Nowadays, imitation for various products such as handwriting documents and famous artworks is prevalent. It is hard to identify fakes from the original one, and acrimonious disputes even happen to argue which one is genuine. Artistic professionals, experts or detectives may identify the authenticity of counterfeit products. The famous detective Holmes may solve the case using his renowned skills of astute observation and deductive reasoning with a magnifying glass. However, this procedure is subjective and is based on the accumulated experience of human experts. Human mainly uses the five senses of sight, hearing, smell, taste and touch to perceive the environment. Among the five senses, sight is the most powerful one. It is known that the majority of brain activity is involved with image treatment. A famous Chinese philosopher, Confucius, said, “A picture is worth a thousand words,” which means an image is very effective in capturing the features of products. The new technology makes it possible to record the visual observation of an image. The skills of observation, such as cameras, telescopes and microscopes, can compensate for the limitations of the human eyes (Dougherty, 2009). The development of computers also contributes to the expansion of image analysis. Computers can be utilized for the complex analysis of image data. These tasks include image segmentation, object recognition, 3-D feature analysis and many other tasks associated with the broad area of visual observation (Hesamian
For handwriting analysis, besides the information on the writing tools, materials, self-expression, spacing and shape of stroke, various features such as the character form, slant and size of writing have been used to identify the forgeries. For the objective method, the distinct features can be extracted from an image through the statistical methods, and those features can be utilized. Lyu
In this paper, we propose novel methods based on multiscale methods such as wavelet and wavelet packet transform coupled with a regularized regression approach. The essential features that distinguish the proposed method from existing ones are three-fold: (a) A multiscale framework of wavelets and wavelet packets is adapted to deal with images with various types of spatial structure. The flexible multiscale framework effectively captures the spatial characteristics of an image and extends the applicable scope of an image. (b) A regularized regression technique is applied to find proper relation between wavelet coefficients with a large number of predictors, and at the same time, to avoid over-fitting problems of high-dimensional data. (c) We extract the distinct feature of an image through a regularized regression.
The rest of this paper is organized as follows. In Section 2, we briefly review wavelets and wavelet packets as multiscale methods and ridge regression as a regularized regression technique. The proposed methods are described in Section 3. In Section 4, we conduct numerical studies to investigate the empirical performance of the proposed method. Finally, conclusions are addressed in Section 5.
We remark that in literature various approaches have been proposed to detect similarity between images. Aljanabi
To make the present paper to be a self-contained material, we briefly discuss wavelets and wavelet packets, and a regularized regression technique. In addition, the intraclass correlation coefficient is introduced to measure the similarity between two groups of observations.
Wavelets and wavelet packets have the advantage of obtaining a variety of additional information according to the spatial position, orientation and scale through domain transform. The various information obtained from the transform is utilized for comparative analysis of image data.
Wavelets are a family of orthonormal basis functions with several useful properties that have been popularly used in various fields such as mathematics, statistics and engineering. For analyzing an image, we focus on two-dimensional wavelets which are constructed by taking tensor products of one-dimensional wavelets. By using wavelets, an image is decomposed according to spatial position, orientation and scale. These local properties reflecting the spatial characteristics will be adapted to analyze an image.
To perform a multiscale analysis for images, we generally use a “pyramidal” algorithm consisting of a series of filters. There are various types of filters, and for a simple example, Haar bivariate filters are used. Let
For a 2 × 2 matrix
the wavelet coefficients for the data matrix
where
where the size of
where
The second filtering is applied to only the smoothed wavelet coefficient matrix
where
In general, for a 2
Furthermore, we consider the wavelet packet transform, which performs filtering to all sub-bands, while the wavelet transform does not take any further filtering for detailed coefficients. For the coefficients of
Then the wavelet packet coefficients are given by
For a particular example of two-dimensional wavelet transform and wavelet packet transform, we consider box image in the top panel of Figure 1. By wavelet transform, the box image is decomposed according to eight resolution levels. The middle panel of Figure 1 shows the decomposition results of two most detailed levels in wavelet transform. The first filtering of the box image generates lowpass, vertical, horizontal and diagonal coefficient matrices that represent smooth part, vertical, horizontal and diagonal details of the image, respectively. The next scale of the decomposed image is created by the recursive filtering of the lowpass sub-band. On the other hand, to obtain additional information, the wavelet packet transform in the bottom panel of Figure 1 takes the filtering of four sub-bands (lowpass, vertical, horizontal, and diagonal) recursively for each coefficient matrix of the most detailed level.
The linear regression model for
where
The parameter
The intraclass correlation coefficient (ICC) is a measurement of reliability index between individual ratings or measurements (Shrout and Fleiss, 1979). ICC is a modified statistic for the interclass correlation (Pearson correlation), and reflects not only a degree of relatedness but also an agreement between raters. ICC regards the observations for each rater as groups while Pearson correlation measures an association for the paired observations. It is commonly used to quantify the degree to which raters with assigning scores to the observation are similar to each other in terms of a quantitative trait. ICC can be estimated in terms of the random effects models
where
where
It is necessary to extract a distinct feature representing the characteristics of a given image. The wavelet and wavelet packet transform decompose an image according to the scale, and the detailed scale reflects the subtle, hidden structure of an image. Thus, we focus on the wavelet coefficients of the two most detailed scales and a distinct feature will be extracted based on the relation between these wavelet coefficients. In the proposed algorithm, two-dimensional discrete wavelet transform is implemented according to the Mallat’s pyramidal algorithm with Daubechies’ least-asymmetric wavelets (Mallat, 1989a; Mallat, 1989b). For a given image, we derive certain relation among wavelet coefficients of smooth part and detailed parts of an image. If this relation changes, we can deduce that an image is modified. The regression is used to model the relation among wavelet coefficients. The wavelet coefficients of a block in the most detailed level will be modeled by the wavelet coefficients of surrounding blocks at the same level and the wavelet coefficients in the next coarser level. To find a proper relation between the coefficients, the ridge regression is adapted to cope with the high-dimension problem at this step. The regression coefficient estimates and residuals are used to assess the quality of the regression model, and the skewness based on the estimates and residuals will represent the characteristics of the given image. If the skewness is similar to each other for given two images, then it is identified that two images are identical. The similarity is measured by ICC of skewness. Suppose that we have two images of interest. The proposed method is implemented by the following steps when wavelet packet transform is used.
1. Feature extraction : For each image, a distinct feature is extracted.
(1) (Transformation) Take wavelet packet transform (wavelet transform) of an image and then obtain the coefficients corresponding to each sub-band of two most detailed levels.
(2) (Regression) The coefficients corresponding to each sub-band of the most detailed level are partitioned into certain blocks. Let
(3) (Measurement) Obtain the estimated regression coefficients and residuals. Let
2. Comparison : The similarity between two images is judged by eight features.
(1) (Similarity) For example, among eight features, consider the skewness
(2) (Testing) Perform eight tests under each null hypothesis
Since the feature extraction step plays a crucial role, we explain the procedure in detail when wavelet packet transform is used. Suppose that we have a 2
This section reports the case study results to assess the empirical performance of the proposed method. The proposed algorithm is supposed to identify the difference between two images. For the analysis, we consider three numerical examples, including a simulated image, portrait image and handwriting image.
For comparison, the following five methods are studied:
WTR: the proposed method with wavelet transform and ridge regression,
WTL: the proposed method with wavelet transform and conventional regression,
WPR: the proposed method with wavelet packet transform and ridge regression,
WPL: the proposed method with wavelet packet transform and conventional regression and
WT3: the algorithm of Lytle and Yang (2006).
The performance of the methods is measured by the number of rejected tests with the null hypothesis that ICC is zero under a significance level of 0.05. Note that the proposed method uses wavelet coefficients of both smooth part and detailed parts while the algorithm suggested by Lytle and Yang (2006) is based on the skewness only for three directions of detailed wavelet coefficients. Thus, WT3 performs six tests to judge the similarity, while the proposed method runs eight tests. For this study, images of size 256 × 256 are used, and 16 blocks for each sub-band are employed.
Figure 3 shows four different box images in which the thickness of the diagonal lines is different, or there exists additional line. Box image
The proposed methods correctly identity that the difference between images
Image
Figure 5 displays Bengali alphabet samples of handwritings from the database (https://www.isical.ac.in/~ujjwal/download/Banglabasiccharacter.html). This database provides 37,858 handwritten image samples for Bengali basic characters. We consider three characters of ‘DA,’ ‘DDA’ and ‘DDHA’. Four handwriting images are chosen for comparison with two handwriting images
In this paper, we proposed the multiscale-based method for identifying the discrepancies between different images. Its performance is evaluated through a simulated box image, portrait image and handwriting image. The proposed method is implemented by coupling wavelet (packet) transform with regression. The proposed method is efficient in the simulation study when the regression model is coupled with wavelet packet transform rather than wavelet transform. The results from numerical experiments suggest that the proposed method possesses promising empirical properties.
We remark that the regularized regression approaches cope with the high-dimensional problem at the feature extraction step to resolve over-fitting for complex models, and the proposed procedure can be extended utilizing the various regularized regression approaches. The block size for the regression must be pre-determined for the proposed algorithm. The optimal block size for an image may improve its performance. Furthermore, it is possible to expand the algorithm to compare the multiple images by adapting multiple comparison procedure for the test of ICC. We leave these issues for future research since it is beyond the scope of this paper.
The number of rejected tests and the mean of ICCs inside parenthesis for box images
WTR | WTL | WPR | WPL | WT3 | |
---|---|---|---|---|---|
8 (0.823) | 8 (0.845) | 8 (0.945) | 8 (0.927) | 6 | |
4 (0.492) | 2 (0.470) | 0 (0.199) | 0 (0.199) | 5 | |
3 (0.402) | 1 (0.379) | 0 (0.159) | 0 (0.195) | 5 | |
1 (0.326) | 1 (0.258) | 0 (0.114) | 0 (0.053) | 2 |
The number of rejected tests and the mean of ICCs inside parenthesis for Lena images
WTR | WTL | WPR | WPL | WT3 | |
---|---|---|---|---|---|
7 (0.734) | 7 (0.737) | 7 (0.782) | 7 (0.765) | 6 | |
6 (0.612) | 4 (0.578) | 4 (0.528) | 5 (0.531) | 6 | |
6 (0.668) | 3 (0.550) | 3 (0.573) | 3 (0.546) | 6 |
The number of rejected tests and the mean of ICCs inside parenthesis for handwriting analysis
WTR | WTL | WPR | WPL | WT3 | |
---|---|---|---|---|---|
DA1 vs. DA2 | 2 (0.274) | 2 (0.268) | 4 (0.323) | 4 (0.472) | 6 |
DA1 vs. DDA | 0 (0.040) | 0 (0.003) | 0 (0.074) | 0 (0.021) | 0 |
DA1 vs. DDHA | 0 (0.088) | 0 (0.084) | 0 (0.095) | 1 (0.051) | 0 |
DA2 vs. DDA | 0 (0.081) | 0 (0.030) | 0 (0.130) | 0 (0.135) | 4 |
DA2 vs. DDHA | 0 (0.039) | 0 (0.077) | 0 (0.005) | 0 (0.038) | 0 |
DDA vs. DDHA | 0 (0.054) | 0 (0.047) | 0 (0.024) | 0 (0.002) | 3 |