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).
Install the development version from GitHub with:
# install.packages("devtools")
devtools::install_github("jzemmels/cmcR")Cartridge case scan data can be accessed at the NIST Ballisitics Toolmark Research Database
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.
library(cmcR)
library(magrittr)
library(dplyr)
library(ggplot2)
library(x3ptools)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.
fadul1.1_id <- "DownloadMeasurement/2d9cc51f-6f66-40a0-973a-a9292dbee36d"
# Same source comparison
fadul1.2_id <- "DownloadMeasurement/cb296c98-39f5-46eb-abff-320a2f5568e8"
# Code to download breech face impressions:
nbtrd_url <- "https://tsapps.nist.gov/NRBTD/Studies/CartridgeMeasurement/"
fadul1.1_raw <- x3p_read(paste0(nbtrd_url,fadul1.1_id)) %>%
x3ptools::x3p_scale_unit(scale_by = 1e6)
fadul1.2_raw <- x3p_read(paste0(nbtrd_url,fadul1.2_id)) %>%
x3ptools::x3p_scale_unit(scale_by = 1e6)
x3pListPlot(list("Fadul 1-1" = fadul1.1_raw,
"Fadul 1-2" = fadul1.2_raw),
type = "faceted")
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.
fadul1.1_processed <- fadul1.1_raw %>%
preProcess_crop(region = "exterior",offset = -30) %>%
preProcess_crop(region = "interior",offset = 200) %>%
preProcess_removeTrend(statistic = "quantile",
tau = .5,
method = "fn") %>%
preProcess_gaussFilter() %>%
x3ptools::x3p_sample()
fadul1.2_processed <- fadul1.2_raw %>%
preProcess_crop(region = "exterior",offset = -30) %>%
preProcess_crop(region = "interior",offset = 200) %>%
preProcess_removeTrend(statistic = "quantile",
tau = .5,
method = "fn") %>%
preProcess_gaussFilter() %>%
x3ptools::x3p_sample()
x3pListPlot(list("Processed Fadul 1-1" = fadul1.1_processed,
"Processed Fadul1-2" = fadul1.2_processed),
type = "faceted")
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.
cellTibble <- fadul1.1_processed %>%
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 rowsThe 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,
target = fadul1.2_processed))
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 rowsWe 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: 24 x 3
#> cellIndex cellPropMissing regionPropMissing
#> <chr> <dbl> <dbl>
#> 1 1, 6 0.809 0.805
#> 2 2, 7 0.624 0.699
#> 3 2, 8 0.846 0.756
#> 4 3, 8 0.343 0.632
#> 5 4, 8 0.130 0.568
#> 6 5, 1 0.108 0.765
#> 7 5, 7 0.829 0.481
#> 8 5, 8 0.0522 0.508
#> 9 6, 1 0.314 0.689
#> 10 6, 2 0.340 0.602
#> # ... with 14 more rowsWe 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
estimated translation values at which the
CCF
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: 24 x 4
#> cellIndex fft_ccf x y
#> <chr> <dbl> <dbl> <dbl>
#> 1 1, 6 0.242 -6 -24
#> 2 2, 7 0.184 55 32
#> 3 2, 8 0.213 53 54
#> 4 3, 8 0.158 -22 -60
#> 5 4, 8 0.176 -2 13
#> 6 5, 1 0.211 -15 -47
#> 7 5, 7 0.117 21 16
#> 8 5, 8 0.166 8 -48
#> 9 6, 1 0.322 -56 -78
#> 10 6, 2 0.272 8 -46
#> # ... with 14 more rowsBecause so many missing values need to be replaced, the
CCF
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,
target = fadul1.2_processed,
theta = 0,
fft_ccf_df = .data$fft_ccf_df)) %>%
dplyr::mutate(pairwiseCompCor = purrr::map2_dbl(.data$cellHeightValues,.data$alignedTargetCell,
~ cor(c(.x$surface.matrix),c(.y$surface.matrix),
use = "pairwise.complete.obs"))) %>%
tidyr::unnest(.data$fft_ccf_df) %>%
select(cellIndex,x,y,pairwiseCompCor)
#> # A tibble: 24 x 4
#> cellIndex x y pairwiseCompCor
#> <chr> <dbl> <dbl> <dbl>
#> 1 1, 6 -6 -24 0.558
#> 2 2, 7 55 32 0.377
#> 3 2, 8 53 54 0.751
#> 4 3, 8 -22 -60 0.373
#> 5 4, 8 -2 13 0.420
#> 6 5, 1 -15 -47 0.449
#> 7 5, 7 21 16 0.511
#> 8 5, 8 8 -48 0.355
#> 9 6, 1 -56 -78 0.590
#> 10 6, 2 8 -46 0.555
#> # ... with 14 more rowsFinally, 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),
~ comparison_allTogether(reference = fadul1.1_processed,
target = fadul1.2_processed,
theta = .,
returnX3Ps = TRUE))
kmComparisonFeatures %>%
arrange(theta,cellIndex)
#> # A tibble: 506 x 11
#> cellIndex x y fft_ccf pairwiseCompCor theta refMissingCount
#> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 2, 7 -28 -54 0.203 0.399 -30 2970
#> 2 3, 1 -22 38 0.414 0.778 -30 2301
#> 3 3, 8 -17 -21 0.255 0.612 -30 1584
#> 4 4, 1 -11 35 0.287 0.756 -30 1467
#> 5 4, 8 -8 -25 0.230 0.656 -30 610
#> 6 5, 1 -1 35 0.300 0.677 -30 516
#> 7 5, 7 2 19 0.113 0.479 -30 3946
#> 8 5, 8 1 -22 0.227 0.599 -30 245
#> 9 6, 1 7 34 0.433 0.793 -30 1475
#> 10 6, 2 10 28 0.282 0.801 -30 1594
#> # ... with 496 more rows, and 4 more variables: targMissingCount <dbl>,
#> # jointlyMissing <dbl>, cellHeightValues <named list>,
#> # alignedTargetCell <named list>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: 26 x 11
#> # Groups: cellIndex [26]
#> cellIndex x y fft_ccf pairwiseCompCor theta refMissingCount
#> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 7, 6 21 -1 0.147 0.471 -30 341
#> 2 4, 8 -7 -12 0.246 0.702 -27 610
#> 3 7, 4 7 10 0.215 0.721 -27 1908
#> 4 8, 5 10 6 0.247 0.603 -27 244
#> 5 2, 7 -14 -33 0.228 0.432 -24 2970
#> 6 3, 8 -7 5 0.277 0.699 -24 1584
#> 7 4, 1 -4 11 0.375 0.850 -24 1467
#> 8 5, 1 -4 11 0.363 0.852 -24 516
#> 9 6, 1 -3 11 0.437 0.832 -24 1475
#> 10 6, 2 -1 10 0.302 0.849 -24 1594
#> # ... with 16 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
,
,
and
thresholds of the consensus
x, y, and theta values are considered
“close.” If these cells also have a correlation greater than a
user-defined
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: 18 x 12
#> cellIndex x y fft_ccf pairwiseCompCor theta refMissingCount
#> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 4, 8 -7 -12 0.246 0.702 -27 610
#> 2 7, 4 7 10 0.215 0.721 -27 1908
#> 3 8, 5 10 6 0.247 0.603 -27 244
#> 4 3, 8 -7 5 0.277 0.699 -24 1584
#> 5 4, 1 -4 11 0.375 0.850 -24 1467
#> 6 5, 1 -4 11 0.363 0.852 -24 516
#> 7 6, 1 -3 11 0.437 0.832 -24 1475
#> 8 6, 2 -1 10 0.302 0.849 -24 1594
#> 9 6, 8 -4 7 0.324 0.787 -24 1475
#> 10 7, 1 -4 8 0.172 0.771 -24 4026
#> 11 7, 2 -2 10 0.253 0.768 -24 305
#> 12 7, 3 -1 7 0.199 0.736 -24 572
#> 13 7, 7 -3 3 0.366 0.829 -24 331
#> 14 8, 3 -1 12 0.279 0.747 -24 1447
#> 15 8, 4 -2 8 0.247 0.738 -24 235
#> 16 5, 8 -6 14 0.254 0.717 -21 245
#> 17 8, 6 -13 13 0.283 0.705 -21 1475
#> 18 3, 1 5 -9 0.516 0.855 -18 2301
#> # ... 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
,
thresholds of the consensus
x and y values and the correlation is at
least as larges as the
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 %>%
mutate(cmcThetaDistribClassif = decision_highCMC_cmcThetaDistrib(cellIndex = cellIndex,
x = x,
y = y,
theta = theta,
corr = pairwiseCompCor,
xThresh = 20,
yThresh = 20,
corrThresh = .5)) %>%
filter(cmcThetaDistribClassif == "CMC Candidate") %>%
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
where
is the maximum value attained in the CMC-
theta distribution (17 in the
plot shown above) and
is a user-defined constant (Tong et
al. (2015)
use
).
Any
theta value with associated an associated CMC candidate count at
least as large as
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
CMCs.
kmComparisonFeatures %>%
mutate(cmcThetaDistribClassif = decision_highCMC_cmcThetaDistrib(cellIndex = cellIndex,
x = x,
y = y,
theta = theta,
corr = pairwiseCompCor,
xThresh = 20,
yThresh = 20,
corrThresh = .5)) %>%
decision_highCMC_identifyHighCMCThetas(tau = 1) %>%
filter(cmcThetaDistribClassif == "CMC Candidate") %>%
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
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 8, 3 21 24 0.295 0.719 -30 1447
#> 2 8, 4 69 20 0.228 0.481 -30 235
#> 3 8, 5 22 4 0.231 0.556 -30 244
#> 4 8, 6 23 2 0.204 0.565 -30 1475
#> 5 2, 7 -21 -43 0.218 0.421 -27 2970
#> 6 2, 8 3 -56 0.189 0.502 -27 3969
#> 7 3, 1 -14 27 0.443 0.808 -27 2301
#> 8 3, 8 -12 -8 0.269 0.664 -27 1584
#> 9 4, 1 -7 23 0.335 0.816 -27 1467
#> 10 4, 8 -7 -12 0.246 0.702 -27 610
#> 11 5, 1 -3 23 0.342 0.799 -27 516
#> 12 5, 7 53 -62 0.117 0.449 -27 3946
#> 13 5, 8 -2 -10 0.250 0.669 -27 245
#> 14 6, 1 2 22 0.437 0.814 -27 1475
#> 15 6, 2 5 19 0.294 0.827 -27 1594
#> # ... 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 8, 3 21 24 0.295 0.719 -30 1447
#> 2 8, 4 69 20 0.228 0.481 -30 235
#> 3 8, 5 22 4 0.231 0.556 -30 244
#> 4 8, 6 23 2 0.204 0.565 -30 1475
#> 5 2, 7 -21 -43 0.218 0.421 -27 2970
#> 6 2, 8 3 -56 0.189 0.502 -27 3969
#> 7 3, 1 -14 27 0.443 0.808 -27 2301
#> 8 3, 8 -12 -8 0.269 0.664 -27 1584
#> 9 4, 1 -7 23 0.335 0.816 -27 1467
#> 10 4, 8 -7 -12 0.246 0.702 -27 610
#> 11 5, 1 -3 23 0.342 0.799 -27 516
#> 12 5, 7 53 -62 0.117 0.449 -27 3946
#> 13 5, 8 -2 -10 0.250 0.669 -27 245
#> 14 6, 1 2 22 0.437 0.814 -27 1475
#> 15 6, 2 5 19 0.294 0.827 -27 1594
#> # ... 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).
#Compare using Fadul 1-2 as reference and Fadul 1-1 as target
kmComparisonFeatures_rev <- purrr::map_dfr(seq(-30,30,by = 3),
~ comparison_allTogether(reference = fadul1.2_processed,
target = fadul1.1_processed,
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: 18 x 10
#> cellIndex x y fft_ccf pairwiseCompCor theta refMissingCount
#> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 4, 8 -7 -12 0.246 0.702 -27 610
#> 2 7, 4 7 10 0.215 0.721 -27 1908
#> 3 8, 5 10 6 0.247 0.603 -27 244
#> 4 3, 8 -7 5 0.277 0.699 -24 1584
#> 5 4, 1 -4 11 0.375 0.850 -24 1467
#> 6 5, 1 -4 11 0.363 0.852 -24 516
#> 7 6, 1 -3 11 0.437 0.832 -24 1475
#> 8 6, 2 -1 10 0.302 0.849 -24 1594
#> 9 6, 8 -4 7 0.324 0.787 -24 1475
#> 10 7, 1 -4 8 0.172 0.771 -24 4026
#> 11 7, 2 -2 10 0.253 0.768 -24 305
#> 12 7, 3 -1 7 0.199 0.736 -24 572
#> 13 7, 7 -3 3 0.366 0.829 -24 331
#> 14 8, 3 -1 12 0.279 0.747 -24 1447
#> 15 8, 4 -2 8 0.247 0.738 -24 235
#> 16 5, 8 -6 14 0.254 0.717 -21 245
#> 17 8, 6 -13 13 0.283 0.705 -21 1475
#> 18 3, 1 5 -9 0.516 0.855 -18 2301
#> # ... with 3 more variables: targMissingCount <dbl>, jointlyMissing <dbl>,
#> # direction <chr>
#>
#> $originalMethodCMCs[[2]]
#> # A tibble: 17 x 10
#> cellIndex x y fft_ccf pairwiseCompCor theta refMissingCount
#> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 4, 1 -3 11 0.529 0.863 18 2850
#> 2 2, 7 -5 -16 0.262 0.637 21 1034
#> 3 6, 2 6 -2 0.217 0.770 21 2858
#> 4 6, 6 1 -14 0.269 0.796 21 3273
#> 5 7, 2 8 -3 0.463 0.827 21 306
#> 6 8, 5 9 -2 0.348 0.652 21 244
#> 7 3, 8 3 -5 0.294 0.677 24 1447
#> 8 4, 8 2 -6 0.369 0.783 24 235
#> 9 5, 1 -1 -13 0.361 0.837 24 1664
#> 10 5, 8 1 -8 0.387 0.811 24 244
#> 11 6, 1 -2 -11 0.348 0.831 24 1482
#> 12 6, 8 1 -3 0.489 0.885 24 1475
#> 13 7, 3 -1 -10 0.359 0.885 24 993
#> 14 7, 7 -3 -7 0.238 0.720 24 331
#> 15 8, 4 -4 -7 0.270 0.761 24 235
#> 16 7, 5 -11 -6 0.285 0.729 27 1213
#> 17 7, 6 -9 0 0.220 0.682 27 17
#> # ... with 3 more variables: targMissingCount <dbl>, jointlyMissing <dbl>,
#> # direction <chr>
#>
#>
#> $highCMCs
#> # A tibble: 25 x 10
#> cellIndex x y fft_ccf pairwiseCompCor theta refMissingCount
#> <chr> <dbl> <dbl> <dbl> <dbl> <dbl> <dbl>
#> 1 7, 4 7 10 0.215 0.721 -27 1908
#> 2 3, 8 -7 5 0.277 0.699 -24 1584
#> 3 4, 1 -4 11 0.375 0.850 -24 1467
#> 4 5, 1 -4 11 0.363 0.852 -24 516
#> 5 6, 1 -3 11 0.437 0.832 -24 1475
#> 6 6, 2 -1 10 0.302 0.849 -24 1594
#> 7 6, 7 -4 4 0.167 0.592 -24 990
#> 8 7, 1 -4 8 0.172 0.771 -24 4026
#> 9 7, 7 -3 3 0.366 0.829 -24 331
#> 10 8, 3 -1 12 0.279 0.747 -24 1447
#> # ... with 15 more rows, and 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.
cmcPlot(reference = fadul1.1_processed,
target = fadul1.2_processed,
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")