Prof. Bart M. ter Haar Romeny, PhD Multi-scale segmentation Edge focusing, watershed, winding numbers Prof. Bart M. ter Haar Romeny, PhD

Edge focusing toppoints graph theory  scale MR slice hartcoronair

Structures exist at their own scale: Original  = e0 px  = e1 px  = e2 px  = e3 px Noise edges

The graph of the sign-change of the first derivative of a signal as a function of scale is denoted the scale-space signature of the signal. Zero-crossings of the second order derivative = max of first order derivative, as a function of scale

The notion of longevity can be viewed of a measure of importance for singularities [Witkin83]. The semantical notions of prominence and conspicuity now get a clear meaning in scale-space theory. In a scale-space we see the emergence of the hierarchy of structures. Positive and negative edges come together and annihilate in singularity points.

Example: Lysosome segmentation in noisy 2-photon microscopy 3D images of macrophages.

Marching-cubes isophote surface of the macrophage. Preprocessing: - Blur with  = 3 px - Detect N strongest maxima

We interpolate with cubic splines interpolation 35 radial tracks in 35 3D orientations

The profiles are extremely noisy: Observation: visually we can reasonably point the steepest edgepoints.

Edge focusing over all profiles. Choose a start level based on the task, i.e. find a single edge.

Detected 3D points per maximum. We need a 3D shape fit function.

The 3D points are least square fit with 3D spherical harmonics:

Resulting detection:

An efficient way to detect maxima and saddlepoints is found in the theory of vector field analysis (Stoke’s theorem)

Topological winding numbers  is the wedge product (outer product for functionals)

maximum:  = 1 minumum:  = 1 regular point:  = 0 In 2D: the surrounding of the point P is a closed path around P. The winding number  of a point P is defined as the number of times the image gradient vector rotates over 2 when we walk over a closed path around P. maximum:  = 1 minumum:  = 1 regular point:  = 0 saddle point:  = -1 monkey saddle:  = -2

Winding number = +1  extremum Winding number = -1  saddle The notion of scale appears in the size of the path.

Winding number = - 4  monkey saddle Generalised saddle point (5th order): (x+i y)5 The winding numbers add within a closed contour, e.g. A saddle point (-1) and an extremum (+1) cancel, i.e. annihilate. Catastrophe theory

Decrease of structure over scale scales with the dimensionality. The number of extrema and saddlepoints decrease as e-N over scale

Multi-scale watershed segmentation Watershed are the boundaries of merging water basins, when the image landscape is immersed by punching the minima. At larger scale the boundaries get blurred, rounded and dislocated.

Regions of different scales can be linked by calculating the largest overlap with the region in the scales just above.

The method is often combined with nonlinear diffusion schemes E. Dam, ITU

Nabla Vision is an interactive 3D watershed segmentation tool developed by the University of Copenhagen. Sculpture the 3D shape by successively clicking precalculated finer scale watershed details.