Quantification and evaluation of handwritten characters loops

In ForensicDocument: Quantification and evaluation tools for forensic document examiners

Introduction

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This tutorial presents toos for quantification and evaluation of loops contained in character, following work from @marquis2005 and @bozza2008.

The fourier quantification method and likelihood statistical evaluation are respectively implemented in ExtractFourier and TwoLevelLR methods.

• Loop quantification
  • Parameters
  • Example
• Statistical evaluation
  • Parameters evaluation
  • Examples evaluation
  • Examples $H_d$ true
  • Examples $H_p$ true
• References

suppressPackageStartupMessages(library(ForensicDocument))

Loop quantification

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The loop quantification method, for contained in handwritten characters as described in @marquis2005, is implemented in the ExtractFourier method. In short, binary character images are skeletonized and quantified with a function $R(\theta)$, where $R(\theta)$ is the distance of the character skeleton to its barycenter at the angke $\theta$.

We give a more detailed description of this method:

1. The algoritm used for the skeletonisation process is based on the one proposed by @stentiford1983. It has been modified for the particular case of closed loop by removing the end point condition, this tweek avoids to perform prunning on the resulting skeleton (see @stentiford1983 for further details).
2. Skeletons (or loop) are parametrised by a discrete function $R(\theta)$, representing the length of a line joining a point of the contour to the barycenter. $\theta$ being the angle made by this line with the horizontal axis, with $0 \leq \theta < 2\times\pi$.
3. Function $R(\theta)$ are resampled for n.samp $\theta$ values. That is, for the values $\theta=\frac{2\pi n}{n.samp+1}, \, n=0,\dots,n.samp-1$.
4. Selected Fourier parameters are extranted from the signal $R[\theta]$ (using the fft method from stats).

Parameters

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The parameters for this method are:

• files, a vector of string specifying the images files to analyse. There is only one supported file format : the png file format, using the png package.
• n.samp, a integer specifying the loops' resampling size. The default resampling size is n.samp=128.
• n.fourier, a integer specifying the number of fourier harmonics to extract (n.fourier < n.samp).
• skeletonize, a logical value (TRUE or FALSE) indicating if character images should be skeletonized. Default is TRUE.
• character_pixel, a integer value (0 or 1), indicating indicating which pixel value is from the character, the other being the background.
• output, a string specifying the name of the output file.
  • If no output file is specified (when the argument output = NULL, or when it is not assigned), the method will return a list () of table containing the n.fourier first fourier parameters $a_n$ and $b_n$.
  • If the output file is specified, results will be written in a csv type file with a tabulation delimiter ('\t'), and the method will a NULL value.
• verbose a logical value, indicating if progress is to be printed on the console.

Example

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In this example, we use the binarised handwritten character o (\emph{fig-O.png} file supplied in the package). The character image is not that is not sketonized skeletonize=TRUE.

We use a sampling size of 128 (n.samp=128) with 7 fourier harmonics (n.fourier=6+1) as in @marquis2005. We will get the ouput as a data.fame, therefore ouput=NULL. The verbose option is set to FALSE. In this case, the method ExtractFourier is used as follows:

files = system.file("extdata", "fig-O.png", package = "ForensicDocument")
result = ExtractFourier(files = files, 
                        n.fourier = 7, 
                        n.samp = 128, 
                        verbose = FALSE, 
                        output = NULL,
                        character_pixel = 0)
result

As stated above, the result object is a list of length length(files). In this particular case, there is only one input file, thus result is a list of length 1.

exp = c("typeof(result)", "length(result) == length(files)", "names(result) == files")
for(e in exp) cat(sprintf("%s : %s\n", e, eval(parse(text = e))))

Statistical Evaluation

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In forensic science, the evidence $y$ is usually interpreted through the computation of a likelihood ratio: $$LR = \frac{f(y|H_p)}{f(y|H_d)} $$, Where

• $H_p$: is the prosection hypthosesis;
• $H_d$: is the defense hypothesis.

In the context of handwritten expertise suppose that: (i) an anonymous letter (i.e. the questioned document) is available for comparative analysis, and (ii) written material from a suspect is selected for comparative purposes (i.e. the reference document. For the compuation of the likelihood ratio, we consider the following propositions of interest:

• $H_p$: the author of the reference document is the author of the questioned document;
• $H_d$: the author of the reference document is not the author of the questioned document.

In this tutorial and package, the proposed evaluation two-level likelihood ratio is based on the one developed by @bozza2008. It allows to take into account the within- and between-writer variability. The evidence $y$, namely the fourier parameters is supposed to follow a multivariate normal density with unknown mean vector and covariance matrix: $$ y \sim \mathcal{M}\mu,W) $$, $$ W^{-1} \sim \mathcal{IW}(U,n_w) $$, $$ \mu \sim \mathcal{M}(\theta,B) $$. Where $B$ and $U$ are the within and between writer covariance matrices.

The likelihood-ratio is comupted using the TwoLeveLR method, and the background parameters ($\mu$, $B$ and $U$) can be computed using the TwoLeveLR_Background

Parameters

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The parameters for TwoLevelLR method are:

• data1, as data.frame object, the measurements from the 'reference' material.
• data2, as data.frame object, the measurements from the 'reference' material.
• background, a list containing the background parameters (overall and group means, within- and between group covariances matrice). This list can be ontained using the TwoLevelLR_Background method.
• n.iter, a integer value giving the number of MCMC iterations. Default is 11000.
• n.burnin, a integer value giving the number of burn-in iterations. Default is 1000.
• nw, a integer value giving the degrees of freedom for the inverse Wishart distribution. Considering p variables in the data, nw must be $> 2\times p+4$.

Similarly, parameters for TwoLevelLR_Background are:

• data, a $n \times p$ numeric matrix, with $p \geq q\times 2$. The background data containing $n$ measurements on $p$ variables.
• fac, a factor of length $p$, indicating the 'population' of each measurement. In our case, the writer.

Examples

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In the following examples, we will use the characterO dataset. It contains the extracted Fourier (n.fourier = 4) parameters from 554 handwritten character loops, written by 11 writers. It is a subset of the data collected by @marquis2006. For more information on this dataset, see ?characterO.For other applications of this methodologie, see @marquis2011a and @taroni2012.

data(characterO)

In both examples, number of iterations and burn in iterations for the MCMC chain and set to 110 and 10. The inverse Wishart distribution degree of freedom nw=50, as in @bozza2008.

n.iter = 110
n.burnin = 10
nw = 50

Example 1: $H_p$ true

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We present the case were the questioned and reference documents are written by the same author: writer 1 (writer 1 has a total of r sum(characterO$info$writer == 1) characters).

In this example we use the: for the reference document (data_reference), the parameters extracted from the first 23 characters of writer 1; for the questioned document (data_questioned), the parameters extracted from the last 23 characters of writer 1; * the background parameters (background), are computed using the TwoLevelLR_Background method with the remaining 10 writers.

# reference & questioned
data_reference = subset(characterO$measurements[,-1], 
                        subset = (characterO$info$writer == 1))[1:23,]
data_questioned = subset(characterO$measurements[,-1], 
                         subset = (characterO$info$writer == 1))[-(1:23),]
# background
subset = characterO$info$writer != 1
data_back = subset(characterO$measurements[,-1], 
                   subset = subset)
background = TwoLevelLR_Background(data_back, 
                                   fac = as.factor(characterO$info$writer[subset]))

The method TwoLevelLR is used as follows:

LLR = TwoLevelLR(data1 = data_reference,  
                data2 = data_questioned,
                background = background, 
                n.iter = n.iter, n.burnin = n.burnin,
                nw = nw)
LLR

The result object LLR is the numeric value of the log-likelihood ratio: $LLR = log(f(y|H_p))-log(f(y|H_d))$. Here the $LLR$ is positive (r LLR), suggesting that $H_p$ is true (i.e. the author of the reference document is the author of the questioned document).

Example 2: $H_d$ true

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Here, we present the case were the questioned and reference documents are written by different authors: writer 1 and writer 2:

• for the reference document (data1), the parameters extracted from writer 1 (first 20 characters);
• for the questioned document (data2), the parameters extracted from writer 2 (first 20 characters);
• the background parameters (background), are computed using the TwoLevelLR_Background method with the remaining 9 writers

# reference & questioned
data_reference = subset(characterO$measurements[,-1], subset = characterO$info$writer == 1)[1:20,]
data_questioned = subset(characterO$measurements[,-1], subset = characterO$info$writer == 2)[1:20,]
subset = characterO$info$writer > 2
# background
data_back = subset(characterO$measurements[,-1], subset = subset)
background = TwoLevelLR_Background(data_back, fac = as.factor(characterO$info$writer[subset]))

The method TwoLevelLR is used as follows:

LLR = TwoLevelLR(data1 = data_reference,  
                data2 = data_questioned,
                background = background, 
                n.iter = n.iter, n.burnin = n.burnin,
                nw = nw)
LLR

Here the $LLR$ is negative (r LLR), suggesting that $H_d$ is true (i.e. the author of the reference document is not the author of the questioned document).

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References

References cited in this tutorial

ForensicDocument documentation built on May 2, 2019, 5 p.m.