Reconstruction of 3D Models from Intensity Images and Partial Depth L. Abril Torres-Méndez and Gregory Dudek Centre for Intelligent Machines School of Computer Science McGill University

Our Application Automatic generation of 3D maps. Robot navigation, localization - Ex. For rescue and inspection tasks. Robots are commonly equipped with camera(s) and laser rangefinder.  Would like a full range map of the the environment.  Simple acquisition of data

Problem Context Pure vision-based methods –Shape-from-X remains challenging, especially in unconstrained environments. Laser line scanners are commonplace, but –Volume scanners remain exotic, costly, slow. –Incomplete range maps are far easier to obtain that complete ones. Proposed solution: Combine visual and partial depth Shape-from-(partial) Shape

Problem Statement From incomplete range data combined with intensity, perform scene recovery. From range scans like this infer the rest of the map

Overview of the Method Approximate the composite of intensity and range data at each point as a Markov process. Infer complete range maps by estimating joint statistics of observed range and intensity.

What knowledge does Intensity provide about Surfaces? Two examples of kind of inferences: Intensity image Range image surface smoothness variations in depth surface smoothness far close

What about outliers? Changes in intensity from texture-like regions Intensity Range Not critical for reconstruction since very close neighborhoods represent the same type of smooth surface.

What about Edges? Edges often detect depth discontinuities Very useful in the reconstruction process! Intensity Range edges

Range synthesis basis Range and intensity images are correlated, in complicated ways, exhibiting useful structure. - Basis of shape from shading & shape from darkness, but they are based on strong assumptions. The variations of pixels in the intensity and range images are related to the values elsewhere in the image(s). Markov Random Fields

Related Work Probabilistic updating has been used for –image restoration [e.g. Geman & Geman, TPAMI 1984] as well as –texture synthesis [e.g. Efros & Leung, ICCV 1999]. Problems: Pure extrapolation/interpolation: –is suitable only for textures with a stationary distribution –can converge to inappropriate dynamic equilibria

MRFs for Range Synthesis States are described as augmented voxels V=(I,R,E). Z m =(x,y):1≤x,y≤m Z m =(x,y):1≤x,y≤m: mxm lattice over which the image are described. I = {I x,y }, (x,y)  Z m I = {I x,y }, (x,y)  Z m : intensity (gray or color) of the input image E is a binary matrix (1 if an edge exists and 0 otherwise). R={R x,y }, (x,y)  Z m R={R x,y }, (x,y)  Z m : incomplete depth values We model V as an MRF. I and R are random variables. R I v x,y Augmented Range Map I R

Markov Random Field Model Definition: A stochastic process for which a voxel value is predicted by its neighborhood in range and intensity. N x,y is a square neighborhood of size n x n centered at voxel V x,y.

Computing the Markov Model From observed data, we can explicitly compute intensity intensity & range V x,y N x,y This can be represented parametrically or via a table. –To make it efficient, we use the sample data itself as a table.

­ Further, we can do this even with partial neighborhood information. Estimation using the Markov Model From what should an unknown range value be? ¬For an unknown range value with a known neighborhood, we can select the maximum likelihood estimate for V x,y. ® Even further, if both intensity and range are missing we can marginalize out the unknown neighbors. intensity intensity & range

Interpolate PDF In general, we cannot uniquely solve the desired neighborhood configuration, instead assume The values in N u,v are similar to the values in N x,y, (x,y) ≠ (u,v). Similarity measure: Similarity measure: Gaussian-weighted SSD ( sum of squared differences ). Update schedule is purely causal and deterministic.

Order of Reconstruction Dramatically reflects the quality of the result. Our reconstruction sequence is based on the amount of reliable information surrounding each of the voxels to be synthesized. Edge information used to defer reconstruction of voxels with edges as much as possible. Info-edge-driven ordering Correct result With the spiral-ordering

Experimental Evaluation Obtain full range and intensity maps of the same environment. Remove most of the range data, then try and estimate what it is. Use the original ground truth data to estimate accuracy of the reconstruction.

Arbitrary shape of unknown range data Input Intensity Synthesized result Scharstein & Szeliski’s Data Set Middlebury College Compact case Ground truth range Input range image Synthesized result Ground truth range

Input IntensityGround truth range Input range image Arbitrary shape of unknown range data Scharstein & Szeliski’s Data Set Middlebury College Less compact case Synthesized result

Arbitrary shape of unknown range data Compact Distributed Synthesized results Ground truth range Expected quality of reconstruction degrades with distance from known boundaries Need broader distribution of range-intensity combinations in the sampling

Input IntensityEdge map Stripes along the x- and y-axis Data courtesy of Oak Ridge National Labs & Univ. of Florida Ground truth range Approximated depth size: 650cms. Input range 66% missing range Synthesized Result Mean Absolute Residual (MAR) Error: 5.76 cms.

Input intensity and range data Synthesized results Ground truth range

Input range data. 66% of missing range Achromatic vs. Color Results Using achromatic image Using color image Synthesized range images Greyscale image Color image Ground truth range

More Experimental Results

Initial range scans Results on Realistic Sampling Not in the paper Real intensity image Ground truth range Edge map Synthesized range image

What happened? Problems with surfaces that slope  Intensity-range combinations from the initial range do not capture the underlying structure. Solution: Surface normal-based MRF range synthesis – Surface normals from initial range data is added to our MRF model so that depth values are computed according to the surface being reconstructed.

Adding Surface Normals Not in the paper We compute the normals by fitting a plane (smooth surface) in windows of nxn pixels. Normal vector: Eigenvector with the smallest eigenvalue of the covariance matrix. Similarity is now computed between surface normals instead of range values.

Initial range scans Preliminary Results Not in the paper Synthesized range image Ground truth range Edge map Real intensity image Initial range data Real intensity imageEdge map

Conclusions Works very well -- is this consistent? Can be more robust than standard methods (e.g. shape from shading) due to limited dependence on a priori reflectance assumptions. Depends on adequate amount of reliable range as input. Depends on statistical consistency of region to be constructed and region that has been measured.

Discussion & Ongoing Work Surface normals are needed when the input range data do not capture the underlying structure Data from real robot –Issues: non-uniform scale, registration, correlation on different type of data

Questions ?

Input intensityGround truth range Range synthesis using edge info Initial range data (white regions are to be estimated) Range synthesis (no edge info) Animated example

Input intensityGround truth range Range synthesis using edge info Initial range data (white regions are to be estimated) Range synthesis (no edge info) Animated example

I V (x,y) R Algorithm Outline The depth value of the augmented voxel V x,y is synthesized as follows: d Synthesis ordering Synthesis ordering : according to available information. (maximum number of “full” neighbors) argmax(P)

Compactness Reconstruction facilitated by less compact regions. Experimental illustration: keep area constant, vary the shape of area being recovered. Not in the paper

Compactness Average errors are 2.8% and 1.6%. Expected quality of reconstruction degrades with distance from known boundaries. Not in the paper

Residual Error in of Reconstructed Depth (from 0 to 20%) Amount to data missing Not in the paper

Depth histogram: ground truth & reconstructed Not in the paper

Technical Aside Really should have independent relations for R(x,y) and I(x,y).

Augmented Range Map Incorporate appearance values, expressing both spatial occupancy as well as appearance under specific viewing conditions, into augmented range map.

Mode of Absolute Residual Error

The Neighborhood The neighborhood N {x,y} is a square mask of size nxn centered at the augmented voxel. Observations: The neighborhood is causal The size is important to capture relevant characteristics v (x,y) I R Neighborhood of v (x,y) N {x,y}

Initial range (none) Novel View Synthesis (ongoing work) Intensity image Range image Known intensity image Infer range map from intensity of current view and from range and intensity of other views. Other view Current view to estimate range map Synthesized range Ground truth range

Previous results Results using EDGES Ground truth range MAR Errors (cms.)

Case 2 Result Mean Absolute Residual (MAR) Error: cms. 75% missing rangeInput intensity

Case 3 Result Mean Absolute Residual (MAR) Error: cms. 78% missing rangeInput intensity

More examples Synthesized results Ground truth rangeThe input intensity and range data

Order of Reconstruction Dramatically influences the quality of final result. Spiral scan-ordering (a.k.a.“onion peeling”) Disadvantage: strong dependence from the previous assigned depth value. With the spiral-orderingCorrect result

Outline Introduction Problem statement Overview of the method Related work Range Synthesis Algorithm Experimental results Conclusions and future directions

Adding Surface Normals Not in the paper We compute the normals by fitting a plane (smooth surface) in windows of nxn pixels. The eigenvector with the smallest eigenvalue of the covariance matrix is the normal vector. – The smallest eigenvalue measures the quality of the fit. In the range synthesis, the similarity is now computed between surface normals instead of range values.