Boundary Estimation from Intensity/Color Images with Algebraic Curve Models Tolga Tasdizen Division of Engineering Brown University Providence, RI 02906 tt@lems.brown.edu David B. Cooper Division of Engineering Brown University Providence, RI 02906 cooper@lems.brown.edu Abstract A new concept and algorithm are presented for non- iterative robust estimation of piecewise smooth curves of maximal edge strength in small image windows – typically to . This boundary-estimation algorithm has the nice properties that it uses all the data in the window and thus can find locally weak boundaries embedded in noise or texture and boundaries when there are more than two re- gions to be segmented in a window; it does not require step edges – but handles ramp edges well. The curve-estimates found are among the level sets of a d’th degree polynomial fit to ”suitable” weightings of the image gradient vector at each pixel in the image window. Since the polynomial fit- ting is linear least squares, the computation to this point is very fast. Level sets then chosen to be appropriate boundary curves are those having the highest differences in average gray level in regions to either side. This computation is also fast. The boundary curves and segmented regions found are suitable for all purposes but especially for indexing using algebraic curve invariants in this form. Acknowledgment This work was partially supported by NSF Grants #IIS-9802392 and #BCS-9980091. 1. Introduction We present a new concept and algorithm for estimating boundary curves in images. These have two uses: 1) for general applications, 2) for direct extraction from images of algebraic curves to represent image boundaries. Algebraic curve fitting to binary images has been studied extensively [2, 3, 6, 7, 4, 1, 5]. Boundary contours of objects of interest can be easily extracted from binary images. However, ob- taining binary images automatically from real sensory data such as an intensity image involves the unsolved problem of segmentation. Consequently, these approaches to algebraic curve fitting which rely on a prior segmentation step will not work in a general setting. Our objective in this regard is to estimate algebraic curves that represent boundary curves in images without assuming any prior segmentation. Our algorithm can be seen as a combined algebraic curve fit- ting/contour detection which uses the representation power of algebraic curves to robustly detect contours from images. Sec. 2.1 gives a brief overview of algebraic curves. A method for extracting “appropriate” polynomials from im- ages is given in Sec. 2.2. Sec. 3 deals with estimating de- sired contours from images, based on the polynomials from Sec. 2.2. 2. Fitting Algebraic Curves to Images 2.1 Shape Representation by Algebraic Curves A th degree algebraic curve, also called an implicit polynomial curve (IP curve) is the set of points satisfying where is the polynomial . More generally, the polynomial yields a set of IP curves where , the set of real numbers. These are the level sets of . If we choose , we obtain the zero set of . If we let assume values other than , multiple shapes can be represented by a single polynomial . We make use of this in Sec. 2.2 and 3. A subset of level sets for a single polyno- mial is the model we extract for the salient boundary curves in a window of an image. 2.2 Direct Fitting to Intensity/Color Images The polynomial estimation algorithm proposed next is based only on image gradient information. Let be an intensity image and its gradient vector map. Let , the length of the gradient vector at . If the input is a color image, is computed as in [8] to make full use of the color information. A few example gra- dient vectors in a window from an intensity image are drawn