Applications of Haar Wavelet Transforms
During the past decade, wavelet analysis has become a major tool in various applications, mostly related to signal processing, data and image compression. "Wavelet analysis has its roots in the field of mathematics but has been adopted into the fields of science and engineering" . In this paper we would discuss two very interesting concepts, both of them dealing with some form of an analysis of an image using Haar Wavelet Transform.
But before we foray into the applications of wavelets, let us take a look at the transition of image processing techniques. The Fourier transform has been the mainstay of transform-based image processing. Since the late 1950s, the wavelet transform has been instrumental in making it even easier to compress, transmit, and analyze images. "Unlike the Fourier transform, whose basic functions are sinusoids, wavelet transforms are based on small waves, called wavelets, of varying frequency and limited duration" . If we might use the analogy of a musical score for an image, imagine if the musical notes, along with informing us what to play, also had information about when to play them.
2.0 Fingerprint matching on Directional Image Constructed Using Expanded Haar Wavelet Transform 
In the few past decades we have seen an unprecedented growth in automation of the industrialized world along with an exponential rise in the information technology. Consequently, this resulted in a massive immigration drive towards the west. In order to accommodate this sudden surge, a very sound security and intrusion detection system
is mandatory. One of the most effective branches of security systems is the Biometrics.
Biometrics is the science of identifying individuals by particular physical characteristics such as voice, eye color, facial appearance or signature .
Fingerprinting has been used since the late 19th century. Over the period of time, it has been discovered that even identical twins have different fingerprints. This has pushed the fingerprint matching security system to the top and one of the most popular of the biometrics.
A fingerprint can be characterized by the pattern of its ridges and furrows. Studies have shown that a fingerprint image can be transformed into a directional fingerprint image which represents the orientation of the ridges. This can be achieved using a number of techniques. The steps involved in obtaining the directional image of the fingerprint and applying a matching algorithms technique can be illustrated in the block diagram in Figure 1.1.
There are three steps involved. The first step is transforming the original image using the wavelet transform. The mid-range frequencies resulting from the wavelet transform are then used to obtain an estimate of the fingerprint ridges. In order to obtain a smooth directional image, Prewitt edge operator is applied. Lastly the directional image is smoothened to get a better directional image.
2.1 Wavelet Transform
Applying the wavelet transform techniques give us four components. The approximation level is determined by the lower frequency. The horizontal and vertical components are determined by the mid-range frequencies. Whereas the diagonal details of the image is determined by the high frequencies. We however are more keenly interested in the mid-range frequencies because that provides us with the majority of the signal content.
There are many popular techniques from the wavelet family, such as Deubechies wavelets, Mexican hat wavelets and Morlet wavelets. However in this paper Haar wavelet has been used. The other techniques offer better resolution but are computationally more expensive. Whereas Haar wavelet transform has the following four advantages.
- It is conceptually simple
- The algorithm makes it faster
- It does not need any temporary array for computation, thus making it more memory efficient.
- Implementation costs are less expensive.
However Haar wavelet transform has its own disadvantages. Consider a one dimensional signal array of 24 elements. Let there be a large drop of signal content in between elements 6 and 7, and a sharp rise in signal content in between elements 11 and 12. The original Haar wavelet transform would fail to detect the second rise between elements 11 and 12. Haar wavelet transform uses the algorithm in between two elements with the first one being the element corresponding to an even number. Thus the original Haar wavelet transform fails to transform the original image's content in between element 11 and 12. This problem can be rectified using an expanded Haar wavelet transform. This can be explained in reference to the graph shown in Figure 2.1 (a) and 2.1 (b).
Expanded Haar wavelet transform uses three filter coefficients instead of two. Thus, the high pass filter's coefficients can represented by the matrix A= [1/2, 0, 1/2], and the low pass filter's coefficients as B= [1/3, 1/3, 1/3]. Using the expanded coefficients, the major content of the image can be transformed. As illustrated in the graph in Figure 2.1 (c), the transformed image rectifies the sudden rise in signal content in between elements 11 and 12.
The horizontal and the vertical details of the image can be represented by [h]x and [h]y. Using the expanded filter's coefficient, the resulting horizontal and vertical details are shown in expression(1), :
Applying the expanded Haar wavelet transform, we obtain the image with a better directional pattern. A comparison between the transformed images is given in Figures 2.2 (a), (b) and (c), :
The ridges are visible but they are not well directed and thus hinder the process of matching the angles between the ridges. However the other two images have been transformed using the expanded Haar wavelet transform.
2.2 Ridge orientation estimation 
Once the fingerprint has been transformed into a directional pattern, they need to be matched with the original fingerprint using an algorithm called "Ridge orientation Estimation". Human fingerprints do not exhibit any high fluctuation in the ridge or furrow's pattern. Therefore if any such pattern appears in the transformed image, it can be safely assumed to be erroneous and it has to go through a smoothening process.
hy(r, c) and hx(r, c) represents the vertical and the horizontal component of a particular row and column, respectively. The value of (r, c) is quantized into eight grey level values representing eight directions. Now the original image and the obtained image are superimposed and matched using Mean Square Error (MSE) checking method.
2.3 Matching using MSE
The standard and the most commonly used error detection method is the "Mean Square Error". MSE has been employed here. The following expression is used to compute MSE mathematically :
From the above expression it can be derived that squaring the differences between input and the output, gets rid of the negative values and also accentuates the small and the large variations.
3.0 A Full - Reference Image Quality Assessment Algorithm Based on Haar Wavelet Transform 
Assessment of the quality of an image has always been a tough challenge in the field of image processing. To determine the quality of an image obtained from a presumably faulty device, with reference to the original image, involves few image characteristics, such as, luminance, contrast, and structure. These characteristics have been borrowed from the general vision and structure of a "Human Eye". The amazing abilities to distinguish color, notice structural variations and the ability to have multi-resolution vision have been incorporated into a set of algorithmic system known as "The Human Visual System" or simply HVS. "Any image quality assessment algorithm developed so far has been attributed to the HVS. During the years, many different objective metrics were proposed, and we can classify them into three categories based on their formulation approach ".
- Full Reference
- Reduced reference
- No reference
This paper proposed an algorithm for image quality assessment based on "Full - Reference" or simply known as FR. "The objective metrics of FR has been derived from the pixel - wise error, which includes the Peak Signal to Noise Ratio (PSNR) and Mean Squared Error (MSE)" .
These objective metrics must correlate with the HVS based "Structural Similarity Assessment Metric" or SSIM. SSIM extracts the structural variations of a distorted image with reference to the original image. It is a very effective tool. However it was found that, Gaussian Blurred images could not yield better results using SSIM. An advanced and improved SSIM, based on gradient structure similarity (GSSIM) was used to solve this problem. With time, GSSIM was also found to be inadequate in image quality assessment. "To solve this issue, two new object metrics was proposed. One of them is based on single - level Haar wavelet transform (HWSSIM) and the other is the Multi- Resolution Haar wavelet transform (MRHWSSIM) ".
3.1 SSIM 
In the structural similarity objective metric, it adapts to the HVS, thus it takes the general characteristics such as, luminance, contrast and structure, into consideration. Based on these characteristics, an equation was computed as shown in expression (4).
Here, l (x, y) is the luminance, c (x, y) is the contrast, and s (x, y) is the structure
As discussed in section 3.0, it has been found that SSIM is not efficient in the assessment of a blurred image. A blurred image mathematically can be defined as the convolution of a signal with a Gaussian function. When the original signal component goes through a convolution with a Gaussian function, we get a literal blurred image.
An advancement of SSIM, based on edge gradient, was incorporated. GSSIM yielded better results than an SSIM. "A comparison of an image is shown in Figure 3.1" . Here the Mean Square Error is standard for all the three images. The second image has some noise component in it. The third one has been blurred using a Gaussian function.
The object metric results are given below:
- Mean Square Error (MSE) = 225, standard for all the images.
- Figure 3.1 (b) : SSIM = 0.7210, GSSIM = 0.5285
- Figure 3.1 (c) : SSIM = 0.7851, GSSIM = 0.5753
Clearly, GSSIM is yields better results than SSIM, however fails in case of the blurred image.
3.2 Haar Wavelet Transform Based Structure Similarity (HWSSIM)
In order to rectify the shortcomings of the previously used image quality assessment techniques, discussed in Section 3.1, Discrete Haar wavelet transform (DHWT) is used. Wavelet transform can detect the irregular structures . "A two - dimensional discrete Haar wavelet transforms is used to obtain the edge details of an image in the wavelet domain. In order to obtain the luminance and contrast details, Haar wavelet transform is also applied in "Contrast enhancement algorithm". Based on these ideas, a detailed algorithm is given below.
Step 1: A two - dimensional DHWT is used to transform the original image, X into different frequency zones. These are then decomposed into four categories, namely LLx, HLx, LHx, and HHx. Here, LLx is has the lowest resolution and is mainly used to determine approximation of the image. The rest of the components are used to determine edge information horizontally, vertically, and diagonally.
Step 2: The above procedures are repeated on the distorted image, Y and the four components obtained are LLy, HLy, LHy, and HHy.
Step 3: Let x and y be the block vectors of LLx and LLy. Let Luminance comparison be represented by 'ldwt(x, y)' and the contrast comparison by 'cdwt(x, y)'. In the DHWT domain, these are defined as :
Where, is the covariance of vectors x and y, ? is the standard deviation of vectors x and y. C1, C2 are the constants to avoid any irrationality
Step 4: The edge maps of the images, Emapx and Emapy are computed separately using the following equation .
Step 5: Consider x and y to be the block vectors of Emapx and Emapy. Note that these block vectors are separate from the ones used in step 3. The structure similarity comparison Se (x, y) is given as :
Applying the above algorithm in the images shown in Figure 3.1 (a), (b), and (c), we observe the following:
- MSE = 225, standard in all the images
- (b): HWSSIM = 0.6011
- (c): HWSSIM = 0.5033
From these observations, we can infer that HWSSIM is a better technique in assessment of the quality of a blurred image, when compared with SSIM or GSSIM.
Over the past few years, Wavelet Transform has become a major tool in the analysis of images. One of the main reasons is the ability of this technique to finer frequency details. Unlike the Fourier transforms, wavelet analysis is not limited to only sinusoidal signals. Most of the algorithms used are computationally less expensive and fairly easy to implement.
In this paper, I have made an attempt to study and understand two applications of wavelet transforms. Section 2 dealt with "Fingerprint Matching techniques". It has been found that, transforming the circular ridges into orientation patterns and measuring the angle between them, gives a far more acute comparison. This has been implemented by expanded Haar Wavelet Transform. The results have been fairly satisfactory. 96% accuracy has been achieved.
On the other hand Section 3 dealt with the "Full Reference Image Quality assessment". This paper presents various assessment techniques used for different types of images. Some of the techniques such as, SSIM and GSSIM are quite popular and efficient. However it was found that the assessment of a blurry image yielded very poor results from the existing techniques. Wavelet Transform was implemented, giving rise to HWSSIM technique. It was experimentally found out that HWSSIM yielded a much better results.
From the study of both these papers, it can be concluded, that wavelet transformations primarily apply to the study of Image Processing. This technique is still quite young and there is plenty of scope for further research. However, in certain areas of image processing, unparalleled advancement has been achieved using Wavelet Transformation.