The cmcR package provides an open-source implementation of the Congruent Matching Cells method for cartridge case identification as proposed by Song (2013) as well as the “High CMC” method proposed by Tong et al. (2015).

## Installation

Install the development version from GitHub with:

# install.packages("devtools")
devtools::install_github("jzemmels/cmcR@dev")

Cartridge case scan data can be accessed at the NIST Ballisitics Toolmark Research Database

## Example

We will illustrate the package’s functionality here. Please refer to the package vignettes available under the “Articles” tab of the package website for more information.

Consider the known match cartridge case pair Fadul 1-1 and Fadul 1-2. The read_x3p function from the x3ptools package can read scans from the NBTRD given the appropriate address. The two scans are read below and visualized using the x3pListPlot function.

# Same source comparison

nbtrd_url <- "https://tsapps.nist.gov/NRBTD/Studies/CartridgeMeasurement/"

x3ptools::x3p_scale_unit(scale_by = 1e6)
x3ptools::x3p_scale_unit(scale_by = 1e6)

type = "faceted")

### Preprocessing

To perform a proper comparison of these two cartridge cases, we need to remove regions that do not come into uniform or consistent contact with the breech face of the firearm. These include the small clusters of pixels in the corners of the two scans from the microscope staging area, and the plateaued region of points around the firing pin impression hole near the center of the scan. A variety of processing procedures are implemented in the cmcR package. Functions of the form preProcess_* perform the preprocessing procedures. See the funtion reference of the cmcR package for more information regarding these procedures. As is commonly done when comparing cartridge cases, we downsample each scan (by a factor of 4, selecting every other row/column) using the sample_x3p function.

x3ptools::sample_x3p() %>%
preProcess_crop(region = "exterior") %>%
preProcess_crop(region = "interior") %>%
preProcess_erode(region = "exterior",morphRadius = 15) %>%
preProcess_erode(region = "interior",morphRadius = 100) %>%
preProcess_removeTrend(statistic = "quantile",
tau = .5,
method = "fn") %>%
preProcess_gaussFilter()

x3ptools::sample_x3p() %>%
preProcess_crop(region = "exterior") %>%
preProcess_crop(region = "interior") %>%
preProcess_erode(region = "exterior",morphRadius = 15) %>%
preProcess_erode(region = "interior",morphRadius = 100) %>%
preProcess_removeTrend(statistic = "quantile",
tau = .5,
method = "fn") %>%
preProcess_gaussFilter()

type = "faceted")

### Cell-based comparison procedure

Functions of the form comparison_* perform the steps of the cell-based comparison procedure. The data generated from the cell-based comparison procedure are kept in a tibble where one row represents a single cell/region pairing.

The comparison_cellDivision function divides a scan up into a grid of cells. The cellIndex column represents the row,col location in the original scan each cell inhabits. Each cell is stored as an .x3p object in the cellHeightValues column. The benefit of using a tibble structure is that processes such as removing rows can be accomplished using simple dplyr commands such as filter.

comparison_cellDivision(numCells = c(8,8))

cellTibble
#> # A tibble: 64 x 2
#>    cellIndex cellHeightValues
#>    <chr>     <named list>
#>  1 1, 1      <x3p>
#>  2 1, 2      <x3p>
#>  3 1, 3      <x3p>
#>  4 1, 4      <x3p>
#>  5 1, 5      <x3p>
#>  6 1, 6      <x3p>
#>  7 1, 7      <x3p>
#>  8 1, 8      <x3p>
#>  9 2, 1      <x3p>
#> 10 2, 2      <x3p>
#> # ... with 54 more rows

The comparison_getTargetRegions function extracts a region from a target scan (in this case Fadul 1-2) to be paired with each cell in the reference scan.

cellTibble <- cellTibble %>%
mutate(regionHeightValues = comparison_getTargetRegions(cellHeightValues = cellHeightValues,

cellTibble
#> # A tibble: 64 x 3
#>    cellIndex cellHeightValues regionHeightValues
#>    <chr>     <named list>     <named list>
#>  1 1, 1      <x3p>            <x3p>
#>  2 1, 2      <x3p>            <x3p>
#>  3 1, 3      <x3p>            <x3p>
#>  4 1, 4      <x3p>            <x3p>
#>  5 1, 5      <x3p>            <x3p>
#>  6 1, 6      <x3p>            <x3p>
#>  7 1, 7      <x3p>            <x3p>
#>  8 1, 8      <x3p>            <x3p>
#>  9 2, 1      <x3p>            <x3p>
#> 10 2, 2      <x3p>            <x3p>
#> # ... with 54 more rows

We want to exclude cells and regions that are mostly missing from the scan. The comparison_calcPropMissing function calculates the proportion of missing values in a surface matrix. The call below excludes rows in which either the cell or region contain more that 85% missing values.

cellTibble <- cellTibble %>%
mutate(cellPropMissing = comparison_calcPropMissing(cellHeightValues),
regionPropMissing = comparison_calcPropMissing(regionHeightValues)) %>%
filter(cellPropMissing <= .85 & regionPropMissing <= .85)

cellTibble %>%
select(cellIndex,cellPropMissing,regionPropMissing)
#> # A tibble: 26 x 3
#>    cellIndex cellPropMissing regionPropMissing
#>    <chr>               <dbl>             <dbl>
#>  1 1, 6                0.786             0.820
#>  2 2, 7                0.533             0.725
#>  3 3, 7                0.831             0.638
#>  4 3, 8                0.536             0.657
#>  5 4, 1                0.308             0.847
#>  6 4, 7                0.840             0.578
#>  7 4, 8                0.268             0.578
#>  8 5, 1                0.237             0.748
#>  9 5, 2                0.814             0.739
#> 10 5, 7                0.564             0.475
#> # ... with 16 more rows

We can standardize the surface matrix height values by centering/scaling by desired functions (e.g., mean and standard deviation). Also, to apply frequency-domain techniques in comparing each cell and region, the missing values in each scan need to be replaced. These operations are performed in the comparison_standardizeHeightValues and comparison_replaceMissingValues functions.

Then, the comparison_fft_ccf function estimates the translations required to align the cell and region using the Cross-Correlation Theorem. The comparison_fft_ccf function returns a data frame of 3 x, y, and fft_ccf values: the $\dpi{110}&space;\bg_white&space;x,y$ estimated translation values at which the CCF$\dpi{110}&space;\bg_white&space;_\max$ value is attained between the cell and region. The tidyr::unnest function can unpack the data frame into 3 separate columns, if desired.

cellTibble <- cellTibble  %>%
mutate(cellHeightValues = comparison_standardizeHeights(cellHeightValues),
regionHeightValues = comparison_standardizeHeights(regionHeightValues)) %>%
mutate(cellHeightValues_replaced = comparison_replaceMissing(cellHeightValues),
regionHeightValues_replaced = comparison_replaceMissing(regionHeightValues)) %>%
mutate(fft_ccf_df = comparison_fft_ccf(cellHeightValues = cellHeightValues_replaced,
regionHeightValues = regionHeightValues_replaced))

cellTibble %>%
tidyr::unnest(cols = fft_ccf_df) %>%
select(cellIndex,fft_ccf,x,y)
#> # A tibble: 26 x 4
#>    cellIndex fft_ccf     x     y
#>    <chr>       <dbl> <dbl> <dbl>
#>  1 1, 6        0.194    -8   -24
#>  2 2, 7        0.210    52    32
#>  3 3, 7        0.119   -29   -64
#>  4 3, 8        0.153     9    14
#>  5 4, 1        0.228     0   -58
#>  6 4, 7        0.120   -61   -39
#>  7 4, 8        0.135     5   -54
#>  8 5, 1        0.230   -14   -54
#>  9 5, 2        0.230    45    99
#> 10 5, 7        0.169    28    -5
#> # ... with 16 more rows

Because so many missing values need to be replaced, the CCF$\dpi{110}&space;\bg_white&space;_{\max}$ value calculated in the fft_ccf column using frequency-domain techniques is not a very good similarity score (doesn’t differentiate matches from non-matches well). However, the x and y estimated translations are good estimates of the “true” translation values needed to align the cell and region. To calculate a more accurate similarity score, we can use the pairwise-complete correlation in which only pairs of non-missing pixels are considered in the correlation calculation. To calculate this the comparison_alignedTargetCell function takes the cell, region, and CCF-based alignment information and returns a matrix of the same dimension as the reference cell representing the sub-matrix of the target region that the cell aligned to. We can then calculate the pairwise-complete correlation as shown below.

cellTibble %>%
dplyr::mutate(alignedTargetCell = comparison_alignedTargetCell(cellHeightValues = .data$cellHeightValues, regionHeightValues = .data$regionHeightValues,
theta = 0,
fft_ccf_df = .data$fft_ccf_df)) %>% dplyr::mutate(pairwiseCompCor = purrr::map2_dbl(.data$cellHeightValues,.dataalignedTargetCell, ~ cor(c(.xsurface.matrix),c(.y$surface.matrix), use = "pairwise.complete.obs"))) %>% tidyr::unnest(.data$fft_ccf_df) %>%
select(cellIndex,x,y,pairwiseCompCor)
#> # A tibble: 26 x 4
#>    cellIndex     x     y pairwiseCompCor
#>    <chr>     <dbl> <dbl>           <dbl>
#>  1 1, 6         -8   -24           0.486
#>  2 2, 7         52    32           0.423
#>  3 3, 7        -29   -64           0.567
#>  4 3, 8          9    14           0.449
#>  5 4, 1          0   -58           0.519
#>  6 4, 7        -61   -39           0.523
#>  7 4, 8          5   -54           0.314
#>  8 5, 1        -14   -54           0.490
#>  9 5, 2         45    99           0.532
#> 10 5, 7         28    -5           0.514
#> # ... with 16 more rows

Finally, this entire comparison procedure is to be repeated over a number of rotations of the target scan. The entire cell-based comparison procedure is wrapped in the comparison_allTogether function. The resulting data frame below contains the features that are used in the decision-rule procedure

kmComparisonFeatures <- purrr::map_dfr(seq(-30,30,by = 3),

theta = .,
returnX3Ps = TRUE))

kmComparisonFeatures %>%
arrange(theta,cellIndex)
#> # A tibble: 524 x 11
#>    cellIndex     x     y fft_ccf pairwiseCompCor theta refMissingCount
#>    <chr>     <dbl> <dbl>   <dbl>           <dbl> <dbl>           <dbl>
#>  1 3, 1        -26    37   0.381           0.713   -30            2829
#>  2 3, 7        -42   -62   0.147           0.549   -30            4428
#>  3 3, 8          0    -9   0.178           0.396   -30            2858
#>  4 4, 1        -12    35   0.295           0.754   -30            1666
#>  5 4, 7          0   110   0.234           0.640   -30            4479
#>  6 4, 8        -10   -26   0.175           0.598   -30            1427
#>  7 5, 1         -5    36   0.286           0.747   -30            1278
#>  8 5, 2         -2    32   0.154           0.746   -30            4338
#>  9 5, 7        -53   -38   0.135           0.440   -30            3004
#> 10 5, 8          0   -22   0.275           0.661   -30            1548
#> # ... with 514 more rows, and 4 more variables: targMissingCount <dbl>,
#> #   jointlyMissing <dbl>, cellHeightValues <named list>,
#> #   alignedTargetCell <named list>

### Decision rule

The decision rules described in Song (2013) and Tong et al. (2015) are implemented via the decision_* functions. We refer to the two decision rules as the original method of Song (2013) and the High CMC method, respectively. Considering the kmComparisonFeatures data frame returned above, we can interpret both of these decision rules as logic that separates “aberrant” from “homogeneous” similarity features. The two decision rules principally differ in how they define an homogeneity.

The original method of Song (2013) considers only the similarity features at which the maximum correlation is attained for each cell across all rotations considered. Since there is ambiguity in exactly how the correlation is computed in the original methods, we will consider the features at which specifically the maximum pairwiseCompCor is attained (instead of using the fft_ccf column).

kmComparisonFeatures %>%
group_by(cellIndex) %>%
top_n(n = 1,wt = pairwiseCompCor)
#> # A tibble: 27 x 11
#> # Groups:   cellIndex [27]
#>    cellIndex     x     y fft_ccf pairwiseCompCor theta refMissingCount
#>    <chr>     <dbl> <dbl>   <dbl>           <dbl> <dbl>           <dbl>
#>  1 6, 6          2     6   0.193           0.688   -30            3958
#>  2 5, 2         -4    20   0.175           0.825   -27            4338
#>  3 7, 5          7     2   0.172           0.722   -27            1688
#>  4 8, 6          6    -2   0.269           0.650   -27            2874
#>  5 4, 1         -6    10   0.353           0.844   -24            1666
#>  6 4, 8         -7     2   0.190           0.661   -24            1427
#>  7 5, 1         -6    11   0.331           0.840   -24            1278
#>  8 5, 8         -6     3   0.248           0.745   -24            1548
#>  9 6, 7         -6     3   0.222           0.696   -24             313
#> 10 6, 8         -5     7   0.257           0.789   -24            2912
#> # ... with 17 more rows, and 4 more variables: targMissingCount <dbl>,
#> #   jointlyMissing <dbl>, cellHeightValues <named list>,
#> #   alignedTargetCell <named list>

The above set of features can be thought of as the x, y, and theta “votes” that each cell most strongly “believes” to be the correct alignment of the entire scan. If a pair is truly matching, we would expect many of these votes to be similar to each other; indicating that there is an approximate consensus of the true alignment of the entire scan (at least, this is the assumption made in Song (2013)). In Song (2013), the consensus is defined to be the median of the x, y, and theta values in this topVotesPerCell data frame. Cells that are deemed “close” to these consensus values and that have a “large” correlation value are declared Congruent Matching Cells (CMCs). Cells with x, y, and theta values that are within user-defined $\dpi{110}&space;\bg_white&space;T_x$, $\dpi{110}&space;\bg_white&space;T_y$, and $\dpi{110}&space;\bg_white&space;T_{\theta}$ thresholds of the consensus x, y, and theta values are considered “close.” If these cells also have a correlation greater than a user-defined $\dpi{110}&space;\bg_white&space;T_{\text{CCF}}$ threshold, then they are considered CMCs. Note that these thresholds are chosen entirely by experimentation in the CMC literature.

kmComparison_originalCMCs <- kmComparisonFeatures %>%
mutate(originalMethodClassif = decision_CMC(cellIndex = cellIndex,
x = x,
y = y,
theta = theta,
corr = pairwiseCompCor,
xThresh = 20,
yThresh = 20,
thetaThresh = 6,
corrThresh = .5))

kmComparison_originalCMCs %>%
filter(originalMethodClassif == "CMC")
#> # A tibble: 19 x 12
#>    cellIndex     x     y fft_ccf pairwiseCompCor theta refMissingCount
#>    <chr>     <dbl> <dbl>   <dbl>           <dbl> <dbl>           <dbl>
#>  1 5, 2         -4    20   0.175           0.825   -27            4338
#>  2 7, 5          7     2   0.172           0.722   -27            1688
#>  3 8, 6          6    -2   0.269           0.650   -27            2874
#>  4 4, 1         -6    10   0.353           0.844   -24            1666
#>  5 4, 8         -7     2   0.190           0.661   -24            1427
#>  6 5, 1         -6    11   0.331           0.840   -24            1278
#>  7 5, 8         -6     3   0.248           0.745   -24            1548
#>  8 6, 7         -6     3   0.222           0.696   -24             313
#>  9 6, 8         -5     7   0.257           0.789   -24            2912
#> 10 7, 3         -3     8   0.217           0.749   -24             412
#> 11 7, 6         -2     1   0.227           0.774   -24              43
#> 12 7, 7         -4     5   0.360           0.839   -24            1103
#> 13 3, 7         -4    17   0.177           0.748   -21            4428
#> 14 6, 1        -11     1   0.317           0.822   -21            2826
#> 15 6, 2         -6    -4   0.344           0.797   -21             852
#> 16 7, 2        -11    -1   0.312           0.738   -21            1048
#> 17 7, 4        -14     4   0.251           0.805   -21            2132
#> 18 8, 3         -9    -4   0.269           0.755   -21            2753
#> 19 3, 1          4    -9   0.514           0.874   -18            2829
#> # ... with 5 more variables: targMissingCount <dbl>, jointlyMissing <dbl>,
#> #   cellHeightValues <named list>, alignedTargetCell <named list>,
#> #   originalMethodClassif <chr>

Many have pointed out that there tends to be many cell/region pairs that exhibit high correlation other than at the “true” rotation (theta) value.
In particular, a cell/region pair may attain a very high correlation at the “true” theta value, yet attain its maximum correlation at a theta value far from the consensus theta value. The original method of Song (2013) is quite sensitive to this behavior. The original method of Song (2013) only considers the “top” vote of each cell/region pairing, so it is not sensitive to how that cell/region pairing behaves across multiple rotations.

Tong et al. (2015) propose a different decision rule procedure that considers the behavior of cell/region pairings across multiple rotations. This method would come to be called the High CMC method. The procedure involves computing a “CMC-theta” distribution where for each value of theta, the x and y values are compared to consensus x and y values (again, the median) and the correlation values to a minimum threshold. A cell is considered a “CMC candidate” (our language, not theirs) at a particular theta value if its x and y values are within $\dpi{110}&space;\bg_white&space;T_x$, $\dpi{110}&space;\bg_white&space;T_y$ thresholds of the consensus x and y values and the correlation is at least as larges as the $\dpi{110}&space;\bg_white&space;T_{\text{CCF}}$ threshold. This is similar to the original method of Song (2013) except that it relaxes the requirement that the top theta value be close to a consensus. Continuing with the voting analogy, think of this as an approval voting system where each cell is allowed to vote for multiple theta values as long as the x and y votes are deemed close to the theta-specific x,y consensuses and the correlation values are sufficiently high.

The CMC-theta distribution consists of the “CMC candidates” at each value of theta. The assumption made in Tong et al. (2015) is that, for a truly matching cartridge case pair, a large number of CMC candidates should be concentrated around true theta alignment value. In their words, the CMC-theta distribution should exhibit a “prominent peak” close to the rotation at which the two cartridge cases actually align. Such a prominent peak should not occur for a non-match cartridge case pair.

The figure below shows an example of a CMC-theta distribution between Fadul 1-1 and Fadul 1-2 constructed using the decision_highCMC_cmcThetaDistrib function. We can clearly see that a mode is attained around -24 degrees.

kmComparisonFeatures %>%
x = x,
y = y,
theta = theta,
corr = pairwiseCompCor,
xThresh = 20,
yThresh = 20,
corrThresh = .5)) %>%
ggplot(aes(x = theta)) +
geom_bar(stat = "count",
alpha = .7) +
theme_bw() +
ylab("CMC Candidate Count") +
xlab(expression(theta))

The next step of the High CMC method is to automatically determine if a mode (i.e., a “prominent peak”) exists in a CMC-theta distribution. If we find a mode, then there is evidence that a “true” rotation exists to align the two cartridge cases implying the cartridge cases must be matches (such is the logic employed in Tong et al. (2015)). To automatically identify a mode, Tong et al. (2015) propose determining the range of theta values with “high” CMC candidate counts (if this range is small, then there is likely a mode). They define a “high” CMC candidate count to be $\dpi{110}&space;\bg_white&space;CMC_{\text{high}} \equiv CMC_{\max} - \tau$ where $\dpi{110}&space;\bg_white&space;CMC_{\max}$ is the maximum value attained in the CMC-theta distribution (17 in the plot shown above) and $\dpi{110}&space;\bg_white&space;\tau$ is a user-defined constant (Tong et al. (2015) use $\dpi{110}&space;\bg_white&space;\tau = 1$). Any theta value with associated an associated CMC candidate count at least as large as $\dpi{110}&space;\bg_white&space;CMC_{\text{high}}$ have a “high” CMC candidate count while any others have a “low” CMC candidate count.

The figure below shows the classification of theta values into “High” and “Low” CMC candidate count groups using the decision_highCMC_identifyHighCMCThetas function. The High CMC count threshold is shown as a dashed line at $\dpi{110}&space;\bg_white&space;21 - 1$ CMCs.

kmComparisonFeatures %>%
x = x,
y = y,
theta = theta,
corr = pairwiseCompCor,
xThresh = 20,
yThresh = 20,
corrThresh = .5)) %>%
decision_highCMC_identifyHighCMCThetas(tau = 1) %>%
ggplot() +
geom_bar(aes(x = theta, fill = thetaCMCIdentif),
stat = "count",
alpha = .7) +
geom_hline(aes(yintercept = max(cmcCandidateCount) - 1),
linetype = "dashed") +
scale_fill_manual(values = c("black","gray50")) +
theme_bw() +
ylab("CMC Candidate Count") +
xlab(expression(theta))

If the range of High CMC count theta values is less than the user-defined $\dpi{110}&space;\bg_white&space;T_{\theta}$ threshold, then Tong et al. (2015) classify all CMC candidates in the identified theta mode as actual CMCs.

The decision_CMC function classifies CMCs based on this High CMC criterion if a value for tau is given. Note that it internally calls the decision_highCMC_cmcThetaDistrib and decision_highCMC_identifyHighCMCThetas functions (although they are exported as diagnostic tools). A cell may be counted as a CMC for multiple theta values. In these cases, we will only consider the alignment values at which the cell attained its maximum CCF and was classified as a CMC. If the cartridge case pair “fails” the High CMC criterion (i.e., the range of High CMC candidate theta values is deemed too large), every cell will be classified as “non-CMC (failed)” under the High CMC method. When it comes to combining the CMCs from two comparison directions (cartridge case A vs. B and B vs. A), we must treat a cell classified as a non-CMC because the High CMC criterion failed differently from a cell classified as a non-CMC for which the High CMC criterion passed.

kmComparison_highCMCs <- kmComparisonFeatures %>%
mutate(highCMCClassif = decision_CMC(cellIndex = cellIndex,
x = x,
y = y,
theta = theta,
corr = pairwiseCompCor,
xThresh = 20,
yThresh = 20,
thetaThresh = 6,
corrThresh = .5,
tau = 1))
#Example of cells classified as CMCs and non-CMCs
kmComparison_highCMCs %>%
slice(21:35)
#> # A tibble: 15 x 12
#>    cellIndex     x     y fft_ccf pairwiseCompCor theta refMissingCount
#>    <chr>     <dbl> <dbl>   <dbl>           <dbl> <dbl>           <dbl>
#>  1 7, 7         12   -10   0.328           0.750   -30            1103
#>  2 8, 3         27    14   0.252           0.694   -30            2753
#>  3 8, 4         37     4   0.251           0.644   -30            1385
#>  4 8, 5         22     1   0.226           0.542   -30            1399
#>  5 8, 6         18    -8   0.257           0.641   -30            2874
#>  6 2, 7        -24   -46   0.224           0.416   -27            2842
#>  7 3, 1        -18    26   0.409           0.733   -27            2829
#>  8 3, 7        -18    -7   0.168           0.657   -27            4428
#>  9 3, 8          5     4   0.177           0.395   -27            2858
#> 10 4, 1         -9    22   0.331           0.815   -27            1666
#> 11 4, 7          3   110   0.179           0.430   -27            4479
#> 12 4, 8         -9   -12   0.192           0.660   -27            1427
#> 13 5, 1         -5    24   0.311           0.809   -27            1278
#> 14 5, 2         -4    20   0.175           0.825   -27            4338
#> 15 5, 7        -58   -25   0.141           0.494   -27            3004
#> # ... with 5 more variables: targMissingCount <dbl>, jointlyMissing <dbl>,
#> #   cellHeightValues <named list>, alignedTargetCell <named list>,
#> #   highCMCClassif <chr>

In summary: the decison_CMC function applies either the decision rules of the original method of Song (2013) or the High CMC method of Tong et al. (2015), depending on whether the user specifies a value for the High CMC threshold tau.

kmComparison_allCMCs <- kmComparisonFeatures %>%
mutate(originalMethodClassif = decision_CMC(cellIndex = cellIndex,
x = x,
y = y,
theta = theta,
corr = pairwiseCompCor,
xThresh = 20,
thetaThresh = 6,
corrThresh = .5),
highCMCClassif = decision_CMC(cellIndex = cellIndex,
x = x,
y = y,
theta = theta,
corr = pairwiseCompCor,
xThresh = 20,
thetaThresh = 6,
corrThresh = .5,
tau = 1))

#Example of cells classified as CMC under 1 decision rule but not the other.
kmComparison_allCMCs %>%
slice(21:35)
#> # A tibble: 15 x 13
#>    cellIndex     x     y fft_ccf pairwiseCompCor theta refMissingCount
#>    <chr>     <dbl> <dbl>   <dbl>           <dbl> <dbl>           <dbl>
#>  1 7, 7         12   -10   0.328           0.750   -30            1103
#>  2 8, 3         27    14   0.252           0.694   -30            2753
#>  3 8, 4         37     4   0.251           0.644   -30            1385
#>  4 8, 5         22     1   0.226           0.542   -30            1399
#>  5 8, 6         18    -8   0.257           0.641   -30            2874
#>  6 2, 7        -24   -46   0.224           0.416   -27            2842
#>  7 3, 1        -18    26   0.409           0.733   -27            2829
#>  8 3, 7        -18    -7   0.168           0.657   -27            4428
#>  9 3, 8          5     4   0.177           0.395   -27            2858
#> 10 4, 1         -9    22   0.331           0.815   -27            1666
#> 11 4, 7          3   110   0.179           0.430   -27            4479
#> 12 4, 8         -9   -12   0.192           0.660   -27            1427
#> 13 5, 1         -5    24   0.311           0.809   -27            1278
#> 14 5, 2         -4    20   0.175           0.825   -27            4338
#> 15 5, 7        -58   -25   0.141           0.494   -27            3004
#> # ... with 6 more variables: targMissingCount <dbl>, jointlyMissing <dbl>,
#> #   cellHeightValues <named list>, alignedTargetCell <named list>,
#> #   originalMethodClassif <chr>, highCMCClassif <chr>

The set of CMCs computed above are based on assuming Fadul 1-1 as the reference scan and Fadul 1-2 as the target scan. Tong et al. (2015) propose performing the cell-based comparison and decision rule procedures with the roles reversed and combining the results. They indicate that if the High CMC method fails to identify a theta mode in the CMC-theta distribution, then the minimum of the two CMC counts computed under the original method of Song (2013) should be used as the CMC count (although they don’t detail how to proceed if a theta mode is identified in only one of the two comparisons).

kmComparisonFeatures_rev <- purrr::map_dfr(seq(-30,30,by = 3),
theta = .))

kmComparison_allCMCs_rev <- kmComparisonFeatures_rev %>%
mutate(originalMethodClassif = decision_CMC(cellIndex = cellIndex,
x = x,
y = y,
theta = theta,
corr = pairwiseCompCor,
xThresh = 20,
thetaThresh = 6,
corrThresh = .5),
highCMCClassif = decision_CMC(cellIndex = cellIndex,
x = x,
y = y,
theta = theta,
corr = pairwiseCompCor,
xThresh = 20,
thetaThresh = 6,
corrThresh = .5,
tau = 1))

The logic required to combine the results in kmComparison_allCMCs and kmComparison_allCMCs_rev can get a little complicated (although entirely doable using dplyr and other tidyverse functions). The user must decide precisely how results from both directions are to be combined. For example, if one direction fails the High CMC criterion yet the other passes, should we treat this as if both directions failed? Will you only count the CMCs in the direction that passed? We have found the best option to be treating a failure in one direction as a failure in both directions – such cartridge case pairs would then be assigned the minimum of the two CMC counts determined under the original method of Song (2013). The decision_combineDirections function implements the logic to combine the two sets of results, assuming the data frame contains columns named originalMethodClassif and highCMCClassif (as defined above).

decision_combineDirections(kmComparison_allCMCs %>%
select(-c(cellHeightValues,alignedTargetCell)),
kmComparison_allCMCs_rev)
#> $originalMethodCMCs #>$originalMethodCMCs[[1]]
#> # A tibble: 19 x 10
#>    cellIndex     x     y fft_ccf pairwiseCompCor theta refMissingCount
#>    <chr>     <dbl> <dbl>   <dbl>           <dbl> <dbl>           <dbl>
#>  1 5, 2         -4    20   0.175           0.825   -27            4338
#>  2 7, 5          7     2   0.172           0.722   -27            1688
#>  3 8, 6          6    -2   0.269           0.650   -27            2874
#>  4 4, 1         -6    10   0.353           0.844   -24            1666
#>  5 4, 8         -7     2   0.190           0.661   -24            1427
#>  6 5, 1         -6    11   0.331           0.840   -24            1278
#>  7 5, 8         -6     3   0.248           0.745   -24            1548
#>  8 6, 7         -6     3   0.222           0.696   -24             313
#>  9 6, 8         -5     7   0.257           0.789   -24            2912
#> 10 7, 3         -3     8   0.217           0.749   -24             412
#> 11 7, 6         -2     1   0.227           0.774   -24              43
#> 12 7, 7         -4     5   0.360           0.839   -24            1103
#> 13 3, 7         -4    17   0.177           0.748   -21            4428
#> 14 6, 1        -11     1   0.317           0.822   -21            2826
#> 15 6, 2         -6    -4   0.344           0.797   -21             852
#> 16 7, 2        -11    -1   0.312           0.738   -21            1048
#> 17 7, 4        -14     4   0.251           0.805   -21            2132
#> 18 8, 3         -9    -4   0.269           0.755   -21            2753
#> 19 3, 1          4    -9   0.514           0.874   -18            2829
#> # ... with 3 more variables: targMissingCount <dbl>, jointlyMissing <dbl>,
#> #   direction <chr>
#>
#> $originalMethodCMCs[[2]] #> # A tibble: 19 x 10 #> cellIndex x y fft_ccf pairwiseCompCor theta refMissingCount #> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl> #> 1 3, 8 -1 -19 0.230 0.652 21 2583 #> 2 4, 1 0 -4 0.550 0.884 21 3009 #> 3 7, 2 8 -3 0.403 0.801 21 945 #> 4 7, 7 10 -13 0.219 0.569 21 979 #> 5 1, 6 10 -8 0.168 0.665 24 3104 #> 6 2, 7 6 -8 0.216 0.669 24 1298 #> 7 4, 8 3 -8 0.286 0.776 24 1206 #> 8 5, 1 1 -14 0.356 0.833 24 1746 #> 9 5, 8 3 -9 0.324 0.822 24 1160 #> 10 6, 2 2 -12 0.293 0.788 24 1802 #> 11 6, 7 0 -2 0.259 0.665 24 0 #> 12 7, 3 1 -12 0.369 0.878 24 655 #> 13 7, 5 0 -9 0.264 0.795 24 1403 #> 14 7, 6 1 -8 0.197 0.634 24 1 #> 15 8, 4 -2 -9 0.282 0.796 24 1256 #> 16 3, 7 12 7 0.118 0.676 27 2359 #> 17 6, 1 -8 -24 0.268 0.789 27 2698 #> 18 6, 6 -9 -3 0.137 0.704 27 2673 #> 19 7, 4 -12 -9 0.145 0.714 27 2427 #> # ... with 3 more variables: targMissingCount <dbl>, jointlyMissing <dbl>, #> # direction <chr> #> #> #>$highCMCs
#> # A tibble: 19 x 10
#>    cellIndex     x     y fft_ccf pairwiseCompCor theta refMissingCount
#>    <chr>     <dbl> <dbl>   <dbl>           <dbl> <dbl>           <dbl>
#>  1 5, 2         -4    20   0.175           0.825   -27            4338
#>  2 7, 5          7     2   0.172           0.722   -27            1688
#>  3 8, 6          6    -2   0.269           0.650   -27            2874
#>  4 4, 1         -6    10   0.353           0.844   -24            1666
#>  5 4, 8         -7     2   0.190           0.661   -24            1427
#>  6 5, 1         -6    11   0.331           0.840   -24            1278
#>  7 5, 8         -6     3   0.248           0.745   -24            1548
#>  8 6, 7         -6     3   0.222           0.696   -24             313
#>  9 6, 8         -5     7   0.257           0.789   -24            2912
#> 10 7, 3         -3     8   0.217           0.749   -24             412
#> 11 7, 6         -2     1   0.227           0.774   -24              43
#> 12 7, 7         -4     5   0.360           0.839   -24            1103
#> 13 3, 7         -4    17   0.177           0.748   -21            4428
#> 14 6, 1        -11     1   0.317           0.822   -21            2826
#> 15 6, 2         -6    -4   0.344           0.797   -21             852
#> 16 7, 2        -11    -1   0.312           0.738   -21            1048
#> 17 7, 4        -14     4   0.251           0.805   -21            2132
#> 18 8, 3         -9    -4   0.269           0.755   -21            2753
#> 19 3, 1          4    -9   0.514           0.874   -18            2829
#> # ... with 3 more variables: targMissingCount <dbl>, jointlyMissing <dbl>,
#> #   direction <chr>

The final step is to decide whether the number of CMCs computed under your preferred method is large enough to declare the cartridge case a match. Song (2013) originally proposed using a CMC count equal to 6 as the decision boundary (i.e., classify “match” if CMC count is greater than or equal to 6). This has been shown to not generalize to other proposed methods and data sets (see, e.g., Chen et al. (2017)). A more principled approach to choosing the CMC count has not yet been described.

Finally, we can visualize the regions of the scan identified as CMCs.

cmcClassifs = kmComparisonFeatures %>%
mutate(originalMethodClassif = decision_CMC(cellIndex = cellIndex,
x = x,
y = y,
theta = theta,
corr = pairwiseCompCor,
xThresh = 20,
thetaThresh = 6,
corrThresh = .5)) %>%
group_by(cellIndex) %>%
filter(pairwiseCompCor == max(pairwiseCompCor)),
cmcCol = "originalMethodClassif")
#> Warning: Raster pixels are placed at uneven vertical intervals and will be
#> shifted. Consider using geom_tile() instead.