1 Introduction

In recent years, palmprint recognition has gained more interest and attention. Compared with other currently available biometric traits, palmprints contain more distinctive information than fingerprints; palmprints acquisition devices are much cheaper than iris acquisition devices; and palmprint scanner builds a more accurate biometric system than face and voice. At present, research on palmprint recognition have been focused to develop multiple representations of palmprint features extracted from its rich texture information consisting of line, wrinkles, and ridges [18].

Any palmprint identification system consists of four basic modules: palmprint acquisition, region of interest (ROI) extraction, feature extraction, and feature matching. Each module plays an important role to enhance the performance of the recognition system. Capturing device should be constraint free for user friendly palmprint acquisition. However, if the device is constraint free then all palmprint images would not be aligned properly. Capturing a palmprint image and cropping ROI makes it very difficult to align palmprint images on the same precise position, as it brings forth rotation, translation (shift), illumination, and noise. Image may be corrupted with noise during its transmission over the network. In fact, these variations on palmprint images are inevitable and degrade the performance of palmprint recognition system [38].

In literature, transforms based on textural features such as Gabor wavelet [23,32], Fourier transform [20], wavelet transform [11,27], complex wavelet transform [39], Stockwell transform [1], curvelet transform [36], complex directional wavelet [22], contourlet transform [4] and many more image and signal transformations have been used for palmprint recognition. Directional texture features of palmprint are found to be good for palmprint recognition. Gabor transform, Radon transform, finite Radon transform (FRAT), steerable filter, and modified FRAT (MFRAT) are powerful tools to extract line orientation features of palmprint [14,41]. Several orientation based coding methods (palm code, fusion code, competitive code, XOR-SUM code, and Radon line orientation code) with high accuracy have also been proposed for palmprint verification [17,28,29]. Recently, Cui et al. proposed appearance-based bidirectional representation of palm [8].

It is still challenging to find efficient and robust palmprint features invariant to noise, translation, scale, rotation, and occlusion. Limited works are available for noise invariant palmprint recognition system [2,9,19]. Badrinath and Gupta proposed phase-difference as palmprint sub-image feature, and PCA based image reconstruction technique is used to identify each sub-image as good or bad block [2]. Based on this concept noisy palmprints with salt and pepper noise are recognized. Fahmy employed mel frequency cepstral coefficients (MFCCs) and DWT to extract palmprint features by converting 2D plamprint image into 1D [9]. Three layer feed forward back propagation error artificial neural network (FFBPENN) is used for palmprint classification. Li et al. presented a palmprint representation by competitive rule on multiple anisotropy filters and used PCA to extract palmprint features and dimension reduction [19]. Although features are claimed noise invariant but experiments are not performed for noisy palmprints. Peng et al. proposed L1-norm-based tensor analysis to make system robust to outliers and ranking graph embedding algorithm for low dimensional representations of multimedia data [24,25].

Most of the rotation and white noise invariant features proposed for robust pattern recognition are based on Radon transform [10,31,34,35]. Jadhav and Holambe, proposed features based on Radon transform with combination of either wavelet or discrete cosine transform for robust face recognition [12,13,30]. These features are found to be invariant to facial expression, illumination variations, and also robust to zero mean white noise. Radon, 1D DTCWT, and 1D FT are combined to find invariant features for Chinese character and aircraft recognition [6].

The issues may be addressed by either estimating degradation to restore original image before feature extraction, or calculating invariant features. Invariance properties for palmprint recognition can be achieved by combining different feature extraction techniques. The combination of complex directional filter bank (CDFB) and local binary pattern (LBP) has been proposed for shiftable and gray scale invariant palmprint descriptor [33]. Scale invariant feature transform (SIFT) and speed up robust feature (SURF) have been used to make palmprint classification system invariant to rotation, scaling, and translation [3,7,26].

The objective of this work is to develop a novel descriptor for palmprint identification which is invariant to rotation, shift, illumination, and noise. This work proposes RDF descriptor for palmprint recognition using Radon, 1D DTCWT, and 2D Fourier transform in cascaded order one after another. Although these transforms have been utilized separately for palmprint feature extraction, but individually they do not perform well against noise and/or rotation. Combined properties of these transforms as discussed in Section 2 give a white noise and rotation invariant palmprint features. Theoretical foundations and experimental results show that the proposed method performs better in term of correct identification rate (CIR), and is also robust against noise and rotation when compared with some recent methods.

The organization of this paper is as follows. Section 2 discusses invariant properties of the proposed RDF descriptor along with a brief description of Radon, 1D DTCWT, and 2D FT. Section 3 gives an overview of the proposed palmprint identification system. ROI and feature extraction processes are also explained here. Section 4 discusses and summarizes the performance of RDF descriptor on normal, rotated, and noisy palmprints. It also presents a comparison with other existing methods in literature. Finally, Section 5 concludes the work.

2 RDF transform and its invariant properties

The higher values of Radon coefficients represent texture information of an image. 1D DTCWT applied on Radon coefficients in angle direction extracts radial shift coefficients of Radon transform in a multi resolution way and high frequency coefficients are further selected for feature extraction. Magnitude of 2D Fourier transform applied on resultant 1D DTCWT coefficients helps to remove circular shift. It is invariant to translation along spatial variable on the projection as well. This section gives a brief review of Radon, Dual tree complex wavelet and Fourier transforms along with invariant properties of the proposed RDF transform.

2.1 Radon transform

It transforms 2D images into a domain of possible line parameters, where each line in the image gives a peak positioned at the corresponding line parameters. Radon transform preserves variations in pixel intensities. Radon projections are computed by adding pixel intensities along a line. This process improves spatial frequency components in a direction in which Radon projection is computed [12]. Radon transform maps intensity function of the pattern f(x, y) to a function R(t, θ) defined as

$$ R(t,\theta)= {\int}_{-\infty}^{\infty} {\int}_{-\infty}^{\infty} f(x,y) \delta(t-x \cos\theta-y\sin\theta ) dx dy $$
(1)

where δ(.) is Dirac delta function, θ ∈ [0, π], t ∈ [−A/2, A/2], and A is the size of image diagonal. Peak of the coordinate in R(t,θ) space denotes existence of a straight line in the image having (t,θ) as parameters [10]. Since the Radon transform is line integrals of the image, it is highly robust to noise and it is easy to construct scale and rotation invariants. The minimum (NS m i n ) and maximum (NS max ) number of projections required for reconstruction are NS m i n = π M, NS max = 2π M, where M is the diameter of the image [12,13].

2.2 Dual tree complex wavelet transform (DTCWT)

Discrete wavelet transform (DWT) lacks shift invariance. DWT is unable to obtain perfect reconstruction and good frequency characteristics. DTCWT helps to solve these fundamental problems while retaining the properties of nearly shift invariance, directional selectivity, perfect reconstruction, and limited redundancy with low computation cost [5,15]. DTCWT employs two real DWTs in essence, and complex coefficients only appear when the two trees are combined. Its good directional selectivity in 2D improves recognition rate in recognizing similar patterns. In DTCWT, (9, 7)-tap biorthogonal filter is used at level 1 decomposition. Beyond the 1st level, 6-tap Q-shift filter is used for further decomposition. Q-shift filters are no longer strictly linear phase. The extra redundancy of DTCWT enables the signal translation to corresponding phase change in complex value [16].

2.3 Fourier transform

For Fourier transform a shift in the time domain causes no change in the magnitude spectrum. Translation invariance of a 2D pattern can be achieved by taking the magnitude spectrum of the 2D Fourier transform of the pattern. 2D Fourier transform of Radon transform in both angle and radial direction provides translation and rotation invariant feature [4,35]. 2D Fourier transform of f(x, y) is define as

$$ F(u,v)= {\int}_{\infty}^{-\infty}{\int}_{\infty}^{-\infty}{f(x,y)e^{j2\pi(ux+vy)}dxdy} $$
(2)

2.4 Invariant properties of RDF transform

In order to make the system robust to rotation and noise, the present work utilizes translation and rotation invariant properties of Radon and Fourier transforms along with shift invariant property of DTCWT.

Translation and Rotation

Let image \(f^{\phi t_{0}}(x,y)\) is rotated and translated version of an image function f(x, y) with rotation angle ϕ and shift vector t 0(x 0, y 0). Radon transform of this rotated and translated image is

$$\begin{array}{@{}rcl@{}} R(t, \theta)(f^{\phi t_{0}}(x,y))&=&R(t - (x-x_{0})cos(\theta+\phi)\\ &-&(y-y_{0}) sin(\theta+\phi),\theta+\phi))\\ &=&R (t - t_{0}, \theta+ \phi) \end{array} $$
(3)

2D Fourier transform of R(t−t 0,θ + ϕ) is

$$ F(R(t-t_{0}, \theta+\phi))= F(u,v) exp(i(-ut_{0} + v\phi)) $$
(4)

where F(u,v) is 2D Fourier transform of R(t,θ), and

$$ | F(u,v)exp(i(-u t_{0} + v\phi)) |=|F(u,v)| $$
(5)

This means that magnitude of the Fourier transform of Radon coefficient does not change for rotated and translated image [35].

Noise

Radon transform is also robust to white noise. Suppose an image f(x, y) is corrupted by Gaussian white noise \(N_{\sigma^{2}} (x, y)\) with zero mean and σ 2 variance to get a noisy image f ′(x, y).

$$ f^{\prime}(x,y)=f(x,y)+N_{\sigma^{2}}(x,y) $$
(6)

The white Gaussian noise of every pixel in the image is uncorrelated to each other. Radon transform of (6) is

$$ R_{f^{\prime}}(t,\theta)=R_{f} (t,\theta)+R_{\sigma^{2}}(t,\theta) $$
(7)

Since Radon transform is line integrals of the image, for the continuous case Radon transform of noise is constant for all of the points and directions, and equal to the mean value of the noise, which is assumed to be zero [12]. Therefore,

$$ R_{f^{\prime}}(t,\theta)=R_{f}(t,\theta) $$
(8)

This means that the white noise with zero mean has no effect on the Radon transform of the image. However, this is not true for digital images because they are composed of a finite number of pixels. In this case, noise immunity is expressed in terms of signal-to-noise ratio of Radon projection (SNR proj ) which is given in terms of signal-to-noise ratio of an image (SNR image ) as

$$ SNR_{proj}=SNR_{image}+ 1.7(M/2)SNR_{image} $$
(9)

where (M/2) is the radius of the image. For large value of M/2, SNR image << 1.7(M/2)SNR image .

$$ SNR_{proj}=1.7(M/2)SNR_{image} $$
(10)

This shows that SNR has been increased by a factor of 1.7(M/2), which is practically a large quantity. As a result, the method is robust to additive noise.

3 Proposed palmprint identification system

The proposed palmprint identification system consists of three basic steps to compute invariant templates for palmprint images: palmprint acquisition; circular ROI extraction; and RDF feature extraction as illustrated in Fig. 1. Palmprints are generally collected by scanner and during scan it is very much possible that the acquired palmprint images may suffer from rotation, shift and noise effects. Therefore, center part of palm is extracted by aligning the vertical axis of palmprint coordinate system with the line between tip of the middle finger and mid of the wrist. Thus extracted center part of palm is the required ROI (region of interest). Further, feature extraction process is applied on this ROI and nearest neighbor classifier (NN classifier) is used to identify palmprint.

Fig. 1
figure 1

Flow diagram of the proposed robust palmprint identification system

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Performance of the proposed palmprint identification system is tested on two publicly available palmprint databases, PolyU 2D [37] and CASIA [40]; and a self collected database IIITDMJ. 900 gray scale palmprint images of both palms of 75 students of IIITDM Jabalpur are collected with constraint free acquisition device and referred as IIITDMJ palmprint database here after. Acquisition devices and the number of palmprints in these three databases are different. As compared to PolyU 2D, palmprint images in CASIA and IIITDMJ databases suffer more from palm movements and distortion. Figure 2 shows palmprint samples from these three databases.

Fig. 2
figure 2

a PolyU 2D b CASIA and c IIITDMJ palmprints

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3.1 ROI extraction

The center part of palm contains important textural information that helps to discriminate two palms. PolyU 2D palmprint images are captured in a constraint environment. The ROIs of all palmprints of PolyU 2D are extracted using a technique earlier proposed by us [28]. The size of palmprint images in PolyU 2D is 384×284 pixels, while the size of extracted ROI is 128×128 pixels. But CASIA and IITDMJ databases have full palmprint images with all five fingers. CASIA and IIITDMJ palmprints are collected in unconstrained environment, i.e., rotation and translation are allowed during palmprint capturing. Therefore, all palmprints should be aligned in the same direction before feature extraction. Palmprints, and hence extracted ROIs are of different size in these databases. Figure 3 shows ROI extraction from normal and noisy palmprints. The following steps are proposed for ROI extraction, and applied on CASIA and IIITDMJ palmprint databases:

  1. 1.

    At first, the palmprint image is filtered with low pass Gaussian filter to remove irrelevant noise and obtain a smooth image.

  2. 2.

    Smooth palmprint image is converted into binary image using a global threshold. Irrelevant parts of the binary palmprint image are removed by morphological operations.

  3. 3.

    Palmprint is segmented by bounding box operation on the mask of the region having the maximum area in the binary images. So obtained segmented image is rotated perpendicular to the maximum elliptical axis, which is a line between tip of the middle finger and mid of the wrist [21].

  4. 4.

    Boundary of four fingers is extracted by tracing the boundary pixel of segmented palmprint from top left to bottom right, while limiting the height of the fingers to one third height of the palmprint image.

  5. 5.

    The tips of four fingers are selected on the basis of local maximum of y-coordinate on the boundary pixels of fingers. Similarly, the gap between two fingers, labeled as A and B is obtained through the local minimum of the boundary pixels between tips of two fingers.

  6. 6.

    A square of size \(\frac {4}{3}\) of AB is drawn parallel to line AB at a distance of one-tenth of length AB. Figure 3 shows the inner circle of this square region selected as circular ROI of the palmprint.

Fig. 3
figure 3

ROI extraction a Rotated palmprint and b Rotated and Noisy palmprint

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3.2 Palmprint feature extraction with RDF transform

The recognition accuracy strongly depends upon the quality of the extracted features. RDF transform is applied on circular ROI to get a rotation, shift and noise invariant RDF palmprint descriptor. Radon transform can be obtained by summing intensity values of all pixels along the direction of lines within circular ROI. For a given ridge, every pixel within the ROI would be projected along perpendicular direction. This gives rise to one Radon slice in Radon domain. The radial variable in Radon domain is discretized with the same dimension as the diameter of the ROI. Twice of the diameter of this ROI is chosen as dimension of the angle variable θ. Radon slices are arranged in the counter clockwise direction in terms of θ. The same slice is saved twice but in reverse order, one in the normal order for orientation θ (0 ≤ θ ≤ π), and the other one in the reverse order for θ + π orientation. Storing slices in this way gives circularly shifted rows for the Radon slice matrix which is invariant to ROI rotation [6]. 1D DTWCT transformation is applied on the Radon slice matrix in radial direction to select the higher frequency components and to remove the shift along radial direction. The magnitude of 2D Fourier transform performed on the resultant coefficients helps to remove the remaining shift in both angle and radial direction of the Radon transform. The process is shown in Fig. 4 and the steps to extract palmprint features from M × M size ROI are:

  1. 1.

    Project the ROI in 2M different orientations (0 ≤ θ ≤ π) to get the Radon transform coefficients.

  2. 2.

    Apply 1D DTCWT on the Radon coefficients along the angle direction up to (log 2 M−2) level decomposition.

  3. 3.

    Select magnitude of high frequency coefficients in each angle direction at level (log 2 M−2) and (log 2 M−1).

  4. 4.

    Arrange these high frequency coefficients into matrix form where each row represents an angle direction.

  5. 5.

    Normalize these DTCWT coefficients using min-max normalization to remove the effect of illumination.

  6. 6.

    Apply 2D Fourier transform on resultant coefficients to obtain the magnitude of Fourier spectrum.

  7. 7.

    Normalize the Fourier spectrum magnitude using min-max normalization, and arrange into 1D vector which is used as the feature vector.

Fig. 4
figure 4

The steps for obtaining RDF descriptor from circular ROI

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4 Results and discussions

For all experiments performed in this work, NN classifier is used to identify palmprint. Euclidean distance metric is used to calculate distance between two palmprints. Performance of the proposed identification system is measured by correct identification rate (CIR). CIR is the percentage of correctly identified palmprints with the total number of palmprints provided in the testing set. A cumulative match characteristic (CMC) curve is also used to summarize the identification rate at different rank values. Its x-axis represents all possible rank values; and y-axis represents the CIR for each rank value. The rank-x is the percentage of correctly identified palmprints in the top-x candidate identities. CIR is considered mainly for 1st rank.

k-fold cross validation is performed to justify the performance. For PolyU 2D palmprint database, one subset is used for training and remaining k-1 sets are used for testing. As CASIA and IITDMJ databases have less number of palmprints for each individual, k-1 subsets are used for training and one subset is used for testing. For each k-fold cross validation, k number of CIRs are obtained. Average of these CIRs is the final CIR for this k-fold. The following set of experiments are performed to test the robustness of RDF descriptor:

  • Experiment 1: Testing the performance on normal palmprints.

  • Experiment 2: Testing the performance on rotated palmprints.

  • Experiment 3: Testing the performance on noisy palmprints.

  • Experiment 4: Testing the performance on rotated and noisy palmprints.

4.1 Performance on normal palmprints

To understand the potential of the proposed RDF descriptor, its performance on normal palmprints is compared with the performance of other descriptors generated by combining Radon transform with Fourier transform, DWT, and DTCWT separately. These descriptors are referred in this work as RF (Radon with Fourier transform) [35], RDW (DWT approximation coefficients of Radon transform) [12], and RDT (Radon with DTCWT). Radon transform coefficients tested for palmprint identification, does not give good results. Radon transform combined with Fourier transform gives better results. CIR(%) for palmprint identification further improves with RDW and RDT as DWT and DTCWT transforms provide multiresolution analysis.

Table 1 compares RF, RDW, RDT, and proposed RDF descriptor on PolyU 2D, CASIA, and IIITDMJ palmprint databases with k-fold cross validation. It is observed that RDF descriptor outperforms other descriptors for all three databases. In case of PolyU 2D palmprint database, 99.69 % (highest) CIR is obtained for 2-fold cross validation with RDF descriptor and it degrades with increasing value of k. In spite of movements and distortions present in the databases, CASIA and IIITDMJ achieve CIR of 97.32 % and 99.02 % for 4-fold cross validation in which two palmprint images per palm are used for testing.

Table 1 CIR(%) obtained with all four descriptor for k-fold cross validation on normal palmprint databases
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Figure 5 shows CMC curves to compare these descriptors for PolyU 2D, CASIA, and IIITDMJ palmprint databases for 4-fold cross validation. It is observed from CMC curve that the CIR of each rank with RDF descriptor for all databases is higher than that for other descriptors.

Fig. 5
figure 5

CMC curves to compare the performance of descriptors on normal palmprint for 4-fold cross validation

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4.2 Performance on rotated palmprints

Although proposed system attempts to handle palmprint rotation during ROI extraction but still some ROIs may suffer with rotation. The remaining rotation effect has to be taken care by the descriptor used in the system. The proposed RDF descriptor is robust against these small rotations. Its performance is tested on manually rotated palmprints up to 10∘. For the purpose, normal palmprints are provided for training and rotated palmprints are used for testing. The proposed RDF descriptor of the original palmprint and two of its rotated versions (5∘ and 10 ∘) are shown in Fig. 6. For each of these rotation angles, it is found that the correlation distance between descriptors of original palmprint and rotated palmprints of the same palm is approximately one, i.e., feature vector does not change due to rotation. It is observed that feature vector does not change with rotation up to 10∘.

Fig. 6
figure 6

Palmprint ROIs and their RDF descriptors for original and rotated palmprints

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Table 2 shows CIR(%) for normal and rotated test palmprints. It is observed that CIR does not degrade much for palmprints rotated in the range of 2∘– 6∘. However, 4-fold cross validation results show that at 10∘ rotations, CIR(%) drops by 7.55 % for PolyU 2D, 8.99 % for CASIA, and 8.82 % for IIITDMJ palmprint database. Figure 7 compares CMC curves obtained with rotated palmprints for PolyU 2D, CASIA, and IIITDMJ databases. Although CIR decreases with increase in rotation of palmprint, however, the degradation is not that much for higher rank values.

Table 2 CIR(%) obtained for k-fold cross validation on rotated palmprints with RDF descriptor
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Fig. 7
figure 7

CMC curves for 4-fold cross validation on rotated palmprints with RDF descriptor

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4.3 Performance on noisy palmprints

Robustness of the proposed RDF descriptor is tested against manually added noise in the palmprints. In Section 2.4, it is shown that white noise with zero mean has no effect on Radon transform of the image. To verify this, the performance is analyzed with the white Gaussian noise for different values of variance (0.01, 0.03, and 0.05) added to palmprint databases. Figure 8 shows normal and noisy palmprints along with their features. A noisy palmprint with high value of variance almost loses its textural information, i.e., SNR decreases. The correlation between descriptors of normal and noisy palmprints is approximately one, i.e., RDF descriptor is able to extract invariant features from noisy palmprint.

Fig. 8
figure 8

Original and noisy palmprint ROIs along with their RDF descriptors for Gaussian noise with zero mean and different variance

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Table 3 CIR(%) obtained for k-fold cross validation on noisy palmprints with RDF descriptor
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Again only normal palmprints are provided for training and noisy palmprints are used for testing. Table 3 shows CIR for all three palmprint databases with RDF descriptor after inserting Gaussian white noise (zero mean and different values of variance) in test palmprints. It is observed that CIR does not degrade much with increasing noise levels in palmprints. CIR decreases only by 3.47 %, 2.65 %, and 3.9 % for PolyU 2D, CASIA, and IIITDMJ databases, respectively for noise of 0.05 variance on 4-fold cross validation. It is observed that the performance decreases with increasing value of variance since the SNR of projection matrix depends on small factor of SNR of image.

Figure 9 shows CMC curves for 4-fold cross validation to compare the performance of RDF descriptor on PolyU 2D, CASIA and IIITDMJ palmprint databases with Gaussian noise for different values of variance. It is observed from CMC curves that the CIR does not degrade much with increasing noise in the palmprints for higher rank values. The proposed RDF descriptor is capable of identifying a palmprint which had completely lost its texture information.

Fig. 9
figure 9

CMC curves for 4-fold cross validation on noisy palmprints with RDF descriptor

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4.4 Performance on rotated and noisy palmprints

Robustness of the proposed RDF descriptor is tested on noisy (0.01 and 0.03 variance) and rotated (5∘ and 10∘) test palmprints. Table 4 shows CIR for all three palmprint databases with added rotation and Gaussian white noise for zero mean and different values of variance. CIR decreases only by 1.30 %, 2.34 %, and 3.41 % for PolyU 2D, CASIA, and IIITDMJ databases with 5∘ rotation and noise with 0.01 variance for 4-fold cross validation. However, CIR decreases by 10.82 %, 18.39 %, and 6.86 % for PolyU 2D, CASIA, and IIITDMJ databases with 10∘ rotation and noise with 0.03 variance for 4-fold cross validation. It is found that the proposed RDF descriptor performs well for the white Gaussian noisy but its performance decreases with increased rotation. CMC curves shown in Fig. 10 also prove the same. It can be seen that the identification rates are quite close for a particular degree of rotation.

Table 4 CIR(%) obtained for k-fold cross validation on rotated and noisy palmprints with RDF descriptor
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Fig. 10
figure 10

CMC curves for 4-fold cross validation on noisy and rotated palmprints with RDF descriptor

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4.5 Comparison with related works

The proposed work aims to achieve good performance for not only normal but also for rotated and noisy palmprints. In literature, noisy palmprint recognition with white Gaussian noise is not available but nevertheless Jadhav et al. proposed white Gaussian noise and rotation invariant facial features based on Radon transform with either DWT or DCT [12,13]. DCT applied on Radon projections to calculate frequency features is referred as RDC [12] and DWT applied on Radon space to improve directional multiresolution facial features is referred as RDW in Table 6 [13]. The best results are obtained in the same setup for PolyU 2D at 2-fold, and CASIA and IIITDMJ at 4-fold are compared for noisy and rotated palmprints with RDC, RDW, and RDF descriptors.

The performance of RDF descriptor is compared with RDW and RDC descriptors for normal palmprints (referred as 0∘ rotation and without variance), as well as for rotated (2∘ to 10∘ rotation) and white Gaussian noisy palmprints (0.01 to 0.05 variance). It is evident from Tables 5 and 6 that RDW and RDC are rotation and noise invariant, and perform consistently for different rotations and noise. However, RDF descriptor performs better as compared to RDW and RDC for each degree of rotation as well as noise. It achieves more than 94 % CIR for noisy palmprints (with Gaussian white noise of 0.05 variance) in all palmprint databases.

Table 5 Comparison (CIR%) of the proposed RDF descriptor with RDW [12] and RDC [13] for rotated palmprints
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Table 6 Comparison (CIR%) of the proposed RDF descriptor with RDW [12] and RDC [13] for noisy palmprints
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5 Conclusions

This paper introduces RDF descriptor for palmprint identification by using Radon, 1D DTCWT, and 2D Fourier Transforms in cascaded order. Combined properties of these transforms make RDF palmprint descriptor invariant to white noise and rotation. Experimental results show that the proposed RDF descriptor yields a better performance in terms of CIR and relatively high robustness to the variations in orientation, position, and illumination as compared to recent transform based approaches.