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Topography-Preserving, Non-Linear Inpainting for Autonomous Bare Earth Digital Elevation Model (Dem) Reconstruction

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Topography-Preserving, Non-Linear Inpainting for Autonomous Bare Earth Digital Elevation Model (Dem) Reconstruction TOPOGRAPHY-PRESERVING, NON-LINEAR INPAINTING FOR AUTONOMOUS BARE EARTH DIGITAL ELEVATION MODEL (DEM) RECONSTRUCTION Josef DeVaughn Allen, Software Engineer Anthony O’Neil Smith, Software Engineer Mark Rahmes, Software Engineer Harris Corporation Government Communications Systems Division Melbourne, Florida 32905 [email protected] [email protected] [email protected] ABSTRACT Harris describes a novel way to autonomously inpaint missing data into high resolution single reflective surfaces utilizing a variant of the Navier-Stokes equations. One product of this process is a high-resolution bare earth Digital Elevation Model (DEM) with the same resolution as the input data. Inpainting allows generation of high resolution bare earth DEMs in both high and low frequency terrain environments for urban 3-D modeling. Having this bare earth DEM accounts for a dramatic increase in accuracy in all other steps of the urban 3-D modeling process. The LiteSite™ toolkit has the capability to automatically extract buildings and vegetation from an urban scene. The resulting DEM from this step of the process acts as input to the inpainting process. The expected building and vegetation base heights can then be inpainted into the area of extraction where data is now missing. The inpainting process maintains building and vegetation base height consistency in the inpainted regions and performs high accuracy interpolation and edge propagation on DEMs. Inpainting is highly effective for data sets where the occurrence of missing data is common and undesired. Examples of such data include Shuttle Radar Topography Mission (SRTM), LIDAR, IFSAR, correlated DEMs from imagery, or any other single reflectance collection data set. This technology preserves height contours. A more accurate bare earth product allows for better automated building vector extraction and therefore reduces manual building vector editing. Automated texturing of 3D model products with aerial or satellite imagery is accomplished using Harris’ RealSiteTM Toolkit. Keywords: Navier-Stokes’, partial differential equation, anisotropic diffusion, Inpainting INTRODUCTION This paper describes a novel way of automatically inpainting missing data into high resolution single reflective surfaces utilizing a variant of the heat and/or Navier-Stokes’ equations. More specifically, a general methodology is presented to fill in variable sized voids of a high resolution Digital Elevation Model (DEM) and LIDAR derived 3D site models. Evaluation and results of sample models are provided. The goal of this paper is to introduce an algorithm that jointly performs high accuracy interpolation and contour preservation around extracted features on DEMs. While inpainting has commonly been a method of interpolation applied to images we are unaware of any other work proposing to apply this idea to DEMs (Verdera et al. 2003). Inpainting is most commonly used as an image restoration technique whereby missing data is flowed in from the boundaries of the manually identified region where filling is desired in the image. The technique presented in this paper autonomously identifies and restores voided elevation data. Ideally we would like to replenish removed or missing culture from the input data. Harris Corporation’s current LiteSite™ algorithm can be run on input DEMs with data missing from collection and processing, but can also perform automated culture extraction and filling of the voided regions resulting from this extraction. In the following sections more explanation of the algorithm will be provided along with results from its application. MAPPS/ASPRS 2006 Fall Conference November 6 – 10, 2006 * San Antonio, Texas CURRENT STATE In the past many techniques have been used to perform void filling on digital elevation models (DEMs). Most interpolation techniques can leave visual artifacts in the center of the filled region and tend to blur edge contours. Preserving edge contours in DEMs is absolutely vital to generating accurate topography. The sinc function, also known as the “sampling function,” assumes the signal is band-limited; while this is ideal for communications and audio signals, it may not be the case in this paradigm. Polynomial interpolation techniques perform well in smooth regions but are computationally expensive. When using this type of technique boundary conditions can be tricky to handle and may not be exact. There is also a trade-off between the order of the polynomial and the data fitting accuracy. Splines are very popular and achieve higher reconstruction accuracy than polynomial techniques. The major draw back seen when attempting to fit this method to the data is that it is difficult to accurately solve a global spline over the entire DEM. This seems to point to a need for a method that can accurately propagate edge content without producing visual artifacts in a way that is flexible to the DEM paradigm while being relatively computationally inexpensive. We believe the algorithm described in this paper is extremely well-suited for this purpose. INPAINTING METHODOLOGY Inpainting is the process of filling in part of an image or video using information from the surrounding area. We extend the canonical paper by M. Bertalmio et al. (2000) such that it autonomously detects and fills in variable sized holes in Digital Elevation Models (DEM); it uses a variant of the heat equation to propagate the information from the boundary, ∂Ω, into the voided area Ω. Moreover, the numerical partial differential equation for a Digital Elevation Models will yield the non-linear solution: Η n+1 = Η n + Δ Η n ∀ ∈ Ω (i, j) (i, j) t t (i, j), (i, j) (Equation 1) Here Hn+1 represents the new updated DEM at iteration n+1, where n denotes the iteration. Current height, Hn, represents the DEM at iteration n. The crux of the technology centers on update to change in current DEM height, n Δ H t . The t represents the rate of improvement allowed per iteration. The improved image is given by: ⎛ ⎞ n = ⎜∂ n • N(i, j,n) ⎟ ∇ n Ht (i, j) ⎜ L (i, j) ⎟ | H (i, j) | (Equation 2) ⎝ | N(i, j,n) | ⎠ with H = H , where H is the initial DEM. Here the rate of change of the Laplacian, L, is propagated in the |∂Ω o o direction of minimum change. Visual representation is shown in Figure 1. MAPPS/ASPRS 2006 Fall Conference November 6 – 10, 2006 * San Antonio, Texas Propagate height information from outside inpainting region ? along direction of iso-contour (lines of constant contour height values). ∂H Isophote direction = ∇L • N ∂t Obtain iso-contour direction, N, by taking 90o rotation of ∇ DEM gradient. L is discrete Laplacian. Full inpainting equation: nn+1 =+Δ∀∈Ω n Gradient direction H(, ij ) H (, ij ) tHijt (, ), (, ij ) Figure 1. Inpainting Propagation of Height Information. Another partial differential equation was presented by Bertalmıo et al (2001). A relationship between the aforementioned approach and fluid dynamics of incompressible fluids is found. This novel solution was similar to the derived equation for 2-D vorticity stream equation. The 2-D vorticity stream function is the derived equation is calculated by taking the cross product of the primitive Navier-Stokes equation: G G ⎛ ∂v G G G⎞ ∂ω G G G ∇ × ⎜ + ()v • ∇ v = −∇p +υ∇ 2v ⎟ = + ()v • ∇ ω = υ∇ 2ω (Equation 3) ⎝ ∂t ⎠ ∂t G G 2 with ∇2H = ω H = H , where H is the initial DEM andω = ∇ × v = ∇ Ψ . The stream function |∂Ω o o Ψ represents the DEM heights. Taking the perpendicular at the gradient to Ψ gives the minimum rate of change of ⊥ G the heights (i.e. ∇ Ψ = v ). We propagate this minimum rate of change via the non-linear advection term, G G ()v • ∇ ω . Observe that the advection term propagates the Laplacian of the heights. Equations 1& 2 are variants of the geometric heat equation whereas equation 3 is the derived equation of the primitive equation for momentum for fluids. In the LiteSite™ tool we take advantage of both basic methodologies and extend these ideas to autonomously detect and inpaint variable sized holes in a DEM. We also leverage characteristics of the DEMs and LIDAR in our implementation of the two transport equations. Furthermore, anisotropic diffusion is added to both inpainting algorithms such that the data becomes more continuous. The anisotropic diffusion equation is taken from Perona and Malik (1990): MAPPS/ASPRS 2006 Fall Conference November 6 – 10, 2006 * San Antonio, Texas ∂I = div(g(| ∇I |)∇I) (Equation 4) ∂t We have chosen to use our own various anisotropic diffusers that align with the data. The main products from these technologies are bare earth and occlusion removal for 3-D urban scenes. LiteSite™ toolkit current algorithms perform autonomous culture and vegetation extraction. These features are autonomously detected and voids are created and filled in the single reflective surface input DEM in place. LITESITE PROCESS The LiteSite™ algorithm currently reads a list of 3-D LIDAR or IFSAR points (usually several million). A competitive filter is used to take an unordered list of LIDAR points and generate an equally spaced Digital Elevation Model (DEM) at a given resolution. Alternatively, optical imagery may be correlated to create the DEM as shown in Figure 2. The buildings and trees are then extracted to a separate DEM from the ground and can then even be separated from each other. The remaining ground data then has all of its resulting missing data filled using our inpainting algorithm to complete a full bare earth DEM with edge contours propagated. Filtering is then performed to remove any noisy LIDAR returns. A line following algorithm followed by a multi-stage generalization algorithm applies 3-D chord points, to be used as polygon vertices, to the rooftop features. The sides of the buildings are extruded straight down to the void filled bare earth from the rooftops. The accuracy of this stage of the process is highly dependent on the accurate void filling of the inpainting algorithm. The vertices from the building roofs and sides as well as the ground surface are mapped into polygons and projected into geo-spatial coordinates.
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