1 Introduction

In today’s scenario digital image processing is identified as an emerging technology finding its application in many fields. This technology is used in many areas like engineering, computer science, and many other fields. One of the most interesting areas where image processing is used is forensic science. In forensic science and archaeology, the most alluring method in image processing uses facial reconstruction method. The carver who is facial anatomy expert performs facial reconstruction. For detection of gender, age, ancestry of gopher, forensic anthropologists along with the carver analyze skeleton features. For positive recognition of individuals, and to divulge anatomical features including affirmation of wounds like broken nose, teeth and facial asymmetry, the carver uses either the 2D or 3D reconstruction techniques [1, 12, 47, 48].

If the carver employs a 3D technique for recognition, the tissue is placed on the skull at predefined points. This ends in a good reconstruction as the clay when placed seems to be at the maximum possible nearest location to the gopher increasing the probability of gopher identification. Conventional methods of depth measurements based on gender, age and ethnicity were used to determine the points where the markers are placed [19,20,21, 46]. Fake eyes also play a part in efficient reconstruction. To determine position of eyes, length/width of nose and mouth, various iteratively measurement has been taken into consideration. When the tissue markers are affixed on the skull, the sculptor starts to place clay on the skull. This initiate sculpting and finally a face are formed [22, 23]. When the elementary or primary shape is constructed, the sculptor can work on the skull, the aim being making the skull look alike a gopher. All these steps are facilitated by means of the information provided by the forensic anthropologist, including the lifestyle of gopher and the geographical location where the gopher lived. In addition to this, for better identification of the gopher, the sculptors may add clay or wig representing hair [13, 14, 16, 17].

The properties are not limited to the above-mentioned ones but can be anything such as articles of clothing, glasses or any other aspects that could end up in a good identification. The basics of the 2D reconstruction techniques are similar to 3D techniques involving the placement of tissue markers on the skull at pre-defined depths and particular places by means of dome measurements for finding age, sex and ancestry [15, 18, 24, 25]. As mentioned above, once the skull is in the Frankfort horizontal position, an 1X1 image of the skull is photographed from both the profile view and frontal view.

The artist follows the boundaries or to be specific skull contours and by referring the tissue markers sketches the skull. By means of 3-D reconstruction techniques size and position of mouse, nose and eyes can be found. The type of hair and style is found by estimating sex and ancestry, data given by the forensic anthropologist, from scene evidences or from other methods. Another type of reconstruction technique involves reconstruction of a human face from a decaying body. Here the artist makes use of their knowledge regarding the position of the soft tissue of the skin on the skull and the process of body decomposition. The goal is to obtain an exact imagery of the gopher during the lifetime. The 2D techniques are advantageous over 3D techniques in terms of time even though they both end up in the same results. The rest of the paper is arranged as follows, in Section 2, the literature review is analyzed and the motivation of the research. The proposed Methodology with mathematical formulations is explained in Section 3, Results and Discussion are given in Section 4 and the conclusion and Future work section is presented in Section 5.

2 Literature review

A 3D face can be reconstructed using the anatomical condition of human head. Deep Neural Networks can be used for detecting such kind of models. In this project, a novel method for recovery of face recognition is put forward by the researcher using deep neural networks [26, 27]. Deep learning is a branch of machining learning. It helps in learning hierarchical representations of the given data. Multiple nonlinear processing layers are a major part of it and by stacking them pattern classification is enabled. The major architecture of a DNN involves an input layer followed by a huge number of hidden layers and an output layer. The network is initialized at the beginning by means of unsupervised training. Consequently, they are tuned in a supervised manner. Figure 1 shows the difference between deep learning and regular machine learning [28,29,30, 32].

Fig. 1
figure 1

Comparison between Deep Neural Network and Regular Methods

Full size image

Amongst regular machining learning methods and deep learning methods, deep learning methods focus on learning different useful feature representations, the raw material being the available input data. This is done by means of capturing significant statistical irregularities from the input data. Next the representation features can be framed for regression, classification and other information retrieval problems [2, 31, 33,34,35]. Some of the major advantages of deep learning include human effort in feature learning and independence from prior knowledge.

In the starting module, initially the frontal human-face image has been studied. This is done to extract significant features for forming a corpus termed as human-face model. In the successive phase, a face image under test with every possible orientation’s have been captured resulting in the application of high-end computing approach of advanced computing for the successful recognition of the subject’s face. In the current research work based on the created data sets, a proper matching-classification- decision process has been performed [10, 36,37,38]. Keeping this objective, in the present work, a methodology based on deep neural network has been proposed for the reconstruction of destructed human face image for forensic application [39, 41].

Jangir et al., [11], Choubey et al., [3, 9] have used similar data science and machine learning algorithms for the identifications and predictions of medical diabetes.

The idea conceived through the review of many published articles, text and references like comparative analysis of classification methods [7, 32], performance evaluation of classification methods [8, 33], rule-based diagnosis system [6, 31] and classification techniques and diagnosis for diabetes [4, 5, 34, 35] are found to be of great help in accomplishment of the present work.

3 Proposed methodology with mathematical formulations

Generally, two methods namely simple structure analysis and template matching approach are used to compute the estimated outline of the wavelet in noisy environment, and its occurrence time. Figure 2 depicts the architecture of the proposed methodology which is shown below.

Fig. 2
figure 2

Architecture of the proposed methodology

Full size image

In order to compute the wavelet coefficients, present in the noisy images, consider a group of wavelet, Wi(t) belonging to the range i = 0, 1, …. , N − 1, for an entire structural characteristic possibilities. By considering additive noise, the corrupted image can be expressed as [40, 42, 43],

$$ P\left(m,n\right)=q\left(m,n\right)+\mathrm{G}r\left(m,n\right) $$
(1)

In which, q(m, n) represents the original image without noise, r(m, n) illustrates the noise present in the image and G is known as signal-to-noise ratio. Windowing of the image is expressed as represented in Eq. (2) when G is set to 1.

$$ {P}_w\left(m,n\right)={q}_w\left(m,n\right)+{r}_w\left(m,n\right) $$
(2)

Applying the Fourier transform, we get

$$ {P}_w\left({e}^{j\omega 1},{e}^{j\omega 2}\right)={Q}_w\left({e}^{j\omega 1},{e}^{j\omega 2}\right)+{R}_w\left({e}^{j\omega 1},{e}^{j\omega 2}\right)\kern0.5em $$
(3)

Where, \( \left\{\begin{array}{c}{P}_w\left({e}^{j\omega 1},{e}^{j\omega 2}\right)\kern1.5em Fouriertransformofwindowednoisyimages\\ {}{Q}_w\left({e}^{j\omega 1},{e}^{j\omega 2}\right)\kern1em Fouriertransformoforiginalimage\\ {}{R}_w\left({e}^{j\omega 1},{e}^{j\omega 2}\right)\kern0.75em Fouriertransformofnoisyimages\end{array}\right. \)

ɸ(t) signifies the fundamental mother wavelet and is illustrated as

$$ \varphi (t)=e\hat{\mkern6mu} \left(\left(j2\pi ft-t\hat{\mkern6mu} 2/(2)\right)\right. $$
(4)

Applying Continuous Wavelet Transform CWT (a, τ), the above expression is modified as,

$$ CWT\left(a,\tau \right)=\Big(1/\sqrt{a\Big)}\int p(t)\phi \left\{\left(t-\tau \right)/a\right\} $$
(5)

Discretization of CWT is performed by applying discrete parameter wavelet transform (DPWT) and the resultant discretized output is presented in Eq. (6).

$$ \kern12em DPWT\left(m,n\right)={2}^{-m/2}{\sum}_k\sum x\left(k,l\right)\left({2}^{-m}k-n\right) $$
(6)

In which

$$ \left\{\begin{array}{c}m, nintegers\\ {}{a}_0,{\uptau}_0\kern0.15em samplingintervalforand\uptau \\ {}x\left(k,l\right)\kern0.15em enhancedimages\end{array}\right. $$

By substituting a0 = 2 and τ0 = 1, the wavelet coefficient can be calculated for Eq. (6).

Sampling of the enhanced images is done at periodic time interval T to obtain sampled image sequences {q(mT, nT)}, with size X x Y, belonging to ranges m = 0,1, …, X-1 and n = 0,1,…,Y-1 significantly. By applying discrete Fourier transform (DFT) [44, 45], the equation is dictated as below

$$ \kern5.5em I\left(u,v\right)={\sum}_{x=0}^{x-1}{\sum}_{y=0}^{y-1}i\left(x,y\right)\exp \left(-j2\pi \left( um/x+ vn/N\right)\right. $$
(7)

where u = 0,1,…,X-1 and v = 0,1, ……..,Y-1.

The magnitude, phase angle and power spectrum are evaluated by time-domain to frequency-domain transformation. If R(u, v)and A(u, v) signifies the real and imaginary part of Q(u, v)its magnitude spectrum can be expressed as

$$ \left|I\left(u,v\right)\right|={\left[{R}^2\left(u,v\right)+{A}^2\left(u,v\right)\right]}^{1/2} $$
(8)

And the phase angle can be defined as

$$ \phi \left(u,v\right)={\tan}^{-1}\left[\frac{A\left(u,v\right)}{R\left(u,v\right)}\right] $$
(9)

The power spectrum with the inclusion of magnitude and phase angle is depicted as

$$ \kern14em P\left(u,v\right)={\left|I\left(u,v\right)\right|}^2={R}^2\left(u,v\right)+{A}^2\left(u,v\right) $$
(10)

The dynamic range seems to be very high as the power spectrum is expressed as the squared value of the magnitude. Hence, logarithmic transformation is applied for normalization purpose and the normalized equation can be defined as

$$ {\left|I\left(u,v\right)\right|}_{normalize}=\log \left(1+\left|I\left(u,v\right)\right|\right. $$
(11)

The expectation value, variance and its auto covariance of the enhanced images are represented as in Eqs. (12), (13) and (14) respectively.

$$ \kern0.5em E\left[I\left(u,v\right)\right]=\frac{1}{XY}{\sum}_{u=0}^{X-1}{\sum}_{v=0}^{Y-1}I\left(u,v\right) $$
(12)
$$ Var\left[I\left(u,v\right)\right]=E\left\{{\left|I\left(u,v\right)-{I}^{\prime}\left(u,v\right)\right|}^2\right\} $$
(13)
$$ \kern0.5em {C}_{xx}\left(u,v\right)=E\left\{\left[I\left(u,v\right)-{I}^{\prime}\left(u,v\right)\right]\left[I\left(u,v\right)-{I}^{\prime}\left(u,v\right)\right]\right\} $$
(14)

The power spectral density can be calculated as

$$ {P}_E(f)={\sum}_{m=0}^{M-1}{\sum}_{n=0}^{N-1}{C}_{xx}\left(m,n\right)W\left(m,n\right)\exp \left(-j2\pi f\left(m=n\right)\right. $$
(15)

Where, Cxx(m,n) and W(m, n)are auto-covariance and Blackman-window function respectively with ′m′and ′n′ samples.

By applying discrete cosine transform (DCT), the data can be compressed by the given equation.

$$ {\mathrm{DCT}}_{\mathrm{c}}\left(\mathrm{u},\mathrm{v}\right)={\sum}_{x=0}^{X-1}{\sum}_{y=0}^{Y-1}I\left(x,y\right)\cos \left(\frac{2\pi T\left(x+y\right)}{XY}\right) $$
(16)

For the calculation of Eigen-values and respective Eigen-vectors, a pattern vector \( \overline{p_n} \)is considered that is represented by similar low dimension vector\( \overline{q_n} \),which can be expressed in terms of linear transform characteristics.

$$ \overline{p_x}=\left[X\right]\overline{q_x} $$
(17)

Where[X] = [I(x, y)], form = 0 to X − 1&n = 0 to X − 1,

\( \overline{q_x} \) = min([X]), \( where\overline{q_n} \)> 0.

By taking the covariance of Eq. (17), the Eigen-vectors are estimated as shown in Eq. (18),

$$ \overline{P}=\mathit{\operatorname{cov}}\ \Big(\overline{p_n\Big)} $$
(18)
$$ \overline{p}.{X}_i={\lambda}_i.{X}_i $$
(19)

Where, λi denotes the respective Eigen-values.

Although using PCA with covariance finds the components that are useful for representing the data but may not be useful for discriminating the data between classes. Thus, Fisher’s Linear Discriminate Analysis (FLDA), has been applied in present work for reducing the feature vector dimensionality from ‘x’ into A =Y − 1 (where Ydenotes the total class involved). Hence the main idea in adopting FLDA is to project the feature vector of ‘x’ dimension into least dimensional region. These regions are selected in such a manner that the partition among the classes have been at minimal distance from the regression line.

Clustering is an unsupervised method and needs a training set where belong to different classes are known as priori. Image clustering and categorization is a means for description of image features. When no such training set is available one should adopt unsupervised methods such as clustering. Clustering can be used for training as well as classification in an unsupervised manner. In clustering, the basic objectives to partition the feature space data points into several groups that follow a pre-defined set of procedures. Image clustering enables the creation of a user-friendly interface to the database and the implementation of efficient retrieval algorithm. There are two widely used methods namely c-means clustering and k-means clustering. Figure 3 depicts the training stage of the proposed method.

Fig. 3
figure 3

Training Stage of Proposed Methodology

Full size image
Fig. 4
figure 4

Testing Stage of Proposed Methodology on Complete Face

Full size image

Figure 4 depicts the testing stage of the proposed methodology on complete face. K-means is a simplest clustering approach that adopts square-error criteria algorithm where the total partitions are pre-defined. The implementation process of K-means clustering is simple and the time required for computation is minimal. For predefined number of clusters, the cluster centers are randomly initialized and every data point is allocated to any of the neighboring cluster. But the selection of the predefined cluster affects the output of k-means algorithms. If the selection of initial clusters is unclear, then the algorithm outputs wrong cluster location along with improper clustering. C-means is a clustering technique that allows single pixel of data to present in more than two clusters. A machine learning algorithm called as Support Vector Machine is applied for classifying and characterizing the progressive switching pattern of facial test images taken from the side-view (Figs. 5, 6, 7, 8 and 9). Figure 5 depicts the testing stage of the proposed methodology on Destructive Face.

Fig. 5
figure 5

Testing Stage of Proposed Methodology on Destructive Face

Full size image
Fig. 6
figure 6

Original picture, Image of Case 1, Image of Case 2, Image of Case 3

Full size image
Fig. 7
figure 7

Image of Case 1, Image of Case 2, Image of Case 3

Full size image
Fig. 8
figure 8

Comparison of FPR and FNR on Various Database with Proposed Methodology

Full size image
Fig. 9
figure 9

Comparison of Sensitivity and Specificity on Various Databases with Proposed Methodology

Full size image

Various facial image database is collected and the proposed technique is applied for automatic generation of seven numerous facial datasets. Table 1 shows the statistic comparison of each database with various Database Faces given in [14].

Table 1 Statistics for various datasets
Full size table

Algorithm for Recognizing the Reconstructed Face Image

The methodology adopted in the present work has been depicted below.

Algorithm 1
figure a

Recognition algorithm called RRFI

Full size image

4 Results and discussions

In the present work an algorithm is developed for the reconstruction of destructed human face has been represented in Fig. 3. It represents the original face image and almost all possible combination of destruction in 2d face image. The proposed work is carried out in two parts, in first part destructed face image is read and preprocessing techniques is performed after that segmentation is perform, from segmented face image the mirror image part is crop which is almost similar to the destructed part (Table 2). After that the face image will be used for recognition and performance is measured as shown in Fig. 4.

Table 2 Comparison of Accuracy, FPR, FNR, Sensitivity and Specificity on all standard datasets (Fig. 10)
Full size table
Fig. 10
figure 10

Comparison of Accuracy on Various Databases with Proposed Methodology

Full size image

The Table 3 illustrates the achieved percentage for different methods for different datasets (Fig. 11).

Table 3 Accurate percent measures of datasets & approaches (in %)
Full size table
Fig. 11
figure 11

Comparison of Accuracy on Various Database with different Methodology

Full size image

The feature clusters used for the identification of behavioral patterns of human face have been drawn by adopting fuzzy c-means clustering and the outcome is visualized in Fig. 12.

Fig. 12
figure 12

Statistical analysis in papers with numerical results

Full size image

False positive rate (FPR)

The percentile of cases where an image was segmented to the shape, but in fact it did not (Fig. 13).

$$ \kern1.25em FPR=\frac{FP}{FP+ TN}\kern0.5em $$
(20)
Fig. 13
figure 13

Box plot analysis in paper with numerical results

Full size image

False negative rate (FNR)

The percentile of cases where an image was segmented to the shape, but in fact it did.

$$ FNR=\frac{FN}{FN+ TP}\kern0.5em $$
(21)

Sensitivity

The sensitivity is defined as the rate of proportions of true positive that are correctly identified. It correlates the ability of the test to obtain positive outcomes.

$$ \kern5.25em Sensitivity=\frac{Number\ of\ TP}{Number\ of\ TP+ Number\ of\ FN}X100 $$
(22)

Specificity

The specificity is defined as the rate of proportions of true negative that are correctly identified. It correlates the ability of the test to obtain negative outcomes.

$$ \kern4.75em Specificity=\frac{Number\ of\ TN}{Number\ of\ TN+ Number\ of\ FN}X100 $$
(23)

Accuracy

The weight percentile of pose varied facial images is exactly categorized by measuring the accuracy. It can be expressed as given in Eq. (14).

$$ \kern6.5em Accuracy=\frac{TP+ TN}{TP+ FP+ TN+ FN}X100\kern1.25em $$
(24)

5 Conclusion

The present research work we have discussed the results obtained. In this research work knowledge-based model has been formed by considering the relevant features of human face image as case study. After that knowledge-model has been mapped with test reconstructed face image for recognition and classification, The feature clusters used for the identification of behavioral patterns of human face have been drawn by adopting fuzzy c-means clustering and the outcome is visualized, From the experimental results, the application of proposed algorithm has been found very satisfactory.