This paper describes a technique based on the Osher and Shethian level-sets approach of recovering object's shape in ultrasound medical images.
Introduction
This paper describes a technique based on the Osher and Shethian level-sets approach of recovering object’s shape in ultrasound medical images. The level-sets approach is an extension of the classic snakes model, in which a 2D active contour deforms toward the object’s boundary, while trying to minimize an energy function. The minimum of the energy function is reached when the curves lye on the object’s boundary. The energy function includes two elements: the first is a stopping criteria synthesized from the image, which takes advantage of the high gradients above the object boundary for stopping the progress of the contour. The other is a smoothing criteria synthesized from the contour’s curvature without any correlation to the image. (The curvature measures how fast a curve tends to fold at any spot. For example, a circle has a constant curvature because it always is turning at the same rate, a smaller circle has a higher constant curvature because it turns faster). The smoothing is necessary, since a sporadic noise might cause local distortions of the curve as a result of the termination criteria. A typical implementation of this model uses a collection of marked particles, and a collection of finite differential equations, which approximate the advance of each particle. This technique got many problems associated with. This model can’t handle sharp corners of the object (as a result of the dependency between the smoothing element and the object’s curvature) or detect more then one object boundary. Moreover, the numerical method might be unstable, since small computation inaccuracies in the particle position can cause large errors in determining the curvature (the curvature has a huge effect on the progress of the contour). We can’t reduce the damage by increasing the number of the marking points, since the particles might cross each other.
Level Sets
The level set idea is to add a dimension to the problem. Instead of the 2D active contour, the level sets approach uses a 3D active surface. The surface is a collection of sets where all the elements which belong to a specific set, got the same unique value. A special set is the zero level set, which defines the position of the curve. This model is stable, since the computation errors spread all over the
surface, and have minor affect on the zero level set. Moreover, this powerful technique allows splitting and merging the curve naturally, it handles sharp corners, allowing the detection of several object boundaries simultaneously, and detects interior boundaries of a complicated object. The mathematical model of the technique defines a sign distance function instead of the energy function. The level of each point on the surface is determined by its distance from the curve (so the zero level set which defines the curve, has zero distance from itself). The points outside the curve are positive while the points inside the curve have a negative sign. The surface is moving in order to minimize the distance function. The minimum of the distance function is reached when the zero level set is lying on the object’s boundary. Movement of the surface usually causes movement of the curve.
Suppose we start with a circle as an initial curve, the distance function will define a con shape surface. When the surface moves down, the curve grows, and when it moves up the curves contracts. The surface moves under its curvature, according to the famous theorem in differential geometry:
“Any simple closed curve moving
under its curvature collapses nicely to a circle and then disappears”
Blue arrows are where the
curvature is negative
Green arrows are where the
curvature is positive
When the curve gets near to the object’s boundary, a termination function restrains it’s movement, and the contour changes it’s shape, and starts tracking the object’s boundary. The termination function prevents the curve from disappearing.
Algorithm
0. Create an initial surface U, according to an initial curve. The initial curve may be around the object and contract, or may be surrounded by the object and grow out.
1. Synthesize a stopping image G out of the image I, according the stopping function g(I). g can be any monotonous decreasing function which it’s domain is the absolute value of the image gradient and it ranges between zero and one (zero correspond to high gradients).
0. Start an iterative sequence.
1. When the iterative sequence stops, the zero level set is lying on the object’s boundary.
We said that a pixel belongs to the zero level set if, in it’s near area there’s a negative pixel and a positive one. The method of zero crossing is used, since with the discrete model we may not find any zero level set.
Medical Imaging
The main idea here is to isolate and extract individual components from a medical image. This is an important part of medical imaging. Once a shape is found, physicians can measure various quantities, such as the size of tumors and the thickness of heart walls. We will present an application based on the above theorem for isolating gaps in cranium bones as appear in ultrasound image of baby’s head. These gaps usually appear as a homogeneous dark area in a noisy bright background. The technique is initializing a small circle inside the region of interest, and allowing it to grow outwards until it reaches the desired boundary.
