Segmentation-Less, Automated, Vascular Vectorization
Fig 5
One-dimensional simplification of linear filtering step.
To form the energy image E at scales 1…n…N, the original image I(x) is convolved with a LoG filter and an Ideal kernel. The Ideal kernel is a linear combination of spherical and annular pulses to match the fluorescent signal shape of vessels. σ2 is the variance of the Gaussian, r is the radius of the Ideal kernel, so R2 = σ2 + r2 is the square “radius” of the LoG, matched filter. The resulting multiscale energy image is projected along the scale coordinate to form two three-dimensional images that depict energy and size (not shown here, example in Fig 2. In this example, the kernel weighting factor was chosen so that the sums of all the spherical and annular kernels were all equal, the ratio r/σ was chosen to be 1, and a vertex was found at location v with radius Rn and energy En(v).