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 (1986) 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.

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:

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)
```

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)
#> [1] 0.6860161
### comparing base to salt-and-pepper
catsim(basemat, errmat, weights = 1)
#> [1] 0.105825
### looking at accuracy
mean(basemat == shiftmat)
#> [1] 0.9355372
mean(basemat == errmat)
#> [1] 0.9364463
```

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.

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")
#> [1] 0.7972214
catsim(basemat, shiftmat, weights = rep(.5,2), method="NMI")
#> [1] 0.2701349
```

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)
#> [1] 0.7878624
catsim(basemat, shiftmat, weights = rep(1 / 3, 3), window = 5)
#> [1] 0.7424498
catsim(basemat, errmat, weights = rep(1 / 3, 3), window = 20)
#> [1] 0.6766773
catsim(basemat, errmat, weights = rep(1 / 3, 3), window = 5)
#> [1] 0.441829
```

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.

Besag, Julian. 1986. “On the Statistical Analysis of Dirty
Pictures.” *Journal of the Royal Statistical Society: Series B
(Methodological)* 48 (3): 259–79. https://doi.org/10.1111/j.2517-6161.1986.tb01412.x.