Results & Analysis

Table 1:

The results of the programs on the synthetic test images can be seen in Table 1.

Average percentage difference (absolute value, ignoring erroneous data):

  • no interpolation: 20.851%
  • nearest neighbor: 24.659%
  • trilinear: 29.798%
  • tricubic: 30.576%

From this data we can immediately draw several conclusions. The data highlighted in red is erroneous; the calculated threshold for the impulse image is too low and as a result the entire image is set to the high value of 255. This leads to the hyperinflated volume calculations seen above. Visual inspection of the impulse test image shows that the interpolators are performing as expected.

Furthermore, tricubic interpolation is not a good choice for images containing sharp edges. The test images that did not contain noise thus performed extremely poorly under tricubic interpolation. Because the program matches a polynomial to each point in space we experience ringing effects at the edges of objects as the byte value transitions sharply from 0 to 255 or vice versa. We can see this effect in Figure 8 - note the edge effects near the surface of the cube. Combined with the thin wall possessed by the hollow cube case, the objects connected components are not large enough to escape removal by the morphological filter.

When considering the rest of the test images, there appears to be no clear winner among the different interpolators, though the values for tricubic interpolation are skewed due to its poor performance on non-noisy images for the reasons described above. When considering only the noisy images, the average percentage difference for TC interpolated-images drops to a mere 11.825%. However, there are not enough images in this sample for this number to be statistically relevant.

It is difficult to create relevant test images for this program because of the complexity involved in actual data (CT scans). However, it is obvious from the test data that this algorithm is not appropriate for nodules with large hollows or nodules of small size as the morphological filtering step can eliminate significant features. In an effort to combat this, the kernel size is parametrizable to allow for greater control over the coarseness of the output.

When applied to CT images, however, the tricubic algorithm appears to yield a more accurate volume calculation than its lower-order counterparts. This can be seen in Table 2.

Average percentage difference (absolute value, ignoring erroneous data):

    • no interpolation: 16.570%
    • nearest neighbor: 13.820%
    • trilinear: 11.525%
    • tricubic: 10.974%

Though the use of tricubic interpolation does appear to yield a more accurate volume calculation, the advantage of using this method is so slight that it likely does not merit the tradeoff in time and complexity that comes with higher-order interpolators on a full-size CT image the time to process an image via tricubic interpolation is significantly greater than that for the nearest neighbor or trilinear operations.

On visual inspection, however, it appears that a far better use for the tricubic interpolator is to improve the accuracy of the segmentation itself rather than the volume calculation. Qualitatively, using linear and then cubic interpolators lead to successively smoother and less blocky 3-dimensional segmentations of the nodules. Running a fraction correct voxel or similarity calculation to the professionally marked boundaries might thus reveal significant improvement over lower-order interpolators.

Unfortunately, no quantitative analysis was performed in pursuit of this revelation. Future investigations might look into tricubic interpolators as a way to improve segmentation.

Concluding Thoughts

The experiment demonstrated that tricubic interpolation improves the accuracy of volume calculation for pulmonary nodules. We further suspect that it will similarly improve the accuracy of automatic segmentation for the same nodules as well. In terms of volume calculation, however, the extra running time required to process medical CT images is an important consideration as current trilinear methods provide acceptably close volume estimations as well.

This running time may be improved by fully optimizing the lekien-tcinterp program to make use of reusable coefficients - this is an excellent candidate for future work in this area of study. Furthermore, devising an experiment to quantitatively measure the effect of interpolation will allow us to follow up on the observations made during experimentation. It is likely that these algorithms will be more useful when applied towards accurate segmentation

instead of volume calculation.

Table 2:

We were able to successfully implement three 3-dimensional interpolation methods using four different algorithms. Furthermore we successfully implemented programs to filter and segment lung nodules from a medical CT image. The use of such tools could greatly aid a medical professional who might need higher resolution in a CT scan to more accurately diagnose a patient. It will also aid in correctly determining the size and location of lesions in the lung.