Question regarding Uniformity

[Nick Gillingham]

Hello,

I was directed to ask my question here by Andrey Federov at PyRadiomics. 

I wanted to know if the First-Order feature "Uniformity" gets greater with more or less heterogeneous samples. PyRadiomics documentation says uniformity increases with heterogeneity, but the word "uniformity" seems to suggest the opposite. 

Andrey says that his Uniformity feature is the same as "3.4.19 Intensity histogram uniformity" in the IBSI. 

Could you help me understand what this feature reflects?

Thanks,

-Nick

[Martin Vallières]

Hi Nick,

Thanks for your question. I will try to answer it the best I can.

Let us assume that we would define "heterogeneity" as the "spatial variation of sub-regions of different sizes and intensities within a region of interest (ROI)". The short answer to your question is that feature 3.4.19 is not a measure of heterogeneity within a given ROI. It could rather be viewed as the "square of the relative proportion of occurrence of high vs low intensities within a ROI, no matter the spatial distribution of the intensities". In fact, the term "uniformity" is indeed misleading. Note that another term commonly used for that feature is "energy".

For the long answer, let's now look in more details at the four attached images. The range of displayed gray scale colorbar is set to 0-200 for all four images. Let us pretend that these are CT images with Hounsfield Units (HU). Images "im1A" and "im1B" have four definite intensities: [0,33,66,100] HU. Images "im2A" and "im2B" have four other definite intensities: [100,133,166,200] HU. Images "im1B" and "im2B" were created by randomly re-distributing the intensities within "im1A" and "im2A", thus creating "more heterogeneous" versions of "im1A" and "im2A", respectively.

To calculate feature 3.4.19, we need to first perform "fixed bin number" (FBN) discretisation as defined in section 2.7 of the IBSI reference manual: https://arxiv.org/abs/1612.07003. We also need to define the number of bins (or grey levels) that we want in the final image to perform the FBN discretisation process. Let's set that number to 4. After performing the discretisation, note that the number of occurrence in each histogram bin is normalized by the total number of voxels in a given ROI, yielding an histogram of probability of occurrence for each histogram bin. We can then calculate feature 3.4.19 according to its formula in the IBSI reference manual, and we end up obtaining a value of 0.25 for all four images. Thus, even if images "im1B" and "im2B" would be defined as "more heterogeneous", they still have the same feature value as "im1A" and "im2A". This is because feature 3.4.19 is not influenced by the spatial distribution of the intensities within the image, but only the relative proportion of occurrence of these intensities to one another. Furthermore, the feature values for "im1A" and "im1B" are still the same even if the range of HU is not the same (similarly to "im2A" and "im2B"), since the FBN discretisation is influenced by the relative range of intensities within a ROI (range of 100 HU for all images), and not by the absolute values of the intensities.

Now, let's say we calculate the "statistical" counterpart of feature 3.4.19, i.e feature 3.3.17. This feature is calculated in the same way, but without performing the prior FBN discretisation.  Also, feature 3.3.17 should be viewed as what it really is: the sum of the square of the intensities. The values we obtain for images "im1A" and "im1B" are identical: 38612500 HU^2. Similarly, the values we obtain for images "im2A" and "im2B" are also identical: 238112500 HU^2. However, this time, the feature values of "im1A" and "im1B" are not the same as the feature values of "im2A" and "im2B".This is because feature 3.3.17 is influenced by the absolute intensity values within a ROI. Even if we could define the relative heterogeneity of "im1A" and "im2A" as being identical (similarly  to "im1B" and "im2B"), feature 3.3.17 cannot describe this phenomenon as it does not take into account the relative range of intensities within a ROI.

I hope this helps.

Best,

Martin

[Joost van Griethuysen]

Hi Nick, Martin,

First off, my apologies, it there was a small error in the description of the formula. If the energy increases, that means that there are gray values in the ROI that occur more often than others, i.e. more Homogeneous. The lowest possible uniformity is when all gray values seen in the ROI occur equally often (for all i, p(i) = p(0)). Uniformity is also lower when there are more gray values (a larger range), this is because the probability of any one gray value is lower (NB. This only occurs when using a fixed bin width, or when comparing features extracted with different settings).

This bug has now been fixed in the pyradiomics master (commit da7321d).

As Martin already nicely explained, Uniformity in PyRadiomics should not be confused with PyRadiomics Energy (IBSI feature 3.3.17), as the second one is the square of the original intensities, not the square of the probabilities of the discretized gray values (i.e. square probability of each bin in the histogram).

Finally, at PyRadiomics we generally use a fixed bin Width, but that is another discussion...

Regards,

Joost