New AMSC 664: Final Report Constructing Digital Elevation Models …rvbalan/TEACHING/AMSC663Fall2007/... · 2008. 4. 29. · 1/31. Mathematical Description of The Wedgelet Algorythm

Mathematical Description of The Wedgelet AlgorythmUse of Wedgelets On Digital Elevation Data

Testing and ResultsSelected References

AMSC 664: Final ReportConstructing Digital Elevation Models From Urban

Point Cloud Data Using Wedgelets

Christopher Miller email: [email protected]: John Benedetto email: [email protected]

April 29, 2008

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Table Of Contents

Mathematical Description of The Wedgelet Algorythm

Use of Wedgelets On Digital Elevation Data

Testing and Results

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Justification: Defficiencies of WaveletsThe Wedglet Transform Of An Image

Wavelets have demonstrated excellent ability at decomposingone dimensional signals. Particurally, the local support allowswavelets to very efficiently represent disconituities in thesignal.

This propertiy disapears when one talks about approximatingpiecewise smooth functions in R2 with continuous boundariesbetween regions using the standard tensor product wavelets.

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Let f be a piecewise constant function defined on [0, 1]2 withpiecewise C 2 boundaries between regions.

If {ψj , k} is a tensor product wavelet basis for L2([0, 1]2) itcan be shown that

εN(f ) = inf {||f −∑λ∈Λ

< f , ψλ > ψλ||} = O(|Λ|−1)[3]

This result is not as good as the analogous result for onedimensional signals where the approximation rate isεN(f ) = O(|Λ|−2).

Hueristically what is occouring is that the tensor productwavelet functions are detecting edges between regions as setsof isolated single points as opposed to continuous curves.

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We proceed as Fuhr in [3]. Again let f be a piecewise constantfunction defined on [0, 1]2 with piecewise C 2 boundaries betweenregions. Define

The set of dyadic squares on [0, 1]2 at scale j ,Q j = {[2−jk , 2−j(k + 1)]× [2−j l , 2−j(l + 1)] : 0 ≤ k , l ≤ 2j}.Q =

⋃∞j=0 Qj .

A dyadic partition Q of [0, 1]2 is a tiling of [0, 1]2 by disjointdyadic squares.A wedglet partition W = {(Qj , ω1, ω2)} splits each element ofQ into at most two wedges ω1, ω2 along a suitable line.

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Definition

A wedgelet segmentation is a pair (g ,W ) consisting of a wedgepartition W and a function g which is constant on each ω ∈W [3].

Definition

The wedgelet approximation of f is given by

min(g ,W )||f − g ||22 + λ|W |[3]

This leads us to the following theorem.

Theorem (Approximation Properties)

Let f : [0, 1]2 → R be a piecewise constant function withC 2boundary . The approximation rate for the wedgeletapproximation of f is O(|W |−2).[3]

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Justification For the use of WedgeletsData SetsImplementationValidation Data

Wedgelets were developed by Donoho in [1] for use on piecewiseconstant images with C 2 boundaries.

Our justification for using wedgelets on urban terrain data isthat urban data is characterized by very geometric structures(rectilinear buildings etc).

We expand the version of wedgelets proposed by Donoho byallowing the approximating function g to be linear on eachω ∈W rather than constant.

This expansion allows us to efficiently represent structureswith sloped roofs.

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The elevation data is generated by a survey aircraft equipped witha LIDAR (Light Detection and Ranging) system. The datareturned by the aircraft is called a point cloud.

Definition

A point cloud is a n × 3 matrix with coefficients in R where eachrow represents a point in R3.

The data set can be thought of as a discrete random sample of thecontinuous elevation function.

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Note that on any wedge, the approximating function g has threedegrees of freedom. On a particular dyadic square, theapproximating function has eight degrees of freedom.

Three; dx1, dy1, z1, determine the function g |ω1 .

Three; dx2, dy2, z2, determine the function g |ω2 .

Two specify the location of the line in the dyadic square thatseparates ω1 and ω2.

Our algorithm at each step stores the best wedge configuration interms of L1 error for each dyadic square in a quad-tree datastructure. The approximating function g |ω1 is computed usingleast squares minimization to the data points.

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The following parameters are user set:

jmax , The smallest scale for the dyadic squares.

∆k The minimum line translation.

|θ| The number of angles used in parameterizing lines.

jmax and ∆k should be chosen small enough to capture all of theinformation at the scale of interest.

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The data at initialization.

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A line divides the top level into two wedges and the least squaresapproximation is found for ω1 and ω2.

If this approximation minimizes the L1 error over all the otherapproximations observed at this level so far store it at the root ofthe tree.

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Move down one level and find the approximating function on eachwedge.

Note the existence of wedges with no data points and dyadicsquares with only one wedge.

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Subdivide again and repeat.

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Once jmax has been reached return to the top level of the tree andrepeat the above process with the next line in the dictionary oflines.

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For this illustration it was assumed that we were moving breadthfirst through the tree. It is more computationally efficient toperform the algorithm depth first in practice and this is what wehave implemented.

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To verify that the algorithm was working as intended we created asimple piecewise planar function which wedgelets should be able toapproximate reasonably well.

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Here is the wedgelet approximation with λ = 90.

Success!! We verified that the algorithm is using three dyadicsquares at the second level. The vast majority of the coefficientsare being used on the circle.

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Sample Data SetsDEM Quality MetricCompression RatesWedgelet Denoising

To test wedgelets on real LIDAR data we used two data sources.One was taken over Ft. Belvior, VA. The other was taken overNew Orleans, LA.

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We assess the quality of the DEM’s created by wedgelets bycomparing them to a reference DEM using a image quality metriccalled TSSIM. The idea behind TSSIM was first described in [6].TSSIM is a statistical reference measure with the followingproperties.

Symmetry: TSSSIM(X ,Y ) = TSSIM(Y ,X )

Boundedness: TSSIM(X ,Y ) ≤ 1

Unique Maximum: TSSIM(X ,Y ) = 1 ⇐⇒ X = Y

Empirical observation has demonstrated that TSSIM is a bettermeasure of objective image quality than the common mathematicalerror norms (i.e L2, L1, ...).

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The reference models are 5122 pixels. We determine thecompression rate by comparing the number of doubles that need tobe stored to create a given wedgelet approximation to 5122. Theresults are shown below.In practice a TSSIM value of .75 represents a model that issufficiently close to the reference model to be considered highquality.

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Conclusions:

In the case of urban data wedgelets allows compression of themodel to 15-20% of it’s original size.

The wedgelet representation is ineffective at representing ruraldata indicating that it is not suited to terrain models lacking aheavily geometric component.

Increasing the number of angles present in the dictionary oflines improves the quality only marginally.

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Using wedgelets to denoise images containing gaussian noise isdescribed in [2]. The basic idea is to find a sweet spot where thewedges you use are large enough so that the interpolative processremoves the noise but small enough that detail is kept.

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To test this we took our the urban reference DEM and addedgaussian white noise with zero mean and σ = 1

3 . Resultingwedgelet reconstructions are shown below.

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30% Retained Data

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15% Retained Data

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Note the sudden disappearance of noise between the 30% and15% levels.

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Unfortunately this sudden removal of noise did not occur when therural data was subjected to the same treatment. This indicatesagain that wedgelets is highly dependent on the presence of stronggeometrical structures.

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David L. Donoho Wedgelets: Nearly Minimax Estimation of Edges,The Annals of Statistics, Vol. 27, No. 3. (Jun., 1999), pp. 859-897.

Laurent Demaret, Felix Friedrich, Hartmut Fhr, Tomasz Szygowski,Multiscale Wedgelet Denoising Algorithms, Proceedings of SPIE,San Diego, August 2005, Wavelets XI, Vol. 5914, X1-12

Beyond Wavelets: New image representation paradigms H.Fuhr,L.Demaret, F.Friedrich Survey article, in Document and ImageCompression, M.Barni and F.Bartolini (eds), May 2006

Moenning C., Dodgson N. A.: A New Point Cloud SimplificationAlgorithm. Proceedings 3rd IASTED Conference on Visualization,Imaging and Image Processing (2003).

S. Sotoodeh , Outlier Detection In Laser Scanner Point Clouds,IAPRS Volume XXXVI, Part 5, Dresden 25-27 September 2006

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Z. Wang, A. C. Bovik, H. R. Sheikh and E. P. Simoncelli, Imagequality assessment: From error visibility to structural similarity,IEEE Transactions on Image Processing, vol. 13, no. 4, pp.600-612, Apr. 2004.

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New AMSC 664: Final Report Constructing Digital Elevation Models …rvbalan/TEACHING/AMSC663Fall2007/... · 2008. 4. 29. · 1/31. Mathematical Description of The Wedgelet Algorythm