Statistics of range images

The statistics of range images from natural environments is a largely unexplored field of research. It closely relates to the statistical modeling of the scene geometry in natural environments, and the modeling of optical natural images. We have used a 3D laser range-finder to collect range images from mixed forest scenes. The images are here analyzed with respect to different statistics.


Introduction
More and more people realize the importance of the statistics of optical images.Good statistical models of optical images are important to applications, such a s image compression, noise remo val and segmentation.The models may also further our understanding of the encoding of visual information in biological visual systems.Not much research has, however, been done on the statistics of range images.There are two main reasons why range images are interesting.First, they lead to a direct understanding of the stochastic nature of the geometry of the world, and give priors for stereo algorithms ( 1], 2]).Secondly, t h e y m a k e it possible to construct more realistic models of optical images.For example, authors in 11], 3] and 8] h a ve modeled optical images as a perspective view of the 3D world, with objects of random geometry (size, shape, position) and intensity.The object geometries in these models are usually based on assumptions, which h a ve not been directly veri ed in real scenes.There is no doubt, that with a fairly large data base of range images, we will better understand the scene geometry of the 3D world, and thus be able to develop more realistic models for optical images.
We h a ve collected 205 panoramic range images from varying environments (both outdoor and indoor scenes) | 54 of these images were taken in di erent forests in Rhode Island and Massachusetts during August-September.In this paper, we will focus on the forest scenes, because the statistics of these Figure 1: A sample image from our range image data base images appear more stable than those of other categories (for example, residential and interior scenes).Figure 1 shows a sample.
W e used a laser range-nder with a rotating mirror 1 to collect the range images.Each image contains 444 1440 measurements with an angular separation of 0.18 deg.The eld of view is thus 80 0 vertically and 259 0 horizontally.Each measurement is calculated from the time of ight of the laser beam.The operational range of the sensor is typically 2-200m.The laser wavelength of the range-nder is 0:9 m, which is in the near infra-red region.
Throughout this paper, we will work with log(range) instead of range directly, because the former statistics is closer to being shape invariant.be the di erence in log range for the same two pixels.
Then, a scene with 4ABC and a scene with triangle 4A 0 B 0 C 0 will lead to di erent v alues for D (r A ; r B vs. R A ; R B ) but the same value for b D (log( rA rB ) = log( RA RB )).Hence, log(range) is appropriate if we w ant the di erence statistics (or any mean-0-lter reaction) to be \shape invariant".

Single-Pixel Statistics
The solid line in Figure 3 shows the single-pixel statistics of log(range) images.W e observe a sudden change in slope at a range of about 20 meters (or log(range) 3 see vertical line in gure) | this may be related to the accumulation of occlusion effects.In Figure 3, we h a ve also plotted the log(range) histograms for the top half (dashed line) and bottom half (dotted line) of a range image separately.The two h a l v es correspond to di erent distributions of objects | mainly ground for the top half and mainly trees for the bottom half | and displa y quite di erent statistics.The distribution from the top half has an approximately linear tail in a semi-log plot (indicating an exponential decay e ;0:12r ), while the bottom half shows an approximately linear tail in a log-log plot (indicating a power law r ;2:6 ).We c a n qualitatively explain the two di erent behaviors with the following simpli ed models: For the top half, we assume tree trunks (cylinders) uniformly distributed on a plane, according to a Poisson process with density .Figure 4 shows a top view of a randomly generated \forest" scene.Each d i s k represents a cross section of the trunk of a tree.If we assume all disks are of diameter L, a simple calculation will show that the probability that a horizontal beam from the laser range-nder rst hits a tree at distance r is given by a n exponential distribution f(r) Le ; Lr : For the bottom half, we assume at ground only.Let the height of the sensor from the ground be H, as shown in Figure 5. Then at angle , the distance between the sensor and the ground is r = H sin .T h e laser range-nder samples evenly with respect to the polar coordinate(s) (and ), i.e. f( ) = constant for =2 < < .With respect to r, w e then get a power-law distribution 3 Derivative Statistics We n o w look at the marginal distribution of the horizontal derivative D, which in the discrete case, is simply the di erence between two adjacent p i x e l s i n the same row.The solid line in Figure 6 shows the log probability density function of D. As in the studies of optical images, this distribution has a high kurtosis with large tails, and a peak at 0. It is closest to the statistics for optical images of man-made environments 8], but has an even higher peak at 0. This indicates that the structure of range images is simpler than for optical images | an issue that w e will explore further in the next section.The derivative statistic has also been used to test the scale invariance property (see Sinai 13], Field 5], Ruderman 12], and for scale invariance) in natural images.Given an imageI, we de ne a scaled-down image I (N ) by computing the average of N N disjoint blocks.In 12], 14], 3] a n d 8], it has been shown that, for optical natural images, the distribution of D is the same for I and I (N ) .I n this paper, we scale down the images by taking the minimum, instead of the average of N N blocks.This is the appropriate renormalization for range images, because laser range nders measure the distance to the nearest object in each solid angle.Figure 6 shows the distribution of D at di erent scales.The results indicate that range images scale well.

Bivariate Statistics
Below, we study the co-occurrence statistics of two pixels with di erent separation distances.Let K(a bjx) = P r fI(x 1 ) = a I(x 2 ) = b j k x 1 ; x 2 k = xg where I(x 1 ) a n d I(x 2 ) represent the log(range) at pixels x 1 and x 2 .The left column of Figure 7 shows  The authors in 8] propose the following model for the bivariate statistics of optical images: where q is the marginal distribution for a single pixel, h x are distributions similar to q, and g x are distributions highly concentrated at 0. The rst term models the case where the two pixels belong to di erent objects (we assume that di erent object are statistically independent), the second term represents the case where they are on the same object (assume that the sum and the di erence of the pixel values are independent), and (x) is the probability of their being on the same object.
The right column of Figure 7 shows a t of the model above to the empirical bivariate statistics (left column) of range images Figure 8 shows the functions used in the t.The mixture nature | \same" v ersus \di erent" objects | of both of data and the model, is seen in the changing shape of the contour levels in the center versus the tails.W e also nd that the random collage model ts better to the bivariate statistics of range images than to that for optical images (see 8]).Again, this indicates that range images present a simpler, cleaner problem than optical images.F or example, the concept of objects is better de ned for range images where we do not need to take lighting, color variations, texture etc. into account.

Joint Statistics of Haar W avelet Coe cien ts
It has been shown that, the statistics in the wavelet domain can be very informative ( 4], 7], 8]).We h e r e choose the Haar wavelet for its simplicity: Any structure in the statistics can be directly related to pixel values.Note also that the central limit theorem can easily make the statistics of larger support lters appear misleadingly Gaussian.
Figure 9 shows the di erent Haar lters.To d escribe how the wavelet coe cients in the subbands are related to each other, we use the same de nitions as in 4]: Coe cients at adjacent spatial locations in the same subband are brothers, and coe cients in the same level and the same position, but di erent orientations are cousins.
Figure 10 shows contour plots of the joint density functions for di erent w avelet coe cient pairs.W e observe strong cusps in all contour level curves most of which lie on the lines x = 0 , y = 0 , x = y and x = 2y.As a comparison, look at Figure 11 where the corresponding contour plots are calculated for optical images in van Haterens's image database 6].We s e e that many cusps occur along the same lines, but are not as peaked.
Furthermore, we did the following experiments on the range images: 1. W e scaled the images down by taking the block minimum and calculated the joint histograms.
Figure 12 shows the result.2. W e scaled the images down by taking the block average and calculated the joint histograms, Figure 13 shows the result.When w e scale a range image down by b l o c k minimum, the resulting image is approximately the same as the image taken in the same environment at half resolution (as mentioned before, range images are almost scale invariant under block minimum).The second method (block a veraging) is, however, closer to how a digital camera for optical images works: The intensity at each pixel is the average (or some weighted mean) of the intensity a t p o i n ts covered by that pixel.This explains why the statistics shown in Figure 12 is similar to that in Figure 10, and the statistics in Figure 13 is similar to that in Figure 11.
Note that the observed cusps in Figure 10 are not caused by noise, but correspond to real structures in the images.It is important to see what these structures are: For the cousin pairs (horizontal and vertical horizontal and diagonal) this is relatively straight-forward | because the join t distribution of horizontal, vertical and diagonal wavelet coe cients is a su cient statistics for 2 2 blocks modulo mean (3 variables).Figure 14 shows an equiprobable surface of the 3D joint distribution (horizontal, vertical, diagonal).W e see vertices along the lines y = z = 0 , x = z = 0 , x = y = 0 and x = y = z.Simple calculations show that these vertices correspond to the following 2 2 blocks and their rotations: For the horizontal lter and left brother pair, we s e e cusps along y = 0 , x = 0 , y = x, y = 2 x and y = 1  2 x.To n d t h e 2 4 patches that correspond to these vertices, we sample randomly from our database for patches with strong lter reactions along y = 0 , y = x and y = 1

Conclusions
We h a ve i n vestigated several simple statistics of range images of forest scenes.The bivariate statistics of two pixels verify the basic assumption of the random collage model | that the w orld can be broken down into piecewise smooth regions (in, for example, range or intensity) that depend little on each other.The 2D and 3D joint distributions of Haar lter reactions show some striking features, which also indicate the presence of piecewise constant geometric structures and sharp discontinuities in range images.The above results all point to the fact that range images are much simpler to analyze than optical images: The concept of \objects" is often better de ned in terms of changes in range, than in terms of changes in intensity, color, texture, lighting etc. W e believe t h a t segmenting range images from natural scenes and a thorough analysis of the results | for example, the structure within objects, and the sizes, positions, and dependencies between di erent objects | will lead to a better understanding of the scene geometry of the 3D world, as well as more realistic statistical models for optical images.
Figure 8: The values for and the 1D functions g x , h x and q x in the best bivariate t at pixel separation distances x = 2 (solid), 16 (dashed), and 256 (dotted).

Figure 3 :
Figure 3: Semi-log (left) and log-log (right) plots of the single pixel statistics (i.e.range statistics).Solid: distribution for the whole image.Dotted: distribution for the bottom half of a range image.Dashed: distribution for the top half of a range image.

Figure 4 :Figure 5 :
Figure 4: Top view of a randomly generated forest scene.See text.
2 x.The 8 range values in the typical patches fall clearly into 2 and occasionally 3 tight clusters.Calling the clusters a,b, and c, we get for example:For the horizontal lter and upper brother pair, we see cusps along y = 0 , x = 0 , y = x, y = ;2x and y = ;1   2 x.The typical patches in the database, that 1063-6919/00 $10.00 ã 2000 IEEE lead to high lter reactions along these lines are: that all the striking cusps in the contour plots in the Haar wavelet domain relate to the piecewise constant geometric structure in range images.

Figure 11 : 1 −Figure 12 :Figure 13 :Figure 14 :
Figure 11: Contour plot of the log histogram of wavelet coe cient pairs for natural optical images.