Compressed Sensing System for a Fingerprint Image Recognition Using Sensing Matrix with Chaotic Model-Based Deterministic Row Indexes


 This paper proposes a novel design approach for a secured compressed sensing system for fingerprint imaging and its transmission. In the proposed design, the first stage is acquiring the signal followed by sparsely modeling it using Orthogonal Matching Pursuit (OMP) algorithm. In addition to compressing, to guaranty its security, we multiply the sparse modeled data by a novel deterministic partially orthogonal Discrete Cosine Transform (DCT) sensing matrix. Furthermore, the construction of the sensing matrix uses a modified Multiplicative Linear Congruential Generator (MLCG) to select the row index appropriately from chaotically re-arranged rows of DCT pseudo-randomly. On the other hand, the simultaneous recovering and decryption of the compressed image accomplished with the help of a convex optimization method. The proposed system tested by employing different image and security assessment techniques. The results show that we have archived better Peak Signal to Noise Ratio (PSNR) than the recommended value for wireless transmission using samples below 25%.


Introduction
Potentially dangerous security treat forces various facilities and IoT base communications to use a secured authentication. And sensors array-based fingerprint scanning is one of them. Till now several design strategies includes capacitive [1] [2][3] [4], ultrasonic [5], and optical [6] MEMS-based systems have been employed with a full sampling method which includes read and record all sensors outputs. However, in terms of the optimal design approach, full sampling is not an efficient approach. Designing a large sensor system with a reduced sampling method can leverage those shortcomings of the previous design. The strategies employed in this design are known as compressed sensing which works mainly based on the concept developed by Candes et al. [7] and Donoho [8]. This formulation allows us to take samples from few acquired data rather than the whole elements. To implement compressive sensing, the signal must be a sparse signal originally or in its transformed version. The sparsity of the signal in its transformed versions are two types. One of them is, over some dictionary formed analytically by DCT and wavelet transformation of itself [9][10] [11] and the other is over some learned dictionary designed by prior knowledge of several correlated signals [12][13] [14][15] [16]. Compressed sensing based on the latter design approach is less loss and we found it suitable for fingerprint image sampling and compression. The scope of compressed sensing is not limited by analytical computations. It has also hardware implementation. FPGA (Field Programmable Gate Array) device has been intensively used and shows higher performance in [17] [18] [19] ECG, EEG, and other one-dimensional signal processing [20]. It also includes the transformation of a high dimensional signal to a lower one by means of matrix multiplication in parallel with original measurements from input [21]. Transmission of an image that has a biometric feature should be secured and reliable. Therefore, a cryptosystem is securely stored for reproduction, authentication or, identification. Some of the developed cryptosystems are phase encoding schemes using joint transform [22], exclusive-OR encryption [23], and fractional Fourier transformation methods are to mention a few [24]. Not only that, how to encrypt fingerprint images using orthogonal coding also studied in [25]. Double stage chaotic biometric image-like fingerprint encryption scheme by using two maps named Arnold (permutation) and Henon (substitution) for pixel shuffling have been studied in [26]. In this paper, we are going to introduce our major contribution in a novel design strategy of sensing matrix for optimized and secured sensing and transmission of fingerprint image or signal. This means all sensor data will be acquired but not got directly saved or transmitted. To achieve this objective, we developed and implement an algorithms to construct a novel sensing matrix and test its validity by employing an image assessment technique suitable for fingerprint image detection. The overall implementation was done with the help of MATLAB script which runs on a personal computer. We organized our work into six main sections including the introduction which already discussed before. The following section, section two will discuss the theoretical background of compressed sensing including a general introduction about data compression and recovery. Next to section two, the detailed methodology of designing the whole compressing sensing and its parts like dictionary and sensing matrix with their algorithm employed to build them has been discussed. The analysis and reliability of the proposed system tested using the encryption key method and by analyzing security threats in section four, five and six. Finally, by pointing out our main contribution, we summarized our study in section seven of this paper.

Background Studies
The main idea of compressive sensing is to recover signals from fewer measurements that are less than the Nyquist rate [7] [8]. In addition to the sparse input signal, for the successful design of a compressed sensing system, the system must be stable against spatial transformation caused by the change of orientations of the input during contact between fingertip and sensor array. Hence, we select and incorporated the necessary background theories in this section.

Concept of Compressed Sensing
In a compressed sensing system, sparse signal (column vector), measured signal and a sensing matrix provided that M , related by the following equation, eq.1 (1) For a given integers k and with , the sets of k-sparse, Sk vectors in define as (2) Any system of a linear equation similar to eq.1 is known as an under-completed and hence leads us to find a non-unique solution. To get a unique solution, one has to set a constraint and apply it to the following optimization problem defined by .
(3) (4) To get sparse vector from solving p=0 and p=1 is a good optimization choice. In this project, we have applied the well-known greedy type method known as Orthogonal Matching Pursuit (OMP) [28] which is p=0 based optimization method. However, under some special cases, may not be sparse in its native form. In those cases, we need to find the sparse representation ( in most cases) of x to satisfy the condition for compressed sensing formulation and this makes eq. 1 to take the form of eq.5, Where, is sparse basis or dictionary and sparse representation of x. This is what we call it as the signal model which obtained by using pursuit algorithms.

Sensing Matrix
The sensing matrix that we denote as in this paper is a matrix used to govern the selection of sparsely modeled signals by means of a predetermined order of sampling process. Basically, random or deterministic matric [29] can be used as a sensing matrix to solve p=1 by employing the interior point method. Any hardware available for signal processing can generate this sensing matrix, for example FPGA units which are configured as an LFSR (Linear Feedback Shift Register) as already done in [30] for ECG signal. A random or deterministic matric is one of the matrices that satisfy the Gaussian distribution. The deterministic matrix used in this work generated from the DCT matrix whose indexes are arranged based on a Logistics map-based chaotic model [31]. A chaotic system is a dynamic system that oscillates forever without even repeating itself or shows any tendency towards steady-state value [32].

Sparse Recovery Algorithms
The spares solution of the input data can be obtained using the Orthogonal Matching Pursuit (OMP) algorithm at a low computational cost. OMP algorithm is a greedy type algorithm which helps to solve LP0 optimization problem given by (5) Where , and . Since the problem is LP0 type one, we start from b and then look for a column of A which is most correlated with it which gives the minimum dot product value as a reference for comparison. In accordance with [33], this algorithm is non-invariant under an illconditioned dictionary.

Dictionary Learning
Dictionary is a rectangular matrix used to study and obtained the sparse representation of the input signal before undergoing compression. One of the widely used methods to build it known as the K-SVD. This method solves optimization problem eq.(5) iteratively. Set of signals Z known as training signal with initial dictionary given as D, the coefficients sets X can be obtained by solving the problem with and suitable matching pursuit algorithm given in eq.(4 & 5) [34] (6) Where is of the coefficients and is sparsity level. All of the dictionary atoms are generated by updating SVD (Singular Value Decomposition) methods. The training signals are selected based on their structural similarity index (SSIM) [35] which is the most fundamental image quality assessment method. This would be done by setting the threshold value and rejecting the signal whose structural similarity index value is far below the adjusted threshold.
The structural similarity index (SSIM) of an image with respect to its reference image (a and b) of sizes (N, N) is Where are averages, are variances, covariance of respectively. And the remaining terms are two variables to stabilize the division with leak denominator. In this paper, we select an image patch that comprises all fingerprint features as a reference to select the other training patches.

Prime-Dual Interior-Point Method
Once we find that the signal is sparse, we can randomly sample it and transmit it to the receiver where it can be recovered back to its original form. But, this requires solving the LP-0 type optimization problem which is a nondeterministic polynomial hard or NP-Hard problem. To leverage this hardness, transforming the problem to the LP-1 type is the best option. Transformation of the problem leads eq.5 to take the form of eq. 8.
In addition to the sparsity of x and the sensing matrix must satisfy the Restricted Isometric Property (RIP) with restricted isometric constant which given by eq.11 [36] to get solution for eq.8 which is also valid for (1). (9) According to the prime dual method, the solution can be achieved by narrowing the dual gap between the feasible solution of the prime and dual problem. It starts from an arbitrary chosen initial point and searching direction then it going to find the optimal solution by applying the classical Newton method [37]. Implementation of this approach computationally possible as a free software package available to solve the problem iteratively.

Proposed Method
The proposed method focused on the modeling of the input signal and a novel design approach for a proper deterministic sensing matrix to compress and transmit it.

The Design Work Flow
The whole design procedure has been summarized in the fig.1 which is given bellow

The Proposed Sensing Matrix Design for Compressed System
For the better design of a compressive sensing scheme, it is vital to have an efficient sensing matrix. For sensors array which contains M by N elements, the locations at which measurements have to be taken are stored in the sensing matrix derived from a Deseret Cosine Transform Matrix (DCT) defined by eq.10 by a random selection of rows. This means that the proposed matrix is a submatrix of N by N size DCT matrix (eq.11).
(10) (11) Equation 11 helps us to generate the matrix whose row width is not less than, , by arranging selected rows in accordance with the recommended sequence by our algorithm. Prior to sub-matrix derivation, there is a step that we have incorporated into this novel design approach which is known as masking the DCT by converting to its equivalent form of itself by swapping every row of the matrix in chaotic sequence. The sequence generation is a completely chaotic model based on a Logistics map. Logistics map is mathematically given by eq.12 with the population at the time, , , the ratio of the existing population to the maximum population and growth rate The value of is any value between zero and four. Whereas, the values of and are always between zero and one. As depicted in fig. 4, the logistic map is highly chaotic if the value of is 3.57 and more. The detailed step that we followed to re-arrange the matrix rows sequence summarized in algorithm-I. Algorithm-I Chaotic model-based row index re-arrangement of the DCT Input: chaotic control parameters such as chaotic initial state , growth rate and the by DCT matrix Output: Equivalent form of the DCT matrix or masked DCT matrix.
1) Using the recursive logistic map eq. (12)  Once this matrix obtained, the parameters used in this design, such as, , r, t, and the initial value would be taken as part of the encryption keys for the sensing system. The sensing matrix obeys the RIP, if the smallest set which formed by deterministic row selection from with high probability is equal to . A Multiplicative Linear Congruential Generator (MLCG) [38] output sequence-based row selection method with slight modification can be employed to achieve it. The modified MLCG sequence generator has designed with shifting properties and mathematically expressed as in eq. 16.
(14) , and Where a is the multiplier, b is the increment, t is the number of bits and S is an integer used to extend the row index selection. Hence by appropriately the values of S we can generate several sensing matrices from without repetition to make the proposed system resistive against security attack.

Algorithm-II Deterministic sensing matrix generation
Input: control parameters such as S or the shifting, multiplier constant a, b, numbers of bit t, and equivalent form of the DCT matrix. Output: a partial orthogonal deterministic sensing matrix of size. 1) Using the modified MLCG eq.(14) expression generate a sequence of integers of size .
2) Select rows from whose indices belongs to 3) Define them as or the deterministic sensing matrix (eq. 5). 4) Repeat steps 1 through 3 to generate others sensing matrices by using different parameters for MLCG.
By applying different combinations of S, a, b, and t on the equivalent form of the DCT matrix that we already at hand, we can generate many partially orthogonal sensing matrix row index sets. Now, we can consider S, a, b, and t as an additional encryption key for the proposed system.

Analysis and Result for the Proposed Sensing Matrix
This section presents the comprehensive experimental study of the proposed system for two sensing matrices. The first one is a random matrix and the other is the proposed sensing matrix. The experimental test for the validity of the above mathematical formulation has been done using public data sources from the NIST database (https://www.nist.gov/itl/iad/image-group/nist-specialdatabase-302). Three assessment methods employed to check the effectiveness of the methodology that we have used to get its transform version of the input signals. These are Root Mean Square Error (RMSE), Peak Signal to Noise Ratio (PSNR), and Similarity Index of Image (SSIM). The mathematical expression for SSIM already given by eq.7 and for the PSNR of image with respect to reference image of a similar dimension is given by eq.15 (15) Where the Peake Value of the pixel in the image and is Mean Square Error of image with respect to reference image (16)

Learning to Build the Dictionary Matrix
We first organized a data store to keep a set of images which used to train our dictionary. Then we have divided each image elements to generate 8 by 8, 16 by 16, and 24 by 24 image patches. The degree of similarity of each image patches is measured using their SSIM (Similarity Index) with one common reference patch which contains most of fingerprints image features. Then using the K-SVD algorithm, we trained three different dictionary matrices for the purposes of signal transformation to sparse from its native form. The plot of these matrices is shown in the figure fig. 5.

Data Modeling for Compressed Sensing Input
Using the K-SVD algorithm we construct the dictionary matrix to model the data which is an equivalent representation of the input with another 81,400 and 1089 length vector with a few numbers of non-zero components.
The best method to get the compressible model of this data is by employing the Orthogonal Matching Pursuit algorithm upon the measurement data.
(a) Fig. 6. Image assessment index Vs number of sparse coefficient (a) For random sensing matrix (b) For proposed sensing matrix As depicted in the result, our result would be better, if we restrict the number of the sparse coefficient low. This result indicates that the methodology supports a high compressional ratio like 1:10 which is fairly good in terms of the possibility of recovery as we already proved it in the next section.

Compressing and Encryption
The compression of the sparse data is the easiest step in architecture. It is simply multiplying the sparse representation by sensing matrix. Both encryption and compression have been done simultaneously. As long as the sensing matrix is secreted from third-party users, the compressed data remain safe from being accessed by external agents. Furthermore, it can be constructed from the encryption key that already could be obtained in algorithm-I and II. Therefore, there is no need of sending the whole sensing matrix rather securely transferring the encryption keys is enough.

Decompression and Decryption
This step or simultaneous decompression and decryption is the reverse process of the previous one and takes place at the receiver side. Once the sensing matrix keys are delivered to the receiver, the decompression is possible by solving the p-1 optimization, eq. 7, problem using the already described prim-dual interior-point method followed by sparse recovery.
(a) (b) Fig.7 Image assessment index Vs number samples (a) for random sensing matrix (b) for proposed sensing matrix Unlike the previous section, we fix the numbers of the sparse coefficient constant to study the rate of signal recovery with the number of samples taken from the sparse signal. The result shows that still we have a high recovery rate at a small number of samples. This result is better than the recommended compression rate for wireless transmission which is from 20 to 25dB [39] [40]. Our conclusion from this analysis is, the proposed sensing matrix performance is almost the same as the random matrix which possesses the RIP feature.

Security Threat Model
For the safe flow of data from one state to another in our case from sender to receiver, the communication should be secured by means of encrypting the data. In this section, we will start by identifying the expected potential threat that will impact and expose it for risk based on a literature survey.
The key elements of the system that responsible for the loss of data if they accessed by potential adversaries are the dictionary, the sensing matrix, and the encryption key used to build the sensing matrix. The last element cannot be a problem if they securely transfer to a legitimate user but the first two elements need further analysis because there are several alternative ways for the attacker to construct them. The dictionary matrix can be constructed using the same algorithm used in the proposed design from different training signals or fingerprint data. However, any output from it may not be sparse simultaneously, or sparse solution is always unique for the fixed input signal and coherent dictionary matrix once it is computed [41]. Hence, we will focus only on the sensing matrix. When the attacker sent a randomly chosen plane text or image to the oracle and gets the cipher version of text or image, there will be a probability to gain a piece of knowledge about how the system encrypts its data. This attack in known as Chosen Plain text Attack (CPA). Like other works [42] [43], the proposed encryption in this paper is related but not directly to the sensing matrix. As already pointed out in [44] [45], such kind of compressed system is not secured against CPA. We can get rid-off this vulnerability by effectively extracting different sensing matrices using the modified LCG (eq. 14) from a single DCT matrix designed based on a chaotic model (eq. 11). And the rest will be discussed in the next section.

Discussion on Performance Test and Reliability Study
In this section, we will identify the sensitive part of the system that makes it vulnerable to attack and explain how the system will respond to them by analyzing how the model impacts the reliability of the design approach.

Randomness Test Via Correlation Analysis
Correlation analysis involves the evaluation of several randomly selected pairs of adjacent pixels aligned horizontally, vertically, and diagonally. For a particular figure print image with each pixel coordinates (x.y) and for randomly selected numbers of pairs , the correlation is given by eq. 17. (17) We effectively utilized eq.17 for 3000-pixel pairs of plain and compressed cipher images and plot distribution as shown in fig. 19. A comparison of the figure in both plain and cipher image shows that the proposed sensing matrix possesses the expected randomness.

Histogram Analysis for Encrypted Image
The histogram analysis is helpful to identify which data of the securely compressed image easily visible for the attacker. In the proposed scheme, there are several entries whose values are negative. Take this as one advantage and furthermore the uniform distribution of values as depicted on the histogram comparison figure further indicate the system is still safe from histogram-based attacks.

Key Sensitivity Test
Key sensitivity test used to examine how the proposed sensing matrix responds to a slight change in the magnitude of some of the keys used to construct the proposed sensing matrix.
Keeping other keys that affects others except for the logistic map, we have observed the sensing matrix row sequence generated by the growth rate of the Logistic map changed by 2.6333X10 -6 % with the help of our proposed algorithm-I. The growth rate which is equal to 3.7504 selects the rows of the DCT matrix according to the sequences of 8, 29, 2, 15, . Therefore, the partially orthogonal sensing matrix with those selected rows from DCT obviously produces different cipher data. The same precision has been used in all computations of sequence and initial values. This behavior of the sensing matrix shows that the proposed scheme is a key sensitive scheme.

Key Space
When a brute force attack occurs, the adversary constructs the sensing matrix by combining rows using its own technique with large numbers of trials. According to [46][44] [47], the attacker must attempt a maximum of 2 100 rows combinations of the DCT matrix which supposed to exceed the keyspace of the system. In this regard, keyspace of our proposed system can be computed for row dimension 121 and 25 of them are enough to construct the sensing matrix.
(18) The term in the first bracket is the number of options for an attacker to construct the masked DCT whereas the second matrix is the number of option to construct the sensing matric and their product give large enough value to turn the brute force attack in to infeasible for the proposed system.

Differential Attack
Types of attack that we already studied before requiring analysis which involve the whole signal or image at a time. Differential attack analysis more focused on a single pixel. Therefore, we only take the image block consists the pixel of our interest only because different sensing matrices were used to resist brute force and chosen cipher text treat. Therefore, our analysis will be based on a selected specific block using NPCR (Net Pixel Change Rate) and UACI (Unified Average Changing Intensity) which are given by eq. 19 and 20.
(19) (20) Where and are the height and width of the image block. And is the encrypted image block whereas also the encrypted image of the block with one of the pixels changed. The value of is one if there is pixel difference between and otherwise it is zero. Our results for samples taken from eight by eight block is 100% for NCPR and 0.92% for UACI. This shows all sixty four pixel values were undergo changes that would make the design scheme resistive to any attack.

Entropy Analysis
The entropy of the encrypted output of our compressed system is the measurement of the system's ability to generate random encrypted output. The entropy of the image with different probability of total number is mathematically expressed and given by eq.21.
(21) Since our system designed based on DCT matrix, there are high numbers of entries whose values are negative. Hence to compute the logarithmic content of equation (21), we took the absolute value of the cipher image and we have verified that the system has 7.20 which is 90% of the ideal value.

Conclusion
In this work, we aimed at a secured transmission of fingerprint images using a compressive sensing approach. To establish security, we designed a novel deterministic sensing matrix based on a partial orthogonal matrix derived from a Discrete Cosine Transform (DST) matrix. A modified Multiplicative Linear Congruential Generator (MLCG) based sub-matrix has been employed to construct the sensing matrix in a deterministic manner. The added shifting factor in MLCG expression enables us to use any rows of the DCT matrix to build our sensing matrix. The traditional MLCG has a limitations to choose any of the rows until the maximum row index of the DCT matrix. By effectively applying the shifting factor, we successfully select any rows entry in accordance with the requirement of the sub-matrix to be used as a measurement matrix. The comprehensive simulation result of our proposed compressed sensing system shows that the deterministic sensing matrix has better performance than the non-deterministic or random matrix in terms of memory resource usage and number of sensors that effectively used. And this would open a new way for the quest of an optimized alternative fingerprint scanning technology.