# A New Fingerprinting Technique for Engraved Binary Matrix Authentication

Léo Nicollier, Marc Michel Pic, Enric Meinhardt-Llopis, Gabriele Facciolo

► To cite this version:

Léo Nicollier, Marc Michel Pic, Enric Meinhardt-Llopis, Gabriele Facciolo. A New Fingerprinting Technique for Engraved Binary Matrix Authentication. 2024 IEEE International Conference on Image Processing (ICIP), Oct 2024, Abu Dhabi, United Arab Emirates. pp.3250-3256, 10.1109/ICIP51287.2024.10647997. hal-04769392

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# A NEW FINGERPRINTING TECHNIQUE FOR ENGRAVED BINARY MATRIX AUTHENTICATION

Léo Nicollier †‡*, Marc Michel Pic ‡, Enric Meinhardt-Llopis †, Gabriele Facciolo †

Centre Borelli - ENS Paris-Saclay †, Advanced Track and Trace ‡, EPFL *

## ABSTRACT

This paper introduces a new method for authenticating engravings, crucial to fight counterfeiting and related issues in industries like luxury brands. The approach is based on extracting natural minutiae, inspired by human biometrics, called (n, n)-modules. These modules, representing small sub-matrices of the printed pattern, that vary in appearance due to factors like material characteristics and engraving structure. To create a reliable fingerprint, we proposed a voting system, with each vote being a comparison between two (n, n)-modules. This method leverages of the fact that the defects of an engraving or printing are difficult to reproduce and are observable on the scale of a printed or engraved point. We evaluate two ways of comparing the minutiae: a standard one using Euclidean distance and a deep learning-based one employing a convolutional neural network. The process delivers high accuracy and recall in authenticating engravings and shows robustness under various lighting conditions and levels of image blur.

**Index Terms—** Object instance fingerprinting, binary matrix, laser engraving, printing, CNN, deep learning

## 1. INTRODUCTION

In many sectors, and particularly for luxury brands, it is essential to be able to authenticate products to avoid falsification, misappropriation, undue repairs or refunds. Anti-copy mechanisms, printed or laser-engraved on the products, are a common solution to this problem [1], see for instance Figure 1. Some of these techniques are based on the notion of measurable defects, which are uncontrollable but visible degradations of the engraved object. These defects prevent the perfect retrieval of the printed pattern, which result in further loss of information when a copy is attempted [2]. The authentication verifies that only the prescribed amount of information can be retrieved from the authentic product. This process is used in many luxury goods. However, it has certain limitations. Due to the degradation of the object over time, industrial printers tend to leave too much information on the pattern, which paradoxically makes copying easier.

Alternatively, since these degradations are unique to each individual objects, they can be used to identify it [3]. This

<figure><img src="image_2.png" /><figcaption>**Fig. 1.** Engraved binary matrix for a product authentication mechanism (left). The natural minutiae, i.e. (n, n)-module can be seen through a numerical zoom (right-most image). The zoom factor for this acquisition is $b = 4$.</figcaption></figure>

procedure requires to acquire and store a fingerprint for each new product during the production phase. A drawback is that storing and transmitting large fingerprints associated to these goods can easily become impractical for large production volumes.

In our paper, we propose a method for lightweight fingerprinting of all printed or engraved binary matrices (henceforth, we will simply use engraving to describe both printing and engraving). Our method is based on the analogy with human bio-metrics. It consists in extracting the natural minutiae from high resolution images of the engravings, in which the binary matrix is seen with a b-factor of zoom. In these images, we focus on sub-matrices of the binary matrix of size (n × n), which we call (n, n)-modules (see Figure 1). The images of these (n, n)-modules will present unique defects permitting to differentiate one minutia from another.

In this paper we show that a set of (n, n)-modules can be used to build a fingerprint and that is robust to changes in the acquisition conditions (illumination, mild blur), as well as small damages associated to the life cycle of the product (see Figure 2). In addition the proposed method is lightweight as only the images of the minutiae are stored and transmitted.

## 2. RELATED WORKS

Fingerprint identification, tracing back to Herschel for pension payments [4], evolved from early works of Malpighi [5] and Purkinge [6] on the nature of fingerprints and their robustness over time, until Galton's criminal applications [7]. This concept expanded to other biometrics for humans, animals,
<figure><img src="image_3.png" /><figcaption>**Fig. 2.** The main sources of variation of an image of a given engraving. We can distinguish three types of sources. Variations own to each instance of the engraving due to: material in-homogeneity, accuracy of the laser and its causal dependence on the engraving path. Variations due to the wear of the engraved object along time. And variations due to the acquisition conditions mainly due to: lighting of the object and optical blur of the camera.</figcaption></figure>

and objects. Physical objects' identification, once reliant on measurements like artwork dimensions, advanced to more sophisticated methods like bullet markings to identify guns [8]. The advent of digital technologies led to unique digital representations for scanned objects, addressing challenges like pip-rated content's recompression, inspiring new approaches like theraographic in Data Loss Prevention [9]. However, in this context, the term fingerprinting is often incorrectly used in place of watermarking [10], which is different from the topic of our discussion here.

The extraction of characteristics enabling objects to be uniquely identified digitally only appeared later, particularly around paper and packaging. The first notable use on physical media was by Buchanan et al. [11] in 2005 on the speckle response of a laser beam on paper surfaces. Since then, a wide variety of characterization approaches have been developed, beyond paper and packaging by texture analysis [12], e.g. on diamonds by phosphorescence [13], on the identification of molds used in the mass production of plastic parts and for which analysis is possible on the elements without having access to the molds [14], on banknotes using transparency [15], on the sound produced by the use of some products or on audio tracks of recordings [16], on the operation of active products such as Wifi or Bluetooth network cards, for which response time fingerprinting can be used to replace failing MAC identification [17], and the list of application fields and usage variants is growing steadily, with a formalization under the concept of Physically Unclonable Function (PUF) [18].

However, the major drawback of these approaches is the need to adapt feature extraction and comparison processing to each context. The aim of the approach presented in this paper is to free ourselves from individual algorithms adaptations by using a deep learning mechanism.

### 3. METHOD

In this section we formalize the problem of identifying the engraving of a binary matrix from images of it. The problem deals with the relationship between three types of objects, illustrated in Figure 3. First, we have the binary matrix $X$, which is a binary image with black and white squares, each square of size $b \times b$ pixels. Then, we have the physical object i.e. the engraving $E(X)$ that we obtain by engraving the binary matrix into a product tag. Notice that we can engrave the same binary matrix $X$ several times, to obtain different physical objects $E(X), E'(X), E''(X),...$. Lastly, we have the images $A(E(X))$ obtained by scanning the physical objects $E(X)$. For simplicity of the exposition, we assume that the acquisition operators $A$ register each image to the original grid of the image $X$.

If the engraving and scanning operations were perfect, then the images $X$ and $A(E(X))$ would be identical. In practice, they are very different. Most of the difference comes from the instability and unpredictability of the scanning operation $A$, which is affected by camera and object pose, camera blur, lighting conditions, sensor noise and registration errors. A small part of the difference, however, comes from the fact that each engraving $E(X)$ is a slightly different physical object, due to the irregularities in the base material, random vibrations of the engraving head, etc. Our goal is thus to identify each individual physical object $E(X)$ in a way that is robust to successive scans of it. This is a challenging problem, as we must detect tiny variations within significant changes. Yet, as we will see, these tiny variations can be detected in a statistically significant way.

For that, we define a function $\delta$ that allows to distinguish different physical objects, while being robust to acquisition variations. More formally, let $\mathcal{E}$ be the set of all the engraving operators $E$ and $\mathcal{A}$ be the set of the acquisition operators $A$. Then, the function $\delta$ is such that

$$ \Delta(A(E(X), A'(E'(X))) = \begin{cases} 1 & \text{if } E(X) = E'(X) \\ 0 & \text{otherwise,} \end{cases} \quad (1) $$

where $X \in \{0, 1\}^{N \times N}$ a binary matrix of size $N \times N$, with $E, E' \in \mathcal{E}$, and $A, A' \in \mathcal{A}$.

Since in the following sections we will focus on the images resulting from the acquisition of engraved matrices, to simplify the notation, we will denote (with a slight abuse of notation) $A$ and $A'$ the images corresponding two acquisitions of the same engraved object.

#### 3.1. Preprocessing and $(n, n)$-module extraction

We will construct an approximation of the $\delta$ function based on sets of sub-images containing $(n, n)$-modules extracted from images $A$ and $A'$. We denote these sub-images as $A_p$, where $p$ is the position of the $(n, n)$-module in the binary matrix
<figure><img src="image_4.png" /><figcaption>**Fig. 3.** Formalization of the problem. We start from the binary matrix $X$ (as a binary image). This binary matrix is engraved into a physical object $E(X)$. Then, this object is acquired into a new gray-scale image $A(E(X))$, which we assume to be registered to $X$. Each operation $A$ and $E$ introduces defects. We are interested in identifying the defects introduced by $E$, while being robust to the (much larger) defects introduced by $A$. In the red box, we can see (3, 3)-module extracted from $A(E(X))$ that we named $A_{p_1}, A_{p_2}, A_{p_3}$.</figcaption></figure>

(see Figure 3). As we will see, different captures of an $(n, n)$-module of the same engraving are more likely to yield a similar image, i.e. $A_p$ is similar to $A'_p$. To facilitate their comparison, the $(n, n)$-modules are extracted from the images with a constant size of $(n \times b, n \times b)$. This is achieved by aligning the binary matrix contained in it, using the SIFT algorithm [19]. In addition the intensity of each $(n, n)$-module is normalized to fit the range [0, 1]. We will only focus on $(n, n)$-modules with size $n = 3$ as for $n = 1, 2$ the module does not contain enough information, and larger $n$ are more likely to be corrupted by scratches.

## 3.2. Comparison and aggregation of $(n, n)$-modules

Given $x, y$, two aligned and normalized images of $(n, n)$-modules at a position $p$. We define a comparator function $\delta_p^f(x, y)$, which can be based on a simple $\ell_2$ distance between the images, or be the output of a small neuronal network

$$ \hat{\delta}_p^f(x, y) = \begin{cases} 1 & \text{if } f(x, y) \le t_p \\ 0 & \text{otherwise,} \end{cases} \quad (2) $$

where $t_p$ is a threshold that must be calibrated and which may depend on the characteristic of $x$ and $y$ (i.e. it might depend on the position $p$ or the pattern). We denote $\hat{\delta}_p^f(x, y)$ the comparator obtained when the distance $f$ is learned using a small neuronal network and by $\delta_p(x, y)$ the comparator obtained with $f(x, y) = \|x - y\|_2$ (i.e. the $\ell_2$ distance).

For the case of the small network, $f$ is obtained as the average of $g(x, y)$ and $g(y, x)$, where $g(x, y)$ is implemented using a convolutional neural network (CNN). The function $g$ consists of two convolutional layers with 16 and 32 output channels, followed by ReLU activation and $2 \times 2$ max pooling. Subsequently, four fully connected layers with ReLU activations produce outputs of dimension 32, 16, 8, and 1. We

<figure><img src="image_5.png" /><figcaption>**Fig. 4.** Each $i$-line represents a (3, 3)-module coming from the same localization from the $E\#i$ engraving. Each column represents different illumination angles ranging from -30 degrees to +30 degrees. Note that the appearance of the same (3, 3)-module can change across different engravings, while different images of the same engraving, but changing illumination (the rows), is rather stable.</figcaption></figure>

train $g$ using binary cross-entropy loss and optimize it with the AdamW optimizer.

Multiple comparisons of corresponding $(n, n)$-modules can then be combined by a voting system into a global predictor function approximating $\Delta$ in (1) as

$$ \hat{\Delta}^f(A, A', P) = \begin{cases} 1 & \text{if } D^f(A, A', P) \ge \tau \\ 0 & \text{otherwise,} \end{cases} \quad (3) $$

where

$$ D^f(A, A', P) = \frac{1}{m} \sum_{p \in P} \hat{\delta}_p^f(A_p, A'_p) \quad (4) $$

and $P$ is a list of positions corresponding to the $(n, n)$-modules that will be used in the definition of the fingerprint. We denote by $\hat{\Delta}$ and $D$ the predictor and the score obtained with the comparator $\hat{\delta}_p$ and by $\hat{\Delta}^f$ and $D^f$ the predictor and the score obtained with the comparator $\delta_p^f$. In (3) we average the comparators because we consider that each comparator term has the same decision-making power.

It remains to define the set of positions $P$ to be used in the fingerprint and the thresholds $\{t_p\}_{p \in P}$, and $\tau$. These will be the result of a calibration process.
<figure><img src="image_6.png" /></figure>

<figure><img src="image_7.png" /><figcaption>**Fig. 5.** Box plots of the $\hat{\delta}_p(x, y)$ group by pattern, where x, y are two corresponding $(n, n)$-modules. The horizontal axis represent the pattern index, where larger pattern index corresponds to more ones in the binary pattern. The two plots show the first 64 and the last 64 patterns among all the possible $510 \times 3 \times 3$ patterns. Blue boxes show the distances computed on pairs of $(n, n)$-modules extracted from two acquisition of the same engraving. While, for the orange boxes, the $(n, n)$-modules are extracted from images of different engravings. We note that the blue boxes are systematically below the orange ones, indicating that images of the same engravings are more similar.</figcaption></figure>

### 3.3. Calibrating the comparison operators

To complete the definition of our fingerprint function, we need to calibrate the thresholds {$t_p$}$_{p \in P}$, $\tau$, and to select the best $(n, n)$-modules patterns to be compared. We use the $F_1$ score (the harmonic mean of the precision and recall) as measure of success of a comparator function. The $F_1$ score distinguishes between correctly predicted matches given two images of the same engraved $(n, n)$-module, and incorrect matches corresponding to two different engraved $(n, n)$-modules.

For this calibration, we built a dataset composed of 9 engravings of the same binary matrix, engraved on the same materials, with the same laser, and the same settings. For each engraving 8 pictures are taken with a fix illumination sources (as illustrated in Figure 2), using the same device, and consid-

<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /><figcaption>**Fig. 6.** $F_1$ score of $\hat{\delta}_p$ (above) and $\hat{\delta}_p^f$ (below) for all the different (3, 3)-patterns obtained by maximizing it on a training set and evaluate it on the testing set. The horizontal axis represent the pattern index, where larger pattern index corresponds to more ones in the binary pattern.</figcaption></figure>

ering 8 degrees of rotation ranging from -30 degrees to +30 degrees. The dataset is publicly available.<fnref n="1" /> The goal, is to be able to identify the images corresponding to the same engraving just by comparing the images. Figure 4 illustrates how the appearance of the same $(n, n)$-module pattern can change across different engravings (each row correspond to a different engraving). The columns correspond to different images of the same $(n, n)$-module, where the light has changed position, illustrating that the appearance has little dependence on the illumination.

Calibrating the thresholds {$t_p$}$_{p \in P}$ would in theory require a different threshold for each $(n, n)$-module position. This is impractical as it would require many acquisitions of the same module in order to set a statistically meaningful threshold. Instead, here, we assume that the threshold only depends on the pattern and does not depend on the position of an $(n, n)$-module. Thus, we group the modules by them by pattern to compute a single threshold for each pattern.

To confirm that grouping the threshold by pattern is not dependent on the localization, we performed a Kolmogorov-Smirnov test for all patterns to know if the distribution of

<footnote n="1">¹https://zenodo.org/doi/10.5281/zenodo.10600909</footnote>
$\hat{\delta}_p(x, y)$ with $x, y$ two $(n, n)$-module with the same pattern at a given localization is independent of the localization. For almost all patterns we have a p-value less than 0.05, expect for the patterns [1, 1, 0, 0, 1, 0, 1, 0, 0] and [1, 1, 0, 0, 1, 0, 1, 0, 1], where we have a p-value of respectively 0.214 and 0.119, which is probably due to instability in the statistical estimation.

Figure 5 illustrates for different patterns the values of the function $\hat{\delta}_p(x, y)$, where $x, y$ are two corresponding $(n, n)$-modules. For the blue boxes $x, y$ are from two images of the same engraving, while the orange boxes correspond to images of different engravings. We note that the blue boxes are systematically below the orange ones, however the threshold cannot be the same for all the patterns. So we will calculate the thresholds $\{t_p\}_{p \in P}$ by maximizing the $F_1$ score of $\hat{\delta}_p(x, y)$ for each pattern.

Before calibrating $\tau$, we need to determine the best set of $(n, n)$-modules that will form the fingerprint, that is the set of positions $P$ in (3). We use the $F_1$ score for the different patterns to identify the best ones. In Figure 6 we plot the $F_1$ score as function of the pattern index, obtained for the two distances $\hat{\delta}_p$ and $\hat{\delta}_p^f$. We can see from the plot that some patterns are less effective, so we will first use the patterns with high score. Lastly, to choose the number $m$ of $(n, n)$-modules to be retained in the fingerprint, we select the smallest possible $m$ while maximizing the $F_1$ score of $\hat{\Delta}$ (as well as $\hat{\Delta}^f$). From Figure 7 we see that in both cases the $F_1$ scores saturates at values close to 0.99 for $m$ between 80 and 100.

## 3.4. Algorithm

After calibrating the thresholds $\{t_p\}_{p \in P}$, $\tau$ and the set $P$ of $m$ patterns to be used in the fingerprint, we can keep only select $(n, n)$-modules instead of keeping the whole images of the data matrix. These fingerprint occupy only $m(b^2n^2 + 4)$ bytes of information (amounts to about 14kB) per engraving.

For authenticating an engraved tag, an image of it is acquired and the $m$ $(n, n)$-modules are extracted at the positions $P$. Equation (3) (using $\hat{\Delta}$ or $\hat{\Delta}^f$) is then used to compare the stored fingerprint with the newly acquired image, validating or not the authenticity of the product. In addition score $D$ (resp. the score $D^f$) in (4) can be used a confidence of our prediction.

# 4. EXPERIMENTS

To evaluate our approach, we will build a confusion matrix-type representation of our predictors, where we show the scores $D$ (resp. $D^f$) for all the pairs of acquisitions in our dataset. Figure 8 shows this matrix, where the 9 row/column blocks correspond to engravings and the individual pixels within each block correspond to the 8 illumination captures. For calibrating the predictors only the first 3 engravings are used, the remaining 6 engravings are used for validation. We

<figure><img src="image_10.png" /></figure>

### F₁, Accuracy and Recall of $\hat{\Delta}$

<figure><img src="image_11.png" /><figcaption>**Fig. 7.** $F_1$, Accuracy and Recall of $\hat{\Delta}$ (above) and $\hat{\Delta}^f$ (below) along different number of (3, 3)-modules evaluate on all the data. The standard deviation is calculated by choosing $m$ elements through the best $(n, n)$-modules.</figcaption></figure>

see that the predictor score generalizes well to the test engravings, with a high score along the block-diagonal and low off-diagonal scores.

We can also see that the two methods ($\ell_2$ and learned) make some mistakes in the same places. For instance, the fifth block in the diagonal is made up of two distinct sub-blocks. This indicates that for this engraving the predictors are less robust to large illumination changes. Nevertheless, the learned predictor is more robust than $\ell_2$ to these conditions.

## 4.1. Robustness of the predictor to image blur

Due to the small size of the engraving, acquisitions may be locally blurred. To manage this defect we propose to adapt the thresholds $\{t_p\}_{p \in P}$ for each blur, while $\tau$ remains fixed. To measure the blur of each $(n, n)$-module, we use the blur estimation algorithm of [20]. To avoid comparing blurred $(n, n)$-modules with non-blurred ones, we normalize blur level of the least blurred image to match the other one. For the numerical experiment, we assume that the blur is Gaussian with $\sigma^2$ in $[0, 0.5, \dots, 9.5]$ and images contain uniform noise between -2 and 2.

In Figure 9, we can see that for the standard method the predictor reaches a maximum $F_1$ score of 0.988 then stabilizes around 0.88 with $\sigma^2$ between 2 and 7.5. For the method
<figure><img src="image_12.png" /><figcaption>**Fig. 8.** Confusion matrix-type representation of our predictors, where we show the scores *D* (resp. *D*<sup>*f*</sup>) for all the pairs of acquisitions in our dataset. The 9 row/column blocks correspond to engravings and the individual pixels within each block correspond to the 8 illumination captures. For calibrating the predictors only the first 3 engravings are used (the green square), the remaining 6 engravings are used for validation.</figcaption></figure>

with deep-learning, we can observe that it reaches its highest $F_1$ score at 0.995 when $\sigma^2 = 0$, which is expected as $\tau$ is maximized for this score, but when $\sigma^2 \ge 2$ the $F_1$ score drops. If we are working with fully blurred images, it's better to use the deep learning method, but if we're dealing with high blur instability, it is better to work with the classical algorithm.

## 4.2. Comparison of the learned and $\ell_2$ predictors

From Table 1 we see that the predictor $\hat{\Delta}^f$ has better accuracy, better recall and a better $F_1$ score, which makes it better than the standard Euclidian based one. On the other hand, the predictor $\hat{\Delta}$ with the $\ell_2$ comparator is much more stable when it comes to taking locally blurry photos into account. We can also see in Figure 8 that the prediction of the two algorithms is quite similar in the sense that the different places, where they are wrong or are not sure (i.e. the white areas) are located at the same place.

<figure><img src="image_13.png" /><figcaption>**Fig. 9.** Accuracy and Recall and $F_1$ of $\hat{\Delta}$ (above) and $\hat{\Delta}^f$ (below) as function of a Gaussian blur level $\sigma$ applied to the images $\sigma$ (in the x-axis). The blurred images are also contaminated with additive uniform noise between -2 and 2. We optimized $\{t_p\}_{p \in P}$ for every blur level but $\tau$ is optimized only for $\sigma^2 = 0$.</figcaption></figure>

<table><caption>**Table 1.** Score of the different predictors.</caption><thead><tr><td>Predictor name</td><td>$F_1$</td><td>Accuracy</td><td>Recall</td></tr></thead><tbody><tr><td>$\hat{\Delta}$</td><td>0.975</td><td>0.995</td><td>0.951</td></tr><tr><td>$\hat{\Delta}^f$</td><td>0.988</td><td>1.000</td><td>0.997</td></tr></tbody></table>

# 5. CONCLUSION

In this article, we have addressed the problem of authenticating laser-engraved plaques. The proposed method is based on a simple yet powerful statistical analysis of small minutiae of the engravings. Our experiments show that we can distinguish between each individual engraving of the same pattern, even under different acquisition conditions. We have studied to different ways of comparing the minutiae: a traditional comparison based on the Euclidean distance and another one using a CNN. We have demonstrated that the CNN-based comparator achieves a better $F_1$ score when the images are blurred. Future research will focus validating the methodology on larger datasets and considering more acquisition conditions, including the physical degradation of the engravings.
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