efficient simultaneous image deconvolution and upsampling...

Efficient simultaneous image deconvolution andupsampling algorithm for low-resolution

microwave sounder data

Jing Qin,a,* Igor Yanovsky,b,c and Wotao Yina

aUniversity of California, Department of Mathematics, Los Angeles,California 90095, United States

bJet Propulsion Laboratory, California Institute of Technology, Pasadena,California 91109, United States

cUniversity of California, Joint Institute for Regional Earth System Science and Engineering,Los Angeles, California 90095, United States

Abstract. Microwave imaging has been widely used in the prediction and tracking of hurricanes,typhoons, and tropical storms. Due to the limitations of sensors, the acquired remote sensing dataare usually blurry and have relatively low resolution, which calls for the development of fastalgorithms for deblurring and enhancing the resolution. We propose an efficient algorithmfor simultaneous image deconvolution and upsampling for low-resolution microwave hurricanedata. Our model involves convolution, downsampling, and the total variation regularization.After reformulating the model, we are able to apply the alternating direction method of multi-pliers and obtain three subproblems, each of which has a closed-form solution. We also extendthe framework to the multichannel case with the multichannel total variation regularization.A variety of numerical experiments on synthetic and real Advanced Microwave SoundingUnit and Microwave Humidity Sounder data were conducted. The results demonstrate the out-standing performance of the proposed method. © 2015 Society of Photo-Optical InstrumentationEngineers (SPIE) [DOI: 10.1117/1.JRS.9.095035]

Keywords: nonlinear image processing; deconvolution; resolution; microwaves; upsampling;advanced microwave sounding unit; microwave humidity sounder; alternating direction methodof multipliers.

Paper 15400 received May 29, 2015; accepted for publication Nov. 16, 2015; published onlineDec. 24, 2015.

1 Introduction

Severe weather phenomena, such as hurricanes and tropical storms, can be continuously cap-tured using geostationary microwave sensors. These sensors are designed to penetrate throughthick clouds to record geophysical profiles of a storm, including temperature, humidity, watervapor, and cloud properties. The acquired images in a broad bandwidth are valuable for evalu-ating internal processes of the storm and its strength. For example, the Geostationary SyntheticThinned Aperture Radiometer (GeoSTAR) is a microwave spectrometer aperture synthesissystem that has been under development at Jet Propulsion Laboratory since 1998 and willbe used to capture hurricane imagery.1 In principle, an aperture synthesis system is characterizedby convolution with a point spread function (PSF), which is modeled as a two-dimensional (2-D)sinc-like function. Due to the presence of positive and negative excursions in a PSF (cf. Fig. 1),there are ringing artifacts along sharp edges and other transitions in the observed field. It is ofhigh necessity to process low-resolution physical weather systems. In the variational framework,we aim to solve the ill-posed “deconvolution” and “upsampling problem” to produce high-resolution images from blurry low-resolution data, which may be polluted by the additiveGaussian noise.

*Address all correspondence to: Jing Qin, E-mail: [email protected]

1931-3195/2015/$25.00 © 2015 SPIE

Journal of Applied Remote Sensing 095035-1 Vol. 9, 2015

http://dx.doi.org/10.1117/1.JRS.9.095035





mailto:[email protected]

mailto:[email protected]

In this paper, we use the data captured by the Microwave Humidity Sounder (MHS) instru-ment. MHS is a five-channel passive microwave radiometer, with frequencies ranging from 89 to190 GHz, carried aboard meteorological satellites. It is similar in design to the AdvancedMicrowave Sounding Unit (AMSU-B) instrument, with some channel frequencies being slightlydifferent from AMSU-B. The instrument examines several bands of microwave radiation andperforms sounding of atmospheric physical properties. AMSU and MHS data have been usedextensively in weather prediction, and long-term AMSU and MHS records are used in climatestudies.

Motivated by the previous work,2 this paper skillfully applies the alternating directionmethod of multipliers (ADMM) to solve the problem of image deconvolution and upsamplingin a manner that all subproblems have closed-form solutions. The ADMM will be reviewed inSec. 2. It can be applied in many different ways, which result in different forms of subproblems.To our problem, in the single-channel case, we separate three components in the objective func-tional–the total variation regularization, the deconvolution operator, and the upsampling operator—and obtain three subproblems in the ADMM so that each has a closed-form solution. Closelyrelated studies include Ref. 3 where upsampling is not being addressed and Ref. 4 where differ-ent applications are considered. The multichannel case of our method considers the multichanneltotal variation (mTV) regularization and thereby has three subproblems in the ADMM that canbe solved in closed forms. In particular, the solution to the subproblem involving mTV is givenby the generalized shrinkage. Even though more computation is needed at each iteration, theproposed multichannel method takes fewer iterations to achieve convergence than the single-channel one.

The rest of the paper is organized as follows. Section 2 reviews the ADMM and total varia-tion minimization. Section 3 describes the proposed splitting technique, applies the ADMM toderive the corresponding algorithm, and discusses its convergence rates. Section 4 extends theproposed method to the multichannel deconvolution and upsampling model with the mTV regu-larization. Section 5 presents the numerical results on the synthetic AMSU and real MHShurricane data. Finally, Sec. 6 concludes this paper and discusses future work.

2 Preliminaries

Since the ADMM was introduced by Glowinski and Marrocco5 and Gabay and Mercier,6 it hasbeen widely applied to problems in imaging, statistical regression, machine learning, optimalcontrol, and many other areas. To apply the ADMM, one must reformulate a problem into theform of

EQ-TARGET;temp:intralink-;e001;116;136minx;z

fðxÞ þ gðzÞ s:t: Axþ Bz ¼ c; (1)

where x and z are the unknown vectors, f and g are the proper closed convex functions, A and Bare the matrices, and c is a given vector. (We restrict our discussion to Euclidean spaces.)

Fig 1 The GeoSTAR point spread function (PSF). A characteristic of an aperture synthesissystem is that the PSF is a two-dimensional sinc-like function, showing positive and negativeexcursions that produces ringing at sharp edges and other transitions in the observed field.

Qin, Yanovsky, and Yin: Efficient simultaneous image deconvolution and upsampling algorithm. . .


The ADMM is an abstract algorithm with two subproblems, one involving f and A andthe other involving g and B, which must be solved at every iteration. There are many differentapproaches to reformulate a convex optimization problem into the form of Eq. (1), andthey generally give rise to different subproblems (though they reduce to a few equivalenceclasses7).

Let us first review the ADMM. The augmented Lagrangian for the problem Eq. (1) is

EQ-TARGET;temp:intralink-;sec2;116;663Lρðx; z; yÞ ≔ fðxÞ þ gðzÞ þ yTðAxþ Bz − cÞ þ ρ

2kAxþ Bz − ck22;

where y is the dual variable. The ADMM seeks a solution to the problem Eq. (1) by iterating

EQ-TARGET;temp:intralink-;e002;116;606xkþ1 ∈ argminx

Lρðx; zk; ykÞ; (2)

EQ-TARGET;temp:intralink-;e003;116;568zkþ1 ∈ argminz

Lρðxkþ1; z; ykÞ; (3)

EQ-TARGET;temp:intralink-;e004;116;537ykþ1 ¼ yk þ γρðAxkþ1 þ Bzkþ1 − cÞ; (4)

where ρ > 0 and the stepsize γ ∈ ½0; ð ffiffiffi5

p þ 1Þ∕2� (Ref. 8, Chapter 6). Note that we couldconstruct the following augmented Lagrangian:

EQ-TARGET;temp:intralink-;sec2;116;484Lρðx; z; yÞ ≔ fðxÞ þ gðzÞ þ ρ

2kAxþ Bz − cþ yk22;

which yields the same solution for the x-subproblem Eq. (2) and the z-subproblem Eq. (3) withthe scaled update ykþ1 ¼ yk þ γðAxkþ1 þ Bzkþ1 − cÞ. For the sake of simplification, we stick tothe latter version in the entire paper. We use “∈” instead of “=” since the solutions xkþ1 and zkþ1

can be nonunique though different solutions will yield the same Axkþ1; Bzkþ1 and thus ykþ1. TheADMM has recently become a popular method for solving convex optimization problems arisingin various application fields. It turned out to be equivalent to the Douglas–Rachford splittingiteration, a primal-dual iteration, and the ADMM itself applied to the Lagrange dual problemof Eq. (1).7,9,10

Total variation minimization11 is a well-known approach to solving regularized inverseproblems and the recovery of images with sharp edges. In this paper, we use the isotropic version

of the discrete total variation, TVðuÞ ¼ kDuk1 ¼P

j

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðD1uÞ2j þ ðD2uÞ2j

q, with the discrete

finite difference operator D ¼ ½D1; D2� satisfying the periodic boundary conditions, whereDu ¼ ½D1u;D2u�. In addition to the traditional denoising problem,11 the total variation mini-mization has been widely used in recovering images from their incomplete measurements.12,13

Typically, the corresponding optimization problem has the following form:

EQ-TARGET;temp:intralink-;e005;116;250minukDuk1 þ

μ

2kAu − bk22; (5)

where A is a sampling operator, b is the (noisy) observation of the true imaging under A, andμ > 0 is a tuning parameter.

The use of variable splitting and the ADMM to total variation minimization and compressivesensing image recovery started in Refs. 13–15. A dummy variable w is introduced to separateDu from k · k1, giving rise to the equivalent problem

EQ-TARGET;temp:intralink-;e006;116;145minu;w

kwk1 þμ

2kAu − bk22 s:t: w −Du ¼ 0; (6)

which is in the form of Eq. (1). It is recognized in Ref. 14 that when A and D are both con-volution operators and satisfy the same boundary conditions, e.g., the periodic boundary con-ditions, a least-squares subproblem involving Au and Du has a closed-form solution via the



diagonalization by the (discrete) Fourier transform. Such a least-squares subproblem arises ina quadratic penalty model in Refs. 14 and 15 and the augmented Lagrangian method in Ref. 13.It is worth noting that the ADMM has been skillfully applied to other image recovery problemsin Refs. 16–19.

The problem of image deconvolution and upsampling studied in this paper has the form[Eq. (5)] with an operator A that composes sampling with convolution. Due to this composition,the least-squares subproblem that involves Au and Du no longer has a closed-form solution. Weexperimented solving the least-squares subproblem with the gradient descent method in Ref. 3and recognized a strong need for better performance. In this paper, we introduce a much moreefficient method by reformulating the problem and obtaining an ADMM algorithm with threesubproblems, all of which have closed-form solutions.

3 Proposed Image Deconvolution and Upsampling Algorithm

Without loss of generality and for presentation simplicity, we assume a square image domain.Let u0 ∈ Rn2 be the ground truth image to be reconstructed, and K∶Rn2 → Rn2 be a discreteconvolution operator associated with the PSF k. The observed blurry and noisy low-resolutionimage b ∈ Rm (m < n2) satisfies

EQ-TARGET;temp:intralink-;sec3;116;513b ¼ Kpu0 þ noise;

where Kp is the composition of the convolution and the sampling operators, i.e., Kp ¼ PK.The 2-D convolution operator K corresponds to the block circulant matrix in the followingsense:

EQ-TARGET;temp:intralink-;sec3;116;441vecðk � vÞ ¼ K vecðvÞ; v ∈ Rn×n:

To reconstruct u0, we consider the following total variation-based minimization problem:

EQ-TARGET;temp:intralink-;e007;116;393minukDuk1 þ

μ

2kPKu − bk22; (7)

where D∶Rn2 → R2n2 is the finite difference operator following the periodic boundary condi-tions and μ is a parameter related to the noise level. The solution u of Eq. (7) is an estimate of u0.

Unlike the conventional deconvolution models, the noncirculant structure of P will make itdifficult to solve the least-squares u-subproblem if we directly apply the ADMM. By intro-ducing w ¼ Ku, we split P and K (in the sense that they are applied to different variables) asfollows:

EQ-TARGET;temp:intralink-;e008;116;276minu;v;w

kvk1 þμ

2kPw − bk22 s:t Du − v ¼ 0; w − Ku ¼ 0: (8)

Then, the augmented Lagrangian is

EQ-TARGET;temp:intralink-;sec3;116;218Lρ1;ρ2ðu; v; w; x; yÞ ¼ kvk1 þμ

2kPw − bk22 þ

ρ12

�kDu − vþ xk22 þ

ρ2ρ1

kw − Kuþ yk22�;

where x and y are the scaled dual variables, scaled by ρ1; ρ2, respectively, so they appear inthe quadratic penalty terms. The scaled ADMM yields the following iteration:

EQ-TARGET;temp:intralink-;e009;116;148

8>>>>>><>>>>>>:

ðvkþ1; wkþ1Þ ¼ argminv;w

Lρ1;ρ2ðuk; v; w; xk; ykÞ;

ukþ1 ¼ argminu

Lρ1;ρ2ðu; vkþ1; wkþ1; xk; ykÞ;xkþ1 ¼ xk þ γðDukþ1 − vkþ1Þ;ykþ1 ¼ yk þ γðwkþ1 − Kukþ1Þ:

(9)



Due to the separability of variables v and w in the first subproblem, we are able to update v andw individually.

• v-subproblem:

EQ-TARGET;temp:intralink-;sec3;116;693vkþ1 ¼ argminv

kvk1 þρ12kDuk − vþ xkk22

can be solved in the closed-form as

EQ-TARGET;temp:intralink-;sec3;116;640vkþ1 ¼ shrinkðDuk þ xk; 1∕ρ1Þ;

where shrinkðv; σÞ ¼ sgnðvÞ⊙maxðjvj − σ; 0Þ with componentwise multiplication ⊙.• w-subproblem:

EQ-TARGET;temp:intralink-;sec3;116;582wkþ1 ¼ argminw

μ

2kPw − bk22 þ

ρ22kw − Kuk þ ykk22

reduces to the normal equations:

EQ-TARGET;temp:intralink-;sec3;116;529μPTðPw − bÞ þ ρ2ðw − Kuk þ ykÞ ¼ 0;

which again can be solved in the closed form as

EQ-TARGET;temp:intralink-;sec3;116;482w ¼ ðμPTPþ ρ2IÞ−1½μPTbþ ρ2ðKuk − ykÞ�

or equivalently


�w ←Kuk − yk

wP ←ðμbþ ρ2wPÞ∕ðμþ ρ2Þ : (10)

Here, wP represents the subvector of w with the index set specified by the sampling operator P.In the noise-free case, where the w-subproblem has an equality constraint

EQ-TARGET;temp:intralink-;sec3;116;364w ¼ argminw

ρ22kw − Kuk þ ykk22 s:t: Pw ¼ b;

the solution can be obtained by


�w ←Kuk − yk

wP ←b:(11)

In fact, this is the projection of the least-squares solution Kuk − yk to the hyperplanefw∶Pw ¼ bg

• u-subproblem:

EQ-TARGET;temp:intralink-;sec3;116;229ukþ1 ¼ argminu

ρ12kDu − vkþ1 þ xkk22 þ

ρ22kwkþ1 − Kuþ ykk22

reduces to the normal equation

EQ-TARGET;temp:intralink-;sec3;116;175ρ1X2i¼1

DTi ðDiu − vkþ1

i þ xki Þ þ ρ2KTðKu − wkþ1 − ykÞ ¼ 0;

which can be solved in the closed form by applying the 2-D discrete Fourier transform asfollows:

EQ-TARGET;temp:intralink-;e012;116;103u ¼ F��ρ1

P2i¼1 FD

Ti ðvkþ1

i − xki Þ þ ρ2FKTðwkþ1 þ ykÞρ1

P2i¼1 diagðFDT

i DiF�Þ þ ρ2diagðFKTKF�Þ�; (12)



where the operator diag∶Rn2 × Rn2 → Rn2 extracts all diagonal entries of a square matrix to

form a vector, the matrix F ∈ Cn2×n2 is generated by the Kronecker product of the one-dimen-sional discrete Fourier transformation matrix Fn ∈ Cn×n and itself, and the division is appliedcomponentwise. Note that although matrix-vector multiplication is involved in Eq. (12), weapply the 2-D fast Fourier transform (FFT) directly to the image in the numerical implemen-tation. Since K represents the convolution operator under the periodic boundary conditions,i.e., Ku ¼ k ⊗ u, the matrix FKTKF� can be efficiently computed by FFT. Under generalboundary conditions for the convolution operator, especially noncirculant K, we could decom-pose K ¼ K1 þ K2, where K1 is still circulant and K2 is sparse added to correct the boundaryconditions.20

The convergence of the ADMM has been long proved in [Ref. 8, Chapter 6]. We apply theresults from the papers21,22 to obtain the following rates:

• The violation to the constraints in the problem Eq. (8) kDuk − vkk2 þ kwk − Kukk2reduces at the rate of oð1∕kÞ, and if we replace uk by the running averageuk ¼ 1

k

Pki¼1 u

i and replace vk and wk by their running averages as well, then the rateimproves to Oð1∕k2Þ. The latter is known as the ergodic rate. Note that the violationis not monotonic in general.

• Let ηk ≔ kvkk1 þ μ2kPwk − bk2 and η� ≔ kDu�k1 þ μ

2kPKu� − bk2. Then, the objective

error jηk − η�j reduces at the rate of oð1∕ ffiffiffik

p Þ, and the corresponding ergodic rate improvesto Oð1∕kÞ. Note that it is possible that ηk < η� due to constraint violation.

4 Multichannel Extension

When the underlying data has multiple channels, we extend the deconvolution and upsamplingmodel Eq. (7) by choosing the mTV as the regularization term. Instead of processing the datachannel by channel, the mTV couples various channels and ensures consistent regularity acrossthe channels. More specifically, let u ¼ ðu1; : : : ; ucÞT with each ui ∈ Rn2 , and generalize themodel Eq. (7) to the following form:

EQ-TARGET;temp:intralink-;e013;116;372minukukmTV þ μ

2

Xci¼1

kPKui − bik22: (13)

Here, the mTV seminorm of u is defined as

EQ-TARGET;temp:intralink-;sec4;116;311kukmTV ¼Xj

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXci¼1

jðDuiÞjj2s

.

Let D1 and D2 be the difference operators along the x-axis and the y-axis, respectively. Here,we use the isotropic total variation at each pixel defined in Sec. 2. Similar to Eq. (8), we splitthe regularization term and the fidelity term as follows:

EQ-TARGET;temp:intralink-;e014;116;223minu;v;w

kvkmTV þ μ

2

Xci¼1

kPwi − bik22 s:t: Dui − vi ¼ 0; wi − Kui ¼ 0; i ¼ 1; : : : ; c: (14)

With the augmented Lagrangian defined by

EQ-TARGET;temp:intralink-;sec4;116;162Lρ1;ρ2ðu; v;w; x; yÞ ¼ kvkmTV þ μ

2

Xci¼1

kPwi − bik22

þXci¼1

�ρ12kDui − vi þ xik22 þ

ρ22kwi − Kui þ yik22

�;

we obtain the following ADMM iteration:




8>>>>><>>>>>:

ðvkþ1;wkþ1Þ ¼ argminv;w

Lρ1;ρ2ðuk; v;w; xk; ykÞ;

ukþ1 ¼ argminu

Lρ1;ρ2ðu; vkþ1;wkþ1; xk; ykÞ;

xkþ1i ¼ xki þ γðDukþ1

i − vkþ1i Þ; i ¼ 1; : : : ; c;

ykþ1i ¼ yki þ γðwkþ1

i − Kukþ1i Þ; i ¼ 1; : : : ; c:

(15)

Fig. 2 Original microwave channels of the simulated hurricane Rita image. (a)–(f) correspond to50.3, 52.8, 53.6, 54.4, 54.9, and 55.5 GHz.

Fig. 3 Original microwave channels of the simulated hurricane Rita image. (a)–(f) correspond to150, 157, 166, 176, 180, and 182 GHz.



Similar to the analysis in Sec. 3, each subproblem has a closed-form solution. In particular,the v-subproblem has the solution represented by the generalized shrinkage operator

EQ-TARGET;temp:intralink-;sec4;116;132vkþ1i ¼ Duki þ xki

skmax

�sk −

1

ρ1; 0�; i ¼ 1; : : : ; c

where

Fig. 4 Tests on the simulated hurricane Rita data with sampling factors r ¼ 2 and 3. From top tobottom: 50.3, 52.8, 53.6, 54.4, 54.9, and 55.5 GHz channels. If r ¼ 2, then μ ¼ 10−3, ρ1 ¼ 10 andρ2 ¼ 10−2. If r ¼ 3, then μ ¼ 10−4, ρ1 ¼ 10 and ρ2 ¼ 10−4.



EQ-TARGET;temp:intralink-;sec4;116;146sk ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXci¼1

kDuki þ xki k22s

:

Each component wi is updated via Eqs. (10) or (11) depending on the noise level. Similar tothe derivation of Eq. (12), by letting xi ¼ ðxi;1; xi;2ÞT and vi ¼ ðvi;1; vi;2ÞT with xi;j; vi;j ∈ Rn2

for i ¼ 1; : : : ; c and j ¼ 1;2, we obtain the following solution to the u-subproblem:

Fig. 5 Tests on the simulated hurricane Rita data with sampling factors r ¼ 2 and 3. From top tobottom: 150, 157, 166, 176, 180, and 182 GHz channels. Here, we choose μ ¼ 10−4, ρ1 ¼ 10, andρ2 ¼ 10−4.



EQ-TARGET;temp:intralink-;e016;116;318ui ¼ F��ρ1

P2j¼1 FD

Tj ðvkþ1

i;j − xki;jÞ þ ρ2FKTðwkþ1i þ yki Þ

ρ1P

2j¼1 diagðFDT

j DjF�Þ þ ρ2diagðFKTKF�Þ�; i ¼ 1; : : : ; c: (16)

5 Numerical Results

This section presents the numerical results obtained by applying the proposed algorithms onvarious synthetic and real data of severe weather phenomena. The 32 × 32 discretizedGeoSTAR PSF depicted in Fig. 1 was chosen as the blurring kernel for all tests. To measureperformance, we use the relative error defined by

EQ-TARGET;temp:intralink-;sec5;116;186

Relerr ¼ ku − uk2kuk2

;

where u ∈ Rn2 is the ground truth image and u is the recovered image. All experiments were runin MATLAB 2013a on a desktop computer with 8 GB of RAM and a 3.10 GHz Intel i7-4770SCPU. All parameters are chosen to yield the best performance.

In the first experiment, we used simulated microwave images of the 2005 Atlantic hurricaneRita shown in Figs. 2 and 3. By default, all images are displayed in the range [110, 285] inKelvin. Figure 2 shows 50.3, 52.8, 53.6, 54.4, 54.9, 55.5 GHz, and Fig. 3 shows 150, 157,166, 176, 180, 182 GHz channel images. For comparison, GeoSTAR operates at some of

Table 1 Relative error comparison for the simulated hurricane Rita data with sampling factorsr ¼ 2 and 3.

Channel (GHz) r ¼ 2 r ¼ 3 Channel (GHz) r ¼ 2 r ¼ 3

50.3 0.0392 0.0713 150 0.0515 0.0837

52.8 0.0098 0.0167 157 0.0339 0.0538

53.6 0.0038 0.0072 166 0.0140 0.0222

54.4 0.0021 0.0040 176 0.0071 0.0126

54.9 0.0015 0.0026 180 0.0040 0.0074

55.5 0.0014 0.0024 182 0.0024 0.0039

Table 2 Relative error comparison for the simulated hurricane Rita data with various noise levelsand fixed sampling factor r ¼ 2. For all tests, μ ¼ 10−8, ρ1 ¼ 10−6. Here, ρ2 is set to 10−6 whenσ ¼ 5 and 10−7 otherwise.

Channel (GHz) σ ¼ 5 σ ¼ 10 σ ¼ 15 Channel (GHz) σ ¼ 5 σ ¼ 10 σ ¼ 15

50.3 0.0816 0.0926 0.1013 150 0.0882 0.0988 0.1049

52.8 0.0292 0.0325 0.0440 157 0.0594 0.0666 0.0741

53.6 0.0234 0.0258 0.0374 166 0.0321 0.0363 0.0464

54.4 0.0234 0.0250 0.0371 176 0.0262 0.0294 0.0403

54.9 0.0229 0.0257 0.0383 180 0.0247 0.0272 0.0391

55.5 0.0265 0.0289 0.0432 182 0.0251 0.0273 0.0404



the same frequencies of the AMSU-A and AMSU-B temperature and humidity sounders near 55and 180 GHz, respectively. The images have 400 × 400 pixels and were derived from cloudresolving numerical weather prediction model (WRF)23,24 simulations. A standard radiativetransfer model was used to generate the brightness temperatures from the WRF fields. Theresolution of a pixel is 1.3 km. With this grid spacing, we are able to resolve features thatare approximately 5-km wide.

Figures 4 and 5 depict images of Figs. 2 and 3 blurred with GeoSTAR kernel as well as theresults after applying the proposed algorithm to the blurry low-resolution data. We choosethe downsampling factor r ¼ 2 and 3, i.e., downsampling the 402 × 402 blurry images intothe 201 × 201 and 134 × 134 ones, respectively. Note that the selected downsampling factors,i.e., 2 and 3, are most commonly used in applications, as larger downsampling factors produceresults that are not so accurate. The corresponding comparison of relative errors is shown inTable 1. From the results, one can see that as the image has more fine features, the reconstructionbecomes more difficult as reflected in the increase of relative error. On the contrary, if the imageto be recovered is more piecewise constant, the reconstruction has smaller relative error. InTable 2, we show the results by fixing the sampling factor as two and varying the noiselevel as σ ¼ 5;10;15.

In Figs. 6 and 7, we tested the proposed method on real MHS data of the 2005 Atlantichurricane Katrina and the 2005 Pacific typhoon Talim, respectively. Each image has five fre-quency bands at 89;150;183� 1, 183� 3, and 183� 7 GHz. The resolution of each pixel is15 km. We used sampling factor r ¼ 2;3 in our experiments. The corresponding relative errorsare shown in Table 3.

Fig. 6 Tests on hurricane Katrina MHS 89;157;183� 1, 183� 3, 190 GHz channels with sam-pling factor r ¼ 3. From top to bottom: ground truth images, blurry data with downsampling factorr ¼ 3, and the recovered images. Here, we choose μ ¼ 10−4, ρ1 ¼ 10, and ρ2 ¼ 10−4.



Finally, we tested the proposed multichannel algorithm [Eq. (14)] on three sets of data withsampling factors r ¼ 2;3. The performance comparison in terms of relative error for each chan-nel is listed in Tables 4 and 5. Comparing these results with those obtained by the single-channelalgorithm in Tables 1 and 3, one can see that the mTV is able to enhance the reconstructionquality for those channels with more fine features while trading the quality of the channelswith less features. In addition, Table 6 compares the average running time in seconds per iter-ation for each data set using the proposed single-channel algorithm and the multichannel algo-rithm. Note that the multichannel algorithm usually uses far less iterations (200 to 400) to reach adesired result, while its single-channel counterpart uses various numbers of iterations (200 to1000) for each channel and some channels need a lot more iterations to achieve the desiredaccuracy. Nevertheless, both versions are very efficient so that the reconstruction of animage is achieved at the rate of about 2.5 × 10−7 per pixel per iteration.

Fig 7 Tests on typhoon Talim MHS 89;157;183� 1, 183� 3, 190 GHz channels with samplingfactor r ¼ 3. From top to bottom: ground truth images, blurry data with downsampling factor r ¼ 3,and the recovered images. Here, we choose μ ¼ 10−4, ρ1 ¼ 10, and ρ2 ¼ 10−4.



Table 4 Relative error comparison for the simulated hurricane Rita data using the multichannelEq. (9) When r ¼ 2, μ ¼ 10−3 ρ1 ¼ 10, and ρ2 ¼ 10−1; when r ¼ 3, μ ¼ 10−4, ρ1 ¼ 10, andρ2 ¼ 10−3.

Channel (GHz) r ¼ 2 r ¼ 3 Channel (GHz) r ¼ 2 r ¼ 3

50.3 0.0377 0.0696 150 0.0496 0.0828

52.8 0.0095 0.0165 157 0.0320 0.0532

53.6 0.0044 0.0077 166 0.0136 0.0229

54.4 0.0024 0.0042 176 0.0076 0.0138

54.9 0.0017 0.0028 180 0.0046 0.0086

55.5 0.0018 0.0025 182 0.0029 0.0048

Table 5 Relative error comparison of hurricane Katrina and typhoon Talim MHS data using themultichannel Eq. (9). For all tests, μ ¼ 10−4 and ρ1 ¼ 10. We choose ρ2 ¼ 10−2 when r ¼ 2 and10−3 when r ¼ 3.

Katrina Talim

Channel (GHz) r ¼ 2 r ¼ 3 r ¼ 2 r ¼ 3

89 0.0244 0.0391 0.0241 0.0385

157 0.0347 0.0569 0.0189 0.0307

183� 1 0.0200 0.0289 0.0167 0.0223

183� 3 0.0238 0.0393 0.0151 0.0235

190� 7 0.0283 0.0485 0.0163 0.0271

Table 6 Comparison of running time in seconds per iteration.

Rita 402 × 402 Katrina 150 × 90 Talim 210 × 90

Single-channel 0.0450 0.0038 0.0044

Multichannel 0.3383 0.0187 0.0239

Table 3 Relative error comparison of hurricane Katrina and typhoon Talim MHS channels withsampling factors r ¼ 2 and 3. Note that for r ¼ 2, ρ2 ¼ 10−3, and all other parameters are setthe same as those specified in Figs. 6 and 7 when r ¼ 3.

Katrina Talim

Channel (GHz) r ¼ 2 r ¼ 3 r ¼ 2 r ¼ 3

89 0.0242 0.0392 0.0252 0.0396

157 0.0349 0.0567 0.0186 0.0304

183� 1 0.0208 0.0297 0.0170 0.0230

183� 3 0.0248 0.0401 0.0155 0.0240

190 0.0293 0.0490 0.0167 0.0273



6 Conclusions

We proposed an efficient deconvolution and upsampling method for the hurricane microwavedata and extended it to handle multichannel data by using the mTV regularization. Derived fromproper variable splitting and the ADMM, the proposed algorithms provide simple subproblemsand fast convergence, and produce high-resolution images with small reconstruction errors. Theexperiments on the synthetic microwave data demonstrate the great potential of our work. Ourtechnique can be easily adapted or extended to address other ill-posed problems for applicationsdealing with large-scale data sets. For the future work, the weighted mTV with adaptive weightscan be considered to improve the smoothness for each channel. Taking the texture-like featuresin the microwave hurricane images into account, the multiresolution transformation-basedregularization will be worth exploring as well.

Acknowledgments

The research was carried out in part at the Jet Propulsion Laboratory, California Institute ofTechnology, under a contract with the National Aeronautics and Space Administration. I.Yanovsky acknowledges the support from NSF Grant No. DMS-1217239. W. Yin acknowledgesthe support from NSF Grant No. DMS-1317602 and ECCS-1462398.

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Jing Qin is an assistant adjunct professor in the Department of Mathematics, University ofCalifornia, Los Angeles. She received her BS degree in mathematics from Xuzhou NormalUniversity in 2005, her MS degree in mathematics from East China Normal University in 2008,and her PhD in applied mathematics from Case Western Reserve University in 2013. Her currentresearch interests include mathematical image processing and numerical optimization.

Igor Yanovsky received his BS and MA degrees in 2002 and PhD in applied mathematics in2008, all from the University of California, Los Angeles, California, USA. He has been with theJet Propulsion Laboratory, California Institute of Technology, Pasadena, California, since 2008.He is also currently with the Joint Institute for Regional Earth System Science and Engineering,University of California, Los Angeles. His research interests are inverse problems and dataanalysis.

Wotao Yin is a professor of mathematics at the University of California, Los Angeles. He workson numerical algorithms for large-scale optimization, image processing, machine learning, andother inverse problems. He received his BS degree from Nanjing University in 2001, and thenhis MS and PhD degrees from Columbia University in 2003 and 2006, respectively. He wonthe NSF CAREER award in 2008 and the Alfred P. Sloan research fellowship in 2009.



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efficient simultaneous image deconvolution and upsampling...

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