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Efficient Implementation for Spherical Flux Computation (3D)

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Fourier domain based analytical compution of 3D multiradii spherical flux responses



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Compute the analytical 3D multi-radii spherical flux responses for curvilinear structure segmentation. Offer up to 1000 times speed up compared to spatial domain implementation. Support GPU computation for further speed improvement. Implementation based on,
[1] M.W.K. Law and A.C.S. Chung, ``Efficient Implementation for Spherical Flux Computation and Its Application to Vascular Segmentation``, TIP 2009, 18(3), pp. 596--612.

See the following regarding the details of level-set segmentation using the spherical flux response,
[2] A. Vasilevskiy and K. Siddiqi, ``Flux Maximizing Geometric Flows``, PAMI 2002, 24(12), pp. 1565--1578.

Also see the more advanced, orientation-sensitive spherical flux for curvilinear structure detection,

Segmentation can be found by zero-thresholding of the spherical flux response image.
outputresponse = fastflux3(image, radii, sigma, pixelspacing)

outputresponse: The final output response. 3D matrix, size(outputresponse)=size(image).
image: The original image. (3D matrix)
radii: A vector containing all interested radii for computation of 3D spherical flux. (N-D vector)
sigma: For image smoothing in during the computation of spherical flux. (Scalar, optional, default value = 1)
pixelspacing: It specifies the pixel size of the input image. This parameter can be omitted if the values of radii and sigma are given in the unit of pixel-length. The pixel spacing can be anisotropic. (Optional, default value = 1; Scalar for isotropic pixel size; 3D vector, given in [x-length,y-length, z-length] for anisotropic pixels)

result = fastflux3(I, 1:5, 1);
Return the result computed from the radii {1,2,3,4,5} on image I.

result = fastflux3(I, 0.4:0.4:2.8, 1, 0.4);
Return the result computed from the radii (0.4mm,0.8mm,1.2mm,1.6mm,2mm,2.4mm,2.8mm}, where the each pixel is 0.4mmx0.4mm.

result = fastflux3(I, 2:5, 1, [1 1 1.4]);
Return the result computed from the raddi {2mm,3mm,4mm,5mm}, where each pixel has a size of 1mm x 1mm x 1.4mm (w x l x h).

To workaround the Fourier wrap around artifact, use the command "padarray". The return value of this function is the analytical version of the maximum magnitude outward flux computed among the spheres with multiple radii (Section 3.2, first paragraph in pp.1572 in [2]). In [2], this response is adopted directly on a level set segmentation framework ("F" in pp.1573).

Technical details:
The return value of this function is the "Output the multicale spherical flux" stage in Fig. 4b in [1]. This code uses Subband_1.5 Fourier oversampling technique for better computation accuracy.

The following variables used in this code correspond to the symbols used in [1]:
image -> I (After Equation 10)
ranges -> M (Equation 29)
sigma -> \sigma (Equation 7)
pixelspacing -> This is a new feature that is not mentioned in [1].
Note1 : This implementation is based on Subband_1.5 as stated in Fig. 2 in [1]
Note2 : An undersize \sigma value (<0.9 voxel-length) may cause inaccurate computation. Please refer to pp.599-601 in [1] for details.
Note3 : Owing to the exceeding memory requirement, the technique of coefficient buffering [1] is not implemented.
Note4 : fftn in MatLab does not support compact transform (i.e. omitting the conjugate half of Fourier coefficient). Thus, redundant coefficient elimination is not supported.
Note5 : GPU computation is automatically enabled if "image" is a gpuArray.
Note6 : The peak memory comsumption is around 11 times of the input image. The precision (double/single) of all variables follows the type of "image". Generally, single precision strike the balance between memory comsumption and computation accuracy for segmentation.
Note7 : Comment Block2 to rewind back to Subband_1. It significantly increases the computation speed and reduces memory usage in the cost of computation accuracy. Error is generally acceptable in most cases. See [1] for more details.

Please kindly cite the following paper if you use this program, or any code extended from this program.
Max W. K. Law and Albert C. S. Chung, "Efficient Implementation for Spherical Flux Computation and Its Application to Vascular Segmentation”, IEEE Transactions on Image Processing, 2009, Volume 18(3), 596–612.

Author: Max W.K. Law

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