Cores (defined at right) are used to automatically segment, or outline, objects in grayscale images (Fig. 1). A core provides geometric information about the desired object. At each point along the core we know its location, the radius of the object, and the orientation of the core tangent direction. Overview Cores (defined at right) are used to automatically segment, or outline, objects in grayscale images (Fig. 1). A core provides geometric information about the desired object. At each point along the core we know its location, the radius of the object, and the orientation of the core tangent direction. Tracking Tubular Shaped Objects CISMM: Computer Integrated Systems for Microscopy and Manipulation Collaborators: Dorothy Erie, Garrett Matthews, Martin Guthold Project Lead: Russell M. Taylor II Investigators: Yonatan Fridman, Stephen Pizer October 2002 Tracking Tubes in 2D Using Cores Extending Tube Tracking to 3D So that spurious bumps and dimples on the object boundary don’t lead to wiggles in the backbone.So that spurious bumps and dimples on the object boundary don’t lead to wiggles in the backbone. To blur out noise so that the object is easier to locate.To blur out noise so that the object is easier to locate. Then march along the core by taking a small step in the tangent direction and repeating optimization.Then march along the core by taking a small step in the tangent direction and repeating optimization. Figure 1: A grayscale image of a strand of DNA segmented using cores. The red curve is the core. The blue curves are the DNA edges implied by the core location and radius. Solution: Compute the difference between the orientations of the core tangents at the two ends of the strand. The blue lines show the tangents of interest. The red curve shows the computed core. Solution: Find the DNA bending angle at the protein by analyzing the core’s tangent directions (blue arrows); determine whether the DNA wraps around the protein by comparing the expected length of the DNA to the computed length of its core. Applications Problem: Determine the bending angle induced by the heating of a bi- layer strand. Problem: Determine how a strand of DNA is bound to a protein. An object’s core also provides a complete re- presentation of the object (Fig. 2). For each core point, place a disk whose center is the specified core point and whose radius is as indicated – the union of these disks is exactly the desired object. Cores can also be extended to 3D for segmenting objects in volume images. Instead of using a core point with two sails to locate the object of interest, each core point now has a set of radially symmetric sails Figure 2: Reconstruction of a tube from its core. t x Figure 6: A representation of a 3D tube detector. x represents the coordinates of the core point (x,y,z) and t represents the core tangent direction. Figure 7: Slice of a noisy 3D image of a tube whose axis lies entirely in the plane displayed. The red curve is the core of the tube. in an extremely noisy synthetic 3D image of a tube. (Fig. 6). This representation makes the limiting assumption that the object of interest is perfectly tubular (i.e., an extended object circular cross-sections). A one- dimensional derivative of a three- dimensional Gaussian is placed at each sail tip, where the derivative is taken in the direction of the sail. (Fig. 6). This representation makes the limiting assumption that the object of interest is perfectly tubular (i.e., an extended object with circular cross-sections). A one- dimensional derivative of a three- dimensional Gaussian is placed at each sail tip, where the derivative is taken in the direction of the sail. Due to the added information provided by examining the image at more than two sail locations, the 3D algorithm is more robust than the 2D algorithm in the presence of noise. Figure 7 shows an accurately located core Note: 2D tube tracking software is available for download from the web page given below. Figure 4: Blurring a noisy image. The Mathematics of Cores A core is a medial axis (backbone) of an object in a blurred image. Why blur the image? A core is computed using a marching algorithm: A core is computed using a marching algorithm: Start by manually estimating a core point at one end of the object and placing a derivative of a 2D Gaussian at two “sail” points. Each Gaussian derivative acts as an edge detector – when convolved with the image, it gives a strong response if it’s aligned with an object edge.Start by manually estimating a core point at one end of the object and placing a derivative of a 2D Gaussian at two “sail” points. Each Gaussian derivative acts as an edge detector – when convolved with the image, it gives a strong response if it’s aligned with an object edge. Refine the estimated core point by simultaneously optimizing the two derivative of Gaussians’ fits to the image with respect to location (x,y), radius (r), and orientation (t).Refine the estimated core point by simultaneously optimizing the two derivative of Gaussians’ fits to the image with respect to location (x,y), radius (r), and orientation (t). Figure 5: Computing a core. (x,y) r t Figure 3a: A heated carbon nanotube with an aluminum layer. Figure 3b: A protein (purple) bound to a strand of DNA. 1.Aylward, SR, E Bullitt (2002). Initialization, noise, singularities, and scale in height ridge traversal for tubular object centerline extraction. IEEE Transactions on Medical Imaging, 21: Aylward, SR, SM Pizer, E Bullitt, D Eberly (1996). Intensity ridge and widths for tubular object segmentation and description. IEEE Workshop on Mathematical Methods in Biomedical Image Analysis, 56: Pizer, SM, D Eberly, BS Morse, DS Fritsch (1998). Zoom-invariant vision of figural shape: The mathematics of cores. Computer Vision and Image Understanding, 69: This work is built upon other work done in MIDAG, including that of Aylward, Bullitt, Eberly, Fritsch, Furst, Morse, and Pizer. Specifically, Aylward and Bullitt [1], [2] use a multi-scale image intensity ridge traversal method in which they separately search for position and width information, and define orientation implicitly. Also see [3] for more information on the mathematics of cores.