Log Space and Euclidean Space Filtering

Smoothing Diffusion Tensors in Log Euclidean Space versus Euclidean space smoothing. The following images /graphs are some findings which support this idea.

Test Data Used

10x10x10 volume with a Tensor Discontinuity Eigen Values:(3 1 1)and(1 1 3) After applying random gaussian noise (std =0.2)

Anisotropic Filtering in Euclidean and Log Euclidean Space

The images below show that both Euclidean and Symmetric Space Anisotropic filtering preserve the tensor discontinuity and remove noise.
Euclidean Space(500 iterations k=0.1,t=0.625) Log Euclidean Space(500 iterations k=0.1,t=0.625)

Error Images

To be able to visually compare errors,i took the absolute value of the difference in tensor components between the smoothed image and the original image .The following image shows one plane of the volume colormapped by Linear Anisotropy. (blue = low ,orange= medium and red= high).
Euclidean Space Error(after Magnifying) Log Euclidean Space Error(after Magnifying)
Observations:
  1. The left side of the volume clearly shows lesser error for log Space diffusion as compared to the euclidean space. superquadrics).
  2. The quadric shapes are more blobby for the euclidean filtering indicating that the residual noise is isotropic as compared to a more flat /laminar residuals for the logspace filtering.
  3. We can also make out from the color maps that residual noise for logspace filtering is more evenly distributed as compared to logspace where the noise seems to be higher in a preferred direction.

Error As a function of number of Iterations

We used 2 different error measures to quantify performance of the two methods:
  1. Sum Absolute Differences(SAD): Here we take the sum of the absolute differences between all the 6 tensor components of the smoothed and the original image. The graph below plots the fraction of residual error as a function of the number of iterations.
  2. Sum of Absolute Differences of Principle Directions : Here we take the SAD only for the principal directions. The graph below plots the fraction of error residual for this error metric agains the number of iterations.