ifsar processing using variational calculus · ifsar processing using variational calculus kenneth...

ASPRS 2008 Annual Conference Portland, Oregon ♦ April 28-May 2, 2008

IFSAR PROCESSING USING VARIATIONAL CALCULUS

Kenneth Sartor, Ph.D., Chief Scientist of Image Processing Melbourne, Florida 32940

[email protected]

Gnana Bhaskar Tenali, Ph.D., Professor of Mathematics Florida Institute of Technology

Melbourne, Florida 32901 [email protected]

Adrian Peter, Software Engineer

Josef DeVaugn Allen, Ph.D., Software Engineer Mark Rahmes, Software Engineer

Harris Corporation Government Communications Systems Division

Melbourne, Florida 32905 [email protected]; [email protected];[email protected]

ABSTRACT Processing of Interferometric SAR (IFSAR) data presents many interesting challenges. This paper will focus on using global approaches based on variational calculus to accomplish the phase estimation of IFSAR data so that an accurate digital elevation model or subsidence measurement can be generated. The global methods are solved using mathematical techniques from calculus of variations, partial differential equations, and functional analysis. Local approaches based on path following methods are useful for solving some phase estimation problems but they are less likely to have a closed form optimal solution suitable for a large class of problems. Since we are interested in an analytical description of phase estimation, we do not discuss heuristic methods. Application of mathematical formalism greatly aids in the interpretation of results obtains from various research algorithms and commercial products. This paper will also detail the results of our work on RADARSAT-1 data and simulated IFSAR datasets.

Keywords: IFSAR, DEM, subsidence, calculus of variations, functional, partial differential equations, anisotropic diffusion, Mumford-Shah, smoothing filter, phase unwrapping, coherent change detection

INTRODUCTION

We introduce a variational approach to anisostropic diffusion and phase unwrapping of interferometric SAR data using the Mumford-Shah framework. The Mumford-Shah variational framework can handle discontinuities in the scene by unifying the phase unwrapping and anisostropic diffusion in one mathematical formalism.

IFSAR OVERVIEW

The Synthetic Aperture Radar (SAR) image of New Orleans shown will serve as a motivating example. This image is collected by the RADARSAT-1 satellite from Canadian Space Agency. Unlike the images that come from a typical Electro-Optical (EO) aerial camera (much like that digital cameras on the consumer market), this is not a real pixel valued image. Each pixel in this image has both a magnitude and phase component. Typical image processing is incoherent because it involves only magnitude. Complex image processing, on the other hand, is coherent because it involves both magnitude and phase.

After sufficient preprocessing, the September 2005 image is combined with the October 2005 image shown in Figure 1 to obtain an interferogram. The interferogram is formed by multiplying on a pixel by pixel basis the complex value of the September image with the complex conjugate of the complex value in the October image.


Figure 1. Pictorial IFSAR Process (Data Courtesy of Canadian Space Agency). Since the interferogram is a complex valued image, only the phase is shown in Figure 1. Each pixel in the

image corresponds to a phase value so the units of the figure are in radians. The magnitude is a useful aid in weighting the phase data. The interferogram phase shown in Figure 1 is wrapped modulo 2π . For this reason, the original phase that one is attempting to estimate must be reconstructed by phase unwrapping which is one of the primary steps in interferometric SAR (IFSAR) processing.

The primary goal of interferometric SAR processing is to generate products that extract topography or change in topography [3,10]. In order to obtain a topographic product such as a digital elevation model (DEM), the measured phase shown in Figure 1 must undergo phase unwrapping. This unwrapped product undergoes additional processing to obtain a height map where each pixel value corresponds to an earth elevation value. Figure 2 details the IFSAR processing chain.


Figure 2. IFSAR Process.

BOUNDARY SHARPENING/EDGE ENHANCEMENT (ANISOTROPIC DIFFUSION)

Because of the high gradients caused by noise in the data, the interferogram must be smoothed so that the

algorithms which take derivatives of the measured data work properly. The most commonly used smoothing algorithms for complex data processing are blurring functions (i.e. Hanning, Taylor weighting, Gaussian, etc.). Unfortunately, the filters so designed blur the high gradients that we desired to maintain; namely the discontinuities in the data such as buildings and mountain cliffs in the scene. For DEM extraction, the blurring of these features cause inaccuracies in the height of the unwrapped phase relating to the terrain surface.

In general, we need the capability to selectively smooth complex SAR data so that it is more useful for various image processing problems. Current methods of interferometric SAR pre-processing work fine for rural scenes which contain slowly varying terrain. However, urban scenes will suffer resolution loss because of the behavior of the smoothing filters.

Using PDE-based smoothing techniques is one way to overcome this difficulty. Anisotropic diffusion (ANDI), an example of such a technique, preserves edges while smoothing intra-region areas. The first approach applies anisostropic diffusion suited to the complex interferogram. For a more detailed discussion on pre-processing methods needed for phase estimation, we refer to [6].

We recall that the anisotropic diffusion equation is given by

( ) ( ) ( )),,,,(,, tyxutyxcdivt

tyxu∇=

∂∂ (1)

The function c(x,y,t) is usually some monotonically decreasing function that determines how the diffusion will behave along the boundary. Weikert [11] proposed using a diffusion tensor but conceptually, it’s equivalent to anisotropic diffusion.


Finding a suitable convection function c(x,y,t) that pre-processes the data for a wide range of IFSAR data (both smooth and nonsmooth scenes) would be the aim in using this approach. Improved pre-processing of the complex data should lead to better variational phase estimation. Our work here is based on the anisotropic nonlinear diffusion introduced by Perona-Malik [5].

In Figure 3, we simulate a scene with a hill and two buildings. We need ANDI because our data contains these buildings (i.e. discontinuous data). In Figure 4, we show the wrapped phase values of the noiseless urban interferogram.

2a 2b

Figure 3. Two buildings on a hill (Nadir and 3-D view of reference scene).

3a 3b

Figure 4. Simulated urban scene interferogram of 2 building and hill.

Figure 4 is a 2-D and 3-D rendering of the wrapped phase. The reader should note that the range is now

[ )ππ +− , . Next, we do a comparison between a standard smoother and anisotropic diffusion (Figure 5).


Figure 5. Comparison Between Standard Smoothing and Anisotropic Diffusion.

Phase estimation based on least squares causes global smoothing of the image. Anisotropic diffusion, on the other hand, causes only local smoothing behavior; this property is desirable for edge preservation. Here, the anisotropic diffusion is a preprocessor to phase estimation but the two methods compete against one another (Figure 5). If the building boundaries are left in tact while the rest of the image is smoothed during pre-processing, the subsequent least squares phase estimation will cause blurring of those boundaries. This is the problem that our work addresses.

Figure 6. Smoothing Effects of Standard Weighted Least Squares Phase Unwrap.

If the gradient of the image is increased, the smoothing effects of the weighted least squares (WLS) is more pronounced (Figure 7).


Figure 7. Wrapped Phase Image for 10X Increase in Height Gradient.

THE ANISOTROPIC DIFFUSION (ANDI) FUNCTIONAL

The operation of the ANDI pre-processing and subsequent WLS unwrapping is encapsulated by the following functional

(2)

Here, ANDIW is the operation of the anisotropic diffusion which nonlinearly weights in the spatial and temporal

dimensions. The ANDI functional mathematically models the process of applying WLS phase unwrapping after pre-processing with anisotropic diffusion.

Anisotropic diffusion is very useful for processing simulated urban scenery (i.e. scene content with sharp discontinuities). In practice, we want to be able to seamlessly integrate SAR digital elevation models with traditional 3D models. The anisotropic diffusion is more suitable than the standard isotropic smoothing (i.e. Gaussian filters) when it comes to these tasks. Also, the anisotropic diffusion does a good job of removing noise before the phase is unwrapped using a standard weighted least squares algorithms. The least squares phase estimation globally smoothes the image so it defeats some of the boundary sharpening of the PDE-based smoothing. For this reason, a 2D variational phase estimation approach that maintains discontinuities in the scene is developed.

OVERVIEW OF MUMFORD-SHAH VARIATIONAL APPROACH The next approach uses the Mumford-Shah framework [4] to take advantage of the boundary sharpened

interferogram given by anisotropic diffusion IFSAR pre-processing. The Mumford-Shah model has been validated rigorously in a pure mathematics framework. Many numerical methods are available to put the infinite dimensional optimization theory of Mumford-Shah into a finite dimensional subspace so that the ideas can be investigated without researching new numerical methods for implementation.

( )( ) ( )∫∫Ω

Ψ∇−Φ∇= dxdyyxWWJ ANDI ,2


We can construct an energy functional that models the smoothness and discontinuities in the scene. We then minimize this functional using standard variational calculus techniques to obtain the Euler Lagrange equations. Once we have the Euler Lagrange equations, we discretize them and solve.

The Mumford-Shah functional is given by

( ) ( ) ∫∫ ∫ +∇+−=Ω Ω CC

dSdAfgfCfE λαβ\

22, . (3)

The functional consists of three terms whose effect is weighted by the constants α , β , and λ . The first term f is

a piece-wise smooth approximation to g (the image) with discontinuities along a curve C lying in the image plane. This first term asks that f approximate g . We may think of this as a data fidelity term which measures quality of f. The second term is called the smoothness term. This term asks that f and hence g not vary that much on each region. The third term penalizes excessive arc length of the discontinuities. This term asks that the number of closed contours C that accomplish this be as small as possible [4].

These three data terms can be adapted to the phase estimation problem by making a few adjustments. This will be discussed in the next section. A detailed overview of Mumford-Shah theory is given in [8].

THE MUMFORD SHAH PHASE ESTIMATION (MSPE) FUNCTIONAL

Now, we adapt this Mumford-Shah framework to the phase estimation problem by converting from the traditional image domain application to phase domain. The first term is updated so that g is the wrapped phase and f is the unwrapped phase estimate. We minimize the differences between these two on some reference region 0Ω of

domain Ω . The second term is updated from 2f∇ to2gf ∇−∇ . This term can be nonlinearly weighted

according to the quality of the measured phase data or the gradient. The third term is used to model the length of the scene discontinuities. The traditional use of this functional is image segmentation. For interferograms, the objects of interest are the scene discontinuities. The discontinuities are minimized with this term. Combining these notions, we arrive at an updated functional suitable for phase estimation (4) where gradient of g is our wrapped phase observation and f is the unwrapped phase estimate that we wish to

obtain. The characteristic (i.e. indicator) function is given by 0Ωχ . This term is added because we only want the

wrapped and unwrapped phase to match on a specified reference region 0Ω where we have truth data. To be consistent with the notation expressed throughout this work, we re-express the phase estimation

functional from 4 as: ( ) ( ) ( ) ∫∫∫ +Ψ−Φ+Ψ∇−Φ∇=Φ

ΩΩ

Ω CC

dsdAdACE γχβα0

2

\

2, . (5)

In the rest of this work, we’ll call this the MSPE Functional.

THE MUMFORD SHAH ANISOTROPIC DIFFUSION (MSAD) PHASE ESTIMATION FUNCTIONAL

If we desire to input an ANDI processed wrapped phase into the MSPE Functional, then we have

( ) ( ) ( ) ∫∫∫ +Ψ−Φ+Ψ∇−Φ∇=ΦΩΩ C

IC

ANDI dsdAdAWCE γχβα0

2

\

2)(, (6)

∫ ∫∫ΩΩ

Ω +∇−∇+−=C C

dsdAgfdAgfCfE\

220

)(),( γαχβ


We call this the MSAD Phase Estimation Functional. For clarity and to highlight the benefits of the Mumford-Shah framework, we focus our discussion on the MSPE Functional because the results are easily extendable to the MSAD Phase Estimation Functional.

MSPE PHASE DOMAIN PARTITIONING If the region Ω is divided up into two regions R and cR , one inside the contour(s) of discontinuity and one

outside, we obtain

( ) ( ) ( )

( ) ( )∫∫

∫ ∫

Ω

Ω

Ψ−Φ+Ψ∇−Φ∇

+Ψ−Φ+Ψ∇−Φ∇=ΦΦ

c

c

c

c

c

RR

RR

R RRRRRC

dAdA

dAdAE

0

0

2

22

,

χβα

χβα (7)

By definition, the reference region 0Ω does not exist outside of the discontinuity contour (i.e. cR∉Ω0 ) so we have

( ) ( ) ( )

( ) dA

dAdAE

c

c

c

RR

R RRRRRC

2

22

,0

∫

∫ ∫

Ψ∇−Φ∇

+Ψ−Φ+Ψ∇−Φ∇=ΦΦ Ω

α

χβα (8)

Taking the first variation of this expression, we obtain the weak form of the Euler-Lagrange equation with test function RΦδ

( ) ( ) ( )[ ]∫ ∫ ∂Φ∂

Φ+⋅Ψ−Φ−Ψ∇−Φ∇Φ=Φ

ΩR R

RRRRR

RC dsn

dAf

Eδχβαδ

δδ

0

22 . (9)

The Neumann boundary conditions are

0=∂Φ∂n

R on C . (10)

So the last term disappears because the value is zero at the boundary and we have the first variation for region inside of the set of discontinuities C .

( ) ( ) ( )[ ]∫ ΩΨ−Φ−Ψ∇−Φ∇Φ=ΦΦ

RIRRR

R

RC dAE

0

22 χβαδδ

δ (11)

Because we want to find the minimum of the phase estimation functional E , we set the above expression equal to zero to obtain the simplified weak form of the Euler Lagrange equation

( ) ( )[ ] 00

22 =⋅Ψ−Φ−Ψ∇−Φ∇Φ∫ ΩR

RRR dAχβαδ . (12)

At this point, we can solve for the unwrapped phase estimates using finite element methods. However, we proceed to the strong form of the Euler Lagrange equation and use finite differences to solve numerically. The strong form of the first variation is

( ) ( ) ( )0

22ΩΨ−Φ−Ψ∇−Φ∇=

ΦΦ

χβαδ

δRR

R

RCE. (13)

Setting this equal to zero, we obtain the strong form of the Euler Lagrange equation,

( ) ( ) 00

22 =Ψ−Φ−Ψ∇−Φ∇ Ωχβα RR on R . (14)

Finally, we obtain the following Poisson PDE that enables us to find the unwrapped phase estimate inside of the region of discontinuities.


( ) ( )00

22ΩΩ Ψ−Ψ∇=Φ−Φ∇ χ

βαχ

βα

RR on R, (15)

0=∂Φ∂n

R on C . (16)

We obtain the following expression for the phase estimation estimate outside C .

( ) ( )Ψ∇=Φ∇ 22

βα

βα

cR on cR , (17)

0=∂

Φ∂

ncR on C . (18)

If we know the location of the discontinuities, we can stop at this step and use the set of discontinuities as our fixed curve when doing the region based Poisson Solver. Recall that the formulation in [2] assumes that the finite set of discontinuities is known. If we don’t know the discontinuity locations, level set techniques [9] can identify them.

CURVE EVOLUTION VIA LEVEL SETS

If we don’t know where the scene discontinuities are located, we can find them automatically. We do this by minimizing the MSPE Functional with respect to C to obtain the gradient flow equation

( ) ( )[ ] NNNC RRRRt cc γκβα−Φ−Ψ−Φ−Ψ+⎟

⎠⎞⎜

⎝⎛ Ψ∇−Φ∇−Ψ∇−Φ∇=

2222 ˆˆ2

ˆˆ2

. (19)

This equation can be solved by a standard curve evolution technique such as level sets. We use level sets because of its ability to deal with discontinuities in the interferometric scene. They are implemented numerically with gradient descent or gaus-seidel. However, for speed and faster convergence, the conjugate gradient or multi-grid method is recommended.

MSPE FUNCTIONAL MINIMIZATION

The gradient descent based algorithm shown in Figure 8 is used to find the minimum of the MSPE Functional. For our experiment, we ran three iterations of this MSPE algorithm. The curve evolved but did not find the discontinuities. This could be due to a poor choice of weighting coefficients or curvature. More experimentation is needed. The problem could also be in the gradient descent algorithm that solves for the unwrapped phase estimate inside and outside the discontinuities. This algorithm may be converging too slowly. Consequently, an alternate approach is suggested next .


Figure 8. MSPE Functional Minimization Algorithm.

THE CHAN-VESE PHASE ESTIMATION (CVPE) FUNCTIONAL

We apply the functional from [1] originally used in the image domain to the phase domain. We call this the CVPE Functional and it is shown below. (20)

The results of minimizing this functional are the set of contours shown in Figure 9. This becomes a pre-processor to another variatonal approach that we’ll discuss in the next section.

Figure 9. Contours Identified by CVPE Functional Minimization.

( ) ( ) ( )

( ) ( )( )∫∫∫∫

∫∫ ∫∫

ΩΩ

Ω Ω

⋅−⋅−Ψ+⋅⋅−Ψ

+⋅+⋅∇=Φ

dxdyHcdxdyHc

dxdyHdxdyccF

φλφλ

φνφφδµ

εε

εεε

1

,,

222

211

21


THE WEIGHTED LEAST SQUARES MUMFORD SHAH (WLS-MS) FUNCTIONAL Next, the standard WLS phase estimation was incorporated into the Mumford-Shah framework to obtain the

phase estimates inside and outside the closed discontinuity contours. This amounts to minimizing the altered Mumford-Shah functional shown in the next figure.

Figure 10. WLS-MS Phase Estimation Inside and Outside Discontinuity.

For this case, the first variation is

( ) ( )( ) ( )( )

0,, 22

ΩΨ−Φ−Ψ∇−Φ∇=ΦΦ

χβαδ

δRR

R

RC yxyxE

(21)

if ( ) 0, =yxβ in Equation 20, we will arrive at the solution of weighted least squares norm formulation. In this example, we assume there is no available truth data. Therefore, we remove the matching term of the functional by making ( ) 0, =yxβ as shown in Figure 10. The following figure summarizes this idea for the case of the PDE.

Figure 11. WLS-MS Euler-Lagrange Equations.

Our updated algorithm is able to do a region based phase unwrapping inside and outside the region of discontinuity (Figure 11). This is accomplished by minimizing the CVPE Functional to obtain the contours of discontinuity via the level set approach from. Then, using these contours of discontinuity, we minimize the WLS-MS Functional to obtain the region based phase unwrapped solution.


Figure 12. Region-based Phase Estimation. Our approach first applies the CVPE Functional to a bimodal interferogram. The bimodal interferogram is obtained by converting the original interferogram of N levels to 2 levels based on an image intensity threshold. (Note: For our experiment N=256 because we had an 8 bit phase image). Minimizing the CVPE Functional yields the contours of discontinuity. Next, we apply the WLS-MS functional to do the region based phase estimation. The regions are defined by the contours obtained from the CVPE minimization. The original interferogram with all 256 levels is used for this part of the process.

Figure 13. CVPE and WLS-MS Based Phase Estimation.

The details of this CVPE and WLS-MS optimization algorithm are shown in Figure 14.


Figure 14. CVPE and WLS-MS Based Phase Estimation (Detailed Diagram).

RADARSAT-1 RESULTS

In this section, we show the application of ANDI on the RADARSAT-1 interferogram. The original phase surface (top figures) was very noisy. In this example, each phase pixel corresponds to an elevation difference. After application of ANDI, the noise floor is sufficiently suppressed so that the actual shape of the phase surface can be seen (bottom figures). After more IFSAR processing, this will enable the creation of a more accurate digital elevation model.


Figure 15. Anisotropic Diffusion for Digital Surface Model Processing using IFSAR Product.

Ultimately, our aim in exploiting the RADARSAT-1 is to determine and illustrate subsidence of New Orleans. In the next figure we show how ANDI brings added value when processing IFSAR for subsidence measurement. In this example, each phase pixel of the coherent change detection (CCD) product corresponds to an elevation difference.


Figure 16. Anisotropic Diffusion for Subsidence using CCD Product. LiteSite™ is a production system for autonomous 3D site model creation from a wide variety of input DSM

sources. Model generation begins with a preprocessing step such as boundary sharpening or edge enhancement. Voids in the input data sets are filled and a bare earth data set is generated. Cultural features are then separated from vegetation features. The accuracy of key stages in our process is highly dependent on the accuracy of void filling of the DSM inpainting algorithm. A discussion of our system for assessing data accuracy for various fill methods has been presented [6]. The vertices from the building roofs and sides as well as the ground surface are mapped into polygons and projected into geo-spatial coordinates. Textures may then be automatically applied to their corresponding polygons.

Displaying RADARSAT-1 data within the context of an urban 3-D model gives the user more visualization options so we give an example of that using Harris LiteSite software. In Figure 15, we show the capability to illustrate a IFSAR derived subsidence measurement by texturing the scene canopy with actual RADARSAT-1 detected Single Look Complex imagery. The ground surface is color-coded with the measurement that corresponds to subsidence. Although we utilized height data for this LiteSite illustration, the same procedure would work on differential height data (i.e. subsidence). Our goal was to demonstrates that we have a process in place to integrate subsidence measurements with an 3-D urban model


Figure 17. Illustration of Results with Harris LiteSite Urban Model.

SUMMARY

This paper described several new functionals applicable to the phase estimation problem. The Mumford-Shah

based phase estimation framework that we developed unifies the Weighted Least Squares and Anisotropic Diffusion approaches. Because of the nature of the discontinuous and non-smooth phase data, we need this paradigm. Essentially, we removed competition with anisotropic diffusion by incorporating that technique into the variational framework (i.e. the MSAD Phase Estimation Functional). We also developed a region based phase estimation technique utilizing the CVPE and WLS-MS functional minimization. Expressing IFSAR processing in this rigorous mathematical framework and illustrating within the Harris LiteSite urban 3-D model greatly aides in the analysis of the IFSAR data.

ACKNOWLEDGEMENTS

We would like to thank Dr. Emile Ganthier of Harris Corporation whose radar expertise was an essential and

key component in the execution of our IFSAR research. We would also like to thank Marco van der Kooij & Adrian Giles of MDA, Dr. Thomas Bahr of CREASO, & Derrold Holcomb of Leica-Geosystems. They provided technical support during the process of evaluating the IFSAR tools Earthview (MDA), ENVI Sarscape (ITT/CREASO), and Radar Mapping Suite (Leica). This greatly aided in processing and analysis of some of RADARSAT data. All RADARSAT examples in this paper were pre-processed with the Earthview INSAR


software package. Finally, we thank Christine Guiguere & Mike Kirby of CSA for providing the RADARSAT-1 data of New Orleans.

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