bnds). While the previous pproch using BPDN chieved very good performnce in mny cses, the RWL1-SF enhncement in this work shows substntil improvements in the most chllenging test cses. The incresed performnce demonstrted here leds us to conclude tht importnt structure beyond simple spectrl sprsity exists in some types of HSI dt nd cn be exploited for significnt gins in super-resolution performnce. We lso note tht this performnce improvement comes t moderte dditionl cost to the previously reported results (no more thn few EM itertions solving BPDN). While these results re encourging, there re severl further venues to explore. Most importntly, we hve evluted the proposed lgorithms on simulted MSI dt to fcilitte orcle evlution with ground-truth HSI dt (ll in the reflectnce domin). 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