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# Efficient three-dimensional (3D) Gaussian smoothing using convolution via frequency domain

### Max W.K. Law (view profile)

12 Nov 2012 (Updated )

Native Fourier implementation, support GPU computation and anisotropic voxel.

File Information
Description

Three-dimensional Gaussian smoothing in the frequency domain, native
frequency domain implementation. Smoothing is achieved by replacing
the spatial domain convolution with Fourier coefficient
multiplication. This is also an excellent example to implement your
own filters using native Fourier expression.

R = gauss3filter(I);
R = gauss3filter(I, sigma);
R = gauss3filter(I, sigma, pixelspacing);
In a spatial domain representation, R = convn(I, f(x,y,z));
The Gaussian kernel f(x,y,z) is different depending on the function
inputs, see the description below. No image padding is provided, pay
attention to the Fourier wrap-around artifacts.

Anisotropic smoothing is partly supported, anisotropic voxel size is
fully supported. Suband_1.5 frequency oversampling is employed to reduce
numerical erros when sigma is less than the voxel length. Please refer to
following paper for the Subband_x frequency oversampling technique:
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

R = gauss3filter(I);
Smooth the image using isotropic smoothing with sigma = 1 voxel-length,
f(x,y,z) = (2*pi)^(-3/2) * exp(-(x.^2/2 - y.^2/2 - z.^2/2));

R = gauss3filter(I, sigma);
If sigma is a scalar, it smooths the image using isotropic smoothing with
sigma voxel-length,
f(x,y,z) = (2*pi)^(-3/2)/(sigma^3) * exp(-(x.^2/sigma^2/2 - y.^2/sigma^2/2 - z.^2/sigma^2/2));
If sigma is a 3D vector, i.e. sigma = [sigma_x sigma_y sigma_z], it
smooths the image using anisotropic smoothing (oriented anisotropic
Gaussian is not supported),
f(x,y,z) = (2*pi)^(-3/2)/sigma(1)/sigma(2)/sigma(3) * exp(-(x.^2/sigma(1)^2/2 - y.^2/sigma(2)^2/2 - z.^2/sigma(3)^2/2));

R = gauss3filter(I, sigma, pixelspacing);
If sigma is a scalar, smooth the image using isotropic smoothing with
sigma physical-length. pixelspacing is a 3D vector. It defines the size
of a voxel in physical-length,
f(x,y,z) = (2*pi)^(-3/2)/(sigma^3) * exp(-((x*pixelspacing(1)).^2/sigma^2/2 - (y*pixelspacing(2)).^2/sigma^2/2 - (z*pixelspacing(3)).^2/sigma^2/2));
If sigma is a 3D vector, sigma = [sigma_x sigma_y sigma_z],
f(x,y,z) = (2*pi)^(-3/2)/sigma(1)/sigma(2)/sigma(3) * exp(-((x*pixelspacing(1)).^2/sigma(1)^2/2 - (y*pixelspacing(2)).^2/sigma(2)^2/2 - (z*pixelspacing(3)).^2/sigma(3)^2/2));

Remarks
The outputs of gauss3filter(I), gauss3filter(I, 1) and
gauss3filter(I, 1, [1 1 1]) are identical.

To enable GPU computation (Matlab 2012a or later, CUDA 1.3 GPU are required), use
R = gauss3filter(gpuArray(I), sigma, pixelspacing).

The Gaussian kernel in the frequency domain is
exp(-2*pi*pi* (u.^2 *sigma1 + v.^2 *sigma2 + w.^2 * sigma3));

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
Email: max.w.k.law@gmail.com
Page: http://www.cse.ust.hk/~maxlawwk/

Required Products MATLAB
MATLAB release MATLAB 7 (R14)
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12 Mar 2013 UWO