knitr::opts_chunk$set( collapse = TRUE, comment = "#>", tidy = "styler" )

```
library(catsim)
```

The **Cat**egorical **S**tructural Similarity **I**mage **M**etric, or **CatSIM**,
provides a similarity metric for binary or multinary images or maps after the
style of MS-SSIM. That is,
CatSIM compares two different sets of labels of pixels (voxels) in 2-D (or 3-D) images
in an image. It attempts to assess similarity between two images in a way that accounts for the
structural features of the image rather than a per-pixel discrepancy, ideally doing so
similarly to human assessments of similarity. The package implements the methods
described in [@thompson2020catsim].

CatSIM computes a set of similarity metrics on a sliding window in the images, then averages and combines them. In then downsamples the images and repeats the process. Finally, it combines the measures from the different levels together. It can use a variety of different similarity metrics inside of it depending on the specific meaning of the labels in the image. By default, it uses Cohen's kappa inside it as its main comparison between the two images. However, there are several other options available which are more appropriate in some contexts.

Besag [-@besag1986statistical] presents a $100 \times 88$ two-dimensional two-class image with "intentionally awkward" features. Here we blow it up to three times its size and create two distorted versions to demonstrate how the method works.

The first distorted version is shifting the body of the figure over 6 pixels. The second distorted version is adding salt-and-pepper noise so the error rate matches the difference between original and the shifted version.

set.seed(20200308) multfactor <- 3 onesmat <- matrix(1, multfactor, multfactor) exmat <- (besag %% 2) ### converting to binary - it's ones and twos bigmat <- exmat %x% onesmat bigmat <- exmat %x% onesmat shift <- 6 basemat <- bigmat[1:(96 * multfactor), 1:(84 * multfactor)] shiftmat <- bigmat[(1 + shift):(96 * multfactor + shift), 1:(84 * multfactor)] acc <- mean(basemat == shiftmat) errmat <- (matrix( sample(0:1, 96 * 84 * (multfactor^2), replace = TRUE, prob = c(acc, ((1 - acc))) ), (96 * multfactor), (84 * multfactor) ) + basemat) %% 2 par(mfrow = c(1,3), mar = c(0,0,0,0)) image(basemat, xaxt = "n", yaxt = "n"); grid(lwd = 3, nx = 6, col = "black") image(shiftmat, xaxt = "n", yaxt = "n"); grid(lwd = 3, nx = 6, col = "black") image(errmat, xaxt = "n", yaxt = "n"); grid(lwd = 3, nx = 6, col = "black")

Here we have the original, the shifted version, and the salt-and-pepper noise. We argue that though the proportion of pixels that differ from the base image are approximately the same for the two, an image similarity metric should say the shift is better.

catsim(basemat, shiftmat) catsim(basemat, errmat)

As we might desire, the `catsim`

metric judges the shift to be more like the original
than the salt-and-pepper noise.

Because this is a two-class image, you can consider using the Jaccard index as the metric inside it with similar results:

catsim(basemat, shiftmat, method = "Jaccard") catsim(basemat, errmat, method = "Jaccard")

To demonstrate how it works, first we construct a simple multicategory image and blow it up to a larger size. The grid lines are added for reference. This is our "ground truth" image.

multfactor <- 20 onesmat <- matrix(1, multfactor, multfactor) exmat <- matrix(0, 12, 12) exmat[2:10, c(2, 3, 9, 10)] <- 1 exmat[2:10, c(4, 5, 7, 8)] <- 2 exmat[5:7, 2:10] <- 1 exmat[2:10, c(6)] <- 3 par( mar = c(0,0,0,0)) image(exmat[11:1, 1:11], xaxt = "n", yaxt = "n") grid(col = "black", lwd = 3) bigmat <- exmat %x% onesmat

Next, we construct two distorted versions. The first simply shifts the central portion of the image over six pixels. Then, we compute the percentage of pixels that disagree with the baseline image and create a second distorted image by adding salt-and-pepper noise at that rate. Again, grid lines are added only for reference.

set.seed(20200323) shift <- 6 # we are shifting horizontally by six pixels basemat <- bigmat[1:(11 * multfactor), 1:(11 * multfactor)] shiftmat <- bigmat[(1 + shift) : (11 * multfactor + shift), 1: (11 * multfactor)] ### Here we have shifted the matrix slightly par(mfrow = c(1, 2), mar = c(0,0,0,0)) image(shiftmat, xaxt = "n", yaxt = "n") grid(col = "black", lwd = 3) ### computing the error rate acc <- mean(basemat == shiftmat) errmat <- (matrix(sample(0:3, 121 * (multfactor^2), replace = TRUE, prob = c(acc, rep((1 - acc) / 3, 3))), (11 * multfactor), (11 * multfactor)) + basemat) %% 4 ### here we have made an image that matches its accuracy with salt and pepper noise image(errmat, xaxt = "n", yaxt = "n") grid(col = "black", lwd = 3)

Depending on your preferences, you may have reason to prefer one or the other, but, arguably, the shifted version is in some sense better, even though their pixelwise agreement is approximately the same.

If we take only one level of `catsim`

, we get the following results when
comparing to the initial matrix:

library(catsim) ### comparing base to shifted catsim(basemat, shiftmat, weights = 1) ### comparing base to salt-and-pepper catsim(basemat, errmat, weights = 1) ### looking at accuracy mean(basemat == shiftmat) mean(basemat == errmat)

This captures the similarity between the original image and the shifted image, but is too "critical" of the salt-and-pepper error. We can correct this by looking at the image on multiple scales: we downsample by a factor of 2 and then compute the index for that layer, then repeat, and combine the layers at the end into one measure. By default, we use 5 layers of downsampling, but this can be specified based on your application and the size of the image. Downsampling should remove and smooth out a light level of salt-and-pepper error.

With five levels, the difference is smaller, but still prefers the shifted version over the salt and pepper error.

catsim(basemat, shiftmat, weights = rep(.2, 5)) ### comparing base to salt-and-pepper catsim(basemat, errmat, weights = rep(.2, 5))

By default, CatSIM uses Cohen's kappa ($\kappa$)
as the similarity comparison for the images. This measures agreement between the classes with
and adjustment for the probability of chance agreement. However, depending on the application,
a different comparison might make sense. The following options are available (using
the `method`

argument):

- Jaccard index, suitable for binary images where presence of
one class is important (for instance, fMRI activation maps), with
`method="Jaccard"`

. - Dice index,
which is similar to the Jaccard index, with
`method="Dice"`

. - Rand index, a similarity metric for comparing clusterings based
on whether pairs of points are in the same group, with
`method="Rand"`

. - Adjusted Rand index,
a modification of the Rand index which corrects for the probability of chance agreements, with
`method ="AdjRand"`

or`ARI`

. - Accuracy, or pixel-wise agreement, which is simply
`(# pixels agreeing)/(# pixels)`

, with`method="accuracy"`

. - Normalized mutual information,
a information theory similarity metric based on the Kullback-Liebler divergence between the
two distributions, with
`method = "NMI"`

or`MI`

. - Adjusted mutual information, a correction to
the mutual information to account for the probability of chance agreements, with
`method = "AMI"`

.

catsim(basemat, shiftmat, weights = rep(.5,2), method="Rand") catsim(basemat, shiftmat, weights = rep(.5,2), method="NMI")

The window size can also be adjusted. It should be large enough to capture the scale of the features that are relevant in the analysis. In this example, a portion of the image was displaced by 6 pixels. Too large of a window may average out that difference too much while too small may miss the similarity between the images. Here is a comparison between the shifted image and the salt-and-pepper noise at different window sizes.

catsim(basemat, shiftmat, weights = rep(1 / 3, 3), window = 20) catsim(basemat, shiftmat, weights = rep(1 / 3, 3), window = 5) catsim(basemat, errmat, weights = rep(1 / 3, 3), window = 20) catsim(basemat, errmat, weights = rep(1 / 3, 3), window = 5)

The number of levels CatSIM uses can be set by specifying either the `levels`

argument or passing a vector to `weights`

, where the length of the vector
dictates how many levels of downsampling will be performed. By default,
if the number of levels are specified without specifying the weights,
the weights are set to be equal across levels and add up to 1.
Specifying `weights = rep(1/3,3)`

is the same as specifying `levels = 3`

.

The `catsim()`

function works for both 2-D and 3-D images. It is a wrapper
for other functions which give more control over the various options.
On a 2-D image, it is calling `catmssim_2d()`

and on a 3-D image it is
calling either `catmssim_3d_cube()`

or `catmssim_3d_slice()`

, which
treats the image as either an isotropic 3-D image or a stack of 2-D slices.

In images with different properties across dimensions, the sliding window
used to assess the similarity might not need to be square.
This can be specified by passing a vector to the `window`

argument. In the
provided examples, this is not necessary, but some sources of
images may not be isotropic. Alternatively, the image may be
elongated in one direction and narrow in another. This makes
it possible to accommodate these differences.

catsim(basemat, shiftmat, weights = rep(.2, 5), window = c(11, 5))

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