Comparing Images Using the Hausdorff Distance Mark Bouts 6th April 2006

Introduction Efficiently compute the Hausdorff Distance between all relative positions of a model and an image Main topic: Computing the Hausdorff Distance under translation  Extension to rigid motion Introduction Hausdorff distance Comparing portionsHD grid pointsHD rigid motionHD translation examples

Introduction  Point sets in the plane The minimal Hausdorff Distance under translation and how to compute it Distance measure portion of a model and an image  Raster data Present approximation algorithms, which operate on binary rasters making them suited for IP and MV applications Distance measure for Grid points (and portions)  Examples Introduction Hausdorff distance Comparing portionsHD grid pointsHD rigid motionHD translation examples

Hausdorff distance Focus on 2D case Measure the Hausdorff Distance for point sets and not for segments HD is a metric over the set of all closed and bounded sets  Restriction to finite point sets (all that is necessary for raster sensing devices) Introduction Hausdorff distance Comparing portionsHD grid pointsHD rigid motionHD translation examples

Hausdorff distance set A = {a1,….,ap} and B = {b1,….,bq}  Hausdorff distance  Directed Hausdorff distance h(A,B) ranks each point of A based on its nearest point of B and uses the most mismatched point Introduction Hausdorff distance Comparing portionsHD grid pointsHD rigid motionHD translation examples

Minimal Hausdorff distance [Alt, et al. 1991] Can be computed in time O((p+q)log(p+q)) Considers the mismatch between all possible relative positions of two sets Minimal Hausdorff Distance M G Introduction Hausdorff distance Comparing portionsHD grid pointsHD rigid motionHD translation examples

How to compute the M T Again the HD definition  We define Interested in the graph of d(x) which gives the distance from any point x to the nearest point in a set of source points in B Introduction Hausdorff distance Comparing portionsHD grid pointsHD rigid motionHD translation examples

Voronoi surface It is a distance transform It defines the distance from any point x to the nearest of source points of the set A or B Introduction Hausdorff distance Comparing portionsHD grid pointsHD rigid motionHD translation examples Images: Alt H. Discrete Geometric Shapes Huttenlocher D. Comparing images using the Hausdorff distance

Computing the HD T Can be rewritten as: is the maximum of translated copies of d(x) and d’(x) Define: the upper envelope (pointwise maximum) of p copies of the function d(-t), which have been translated to each other by each Introduction Hausdorff distance Comparing portionsHD grid pointsHD rigid motionHD translation examples

Runtime [Huttenlocher et al., 1991] O(pq(p+q)logpq)  L 1,L 2 or L ∞ norm  [Chew et al.] L 1, L ∞ O(pqlogpq)  Complicated to implement, less efficient in practice than the rasterized approximations presented later Introduction Hausdorff distance Comparing portionsHD grid pointsHD rigid motionHD translation examples

Comparing Portions of Shapes Extent the case to Finding the best partial distance between a model set B and an image set A Ranking based distance measure K= Introduction Hausdorff distance Comparing portionsHD grid pointsHD rigid motionHD translation examples

Comparing Portions of Shapes Target is to find the K points of the model set which are closest to the points of the image set. ‘Automatically’ select the K ‘best matching’ points of B  It identifies subsets of the model of size K that minimizes the Hausdorff distance  Don’t need to pre-specify which part of the model is being compared Introduction Hausdorff distance Comparing portionsHD grid pointsHD rigid motionHD translation examples

Comparing Portions of Shapes Does not obey metric properties!  Identity and symmetry hold  Triangle inequality does not always hold only under strict conditions Have to make sure B1 and B2 match the same part of the image. Then we expect both models to be similar Introduction Hausdorff distance Comparing portionsHD grid pointsHD rigid motionHD translation examples

Min HD for Grid Points Now we consider the sets A and B to be binary arrays A[k,l] and B[k,l] F[x,y] is small at some translation when every point of the translated model array is near some point of the image array Introduction Hausdorff distance Comparing portionsHD grid pointsHD rigid motionHD translation examples

Min HD for Grid Points Rasterization introduces a small error compared to the true distance  Claim: F[x,y] differs from f(t) by at most 1 unit of quantization The translation minimizing F[x,y] is not necessarily close to translation of f(t)  So there may be more translation having the same minimum Introduction Hausdorff distance Comparing portionsHD grid pointsHD rigid motionHD translation examples

Compute VS array D[x,y] Many distance transform algorithms  Two-pass serial algorithm proposed by Paglieroni [Paglieroni,1992]  Using rendering or z-buffers Use these to compute the lower envelope of the cone-shapes Run time O(p) time (p number points in E) (rendering and z-buffering is constant time) Introduction Hausdorff distance Comparing portionsHD grid pointsHD rigid motionHD translation examples

Computing HD array F[x,y] can be viewed as the maximization of D’[x,y] shifted by each location where B[k,l] takes a nonzero value Expensive computation!  Constantly computing the new upper envolope Introduction Hausdorff distance Comparing portionsHD grid pointsHD rigid motionHD translation examples

Computing HD array F[x,y] Probing the Voronoi surface of the image Looks similar to binary correlation  Due to no proximity notion is binary correlation more sensitive to pixel purturbation Introduction Hausdorff distance Comparing portionsHD grid pointsHD rigid motionHD translation examples

Matching portions of shapes Matching portions of the image and a model  Partial distance formulation is not ideal! Consider portion of the image to the model More wise to only consider those points of the image that ‘underneath’ the model Introduction Hausdorff distance Comparing portionsHD grid pointsHD rigid motionHD translation examples

HD under Rigid motion Extent the transformation set with rotation  Minimum value of the HD under rigid (Euclidean) motion Ensure that each consecutive rotation moves each point by at most 1 pixel  So Introduction Hausdorff distance Comparing portionsHD grid pointsHD rigid motionHD translation examples

HD under Rigid motion Computation Limitations  Only from the model B to the image A  Complete shapes Method  For each translation Create an array Q in which each element =  For each point in B For each rotation we probe the distance transform and maximize it with values already in the array Introduction Hausdorff distance Comparing portionsHD grid pointsHD rigid motionHD translation examples

Example translation Image: 360 x 240 pixels Model: 115 x 199 pixels f1 = 0.8 and f2 = 0.5 Sun-4 (SPARCstation 2) Runtime ± 20 seconds 2 matches Imagemodeloverlaid Introduction Hausdorff distance Comparing portionsHD grid pointsHD rigid motionHD translation examples Images Huttenlocher D. Comparing images using the Hausdorff distance

Example Rigid motion Image: 360 x 240 pixels Model: 31 x 31 pixels f 1 = 0.8 and f 2 = 0.5 Sun-4 (SPARCstation 2) Runtime ± 216 seconds 15 matches Imagemodel overlaid Introduction Hausdorff distance Comparing portionsHD grid pointsHD rigid motionHD translation examples Images Huttenlocher D. Comparing images using the Hausdorff distance

Summary Hausdorff Distance Minimal Hausdorff as a function of translation  Computation using Voronoi surfaces Compared portions of shapes and models The minimal HD for grid points  Computed the distance transform  The minimal HD as a function of translation  Comparing portions of shapes and models The Hausdorff distance under rigid (euclidean) motion Examples Introduction Hausdorff distance Comparing portionsHD grid pointsHD rigid motionHD translation examples

References  [Huttenlocher et al., 1991] Huttenlocher Daniel P, Kedem K and M. Sharir The Upper Envelope of Voronoi Surfaces and Its Applications, ACM symposium on Computational Geometry, , 1991  [Alt et al., 1991] Alt H., Behrends B., Blomer J. Measuring the resemblance of polygonal shapes. In Proc. Seventh ACM Symposium on Computational Geometry  [Paglieroni, 1992] Paglieroni D.W, Distance transforms: Properties and machine vision applications. Computer Vision, Graphics and Image Proc: Graphical Models and Image Processing, 54(1):56-74,1992 Introduction Hausdorff distance Comparing portionsHD grid pointsHD rigid motionHD translation examples