The aggregation of data in recent years has been expanding at an exponential rate. There are various data generating sources that are responsible for such a tremendous data growth rate. Some of the data origins include data from the various social media, footages from video cameras, wireless and wired sensor network measurements, data from the stock markets and other financial transaction data, supermarket transaction data and so on. The aforementioned data may be high dimensional and big in Volume, Value, Velocity, Variety, and Veracity. Hence one of the crucial challenges is the storage, processing and extraction of relevant information from the data. In the special case of image data, the technique of image compressions may be employed in reducing the dimension and volume of the data to ensure it is convenient for processing and analysis. In this work, we examine a proof-of-concept multiresolution analytics that uses wavelet transforms, that is one popular mathematical and analytical framework employed in signal processing and representations, and we study its applications to the area of compressing image data in wireless sensor networks. The proposed approach consists of the applications of wavelet transforms, threshold detections, quantization data encoding and ultimately apply the inverse transforms. The work specifically focuses on multi-resolution analysis with wavelet transforms by comparing 3 wavelets at the 5 decomposition levels. Simulation results are provided to demonstrate the effectiveness of the methodology.
High dimensional image data consist of huge pixels of information that requires large storage space and significantly broad bandwidth resources for data transmissions. Storing and analyzing such data is expensive and the expenses increase as the size of data grows. It is a fact that there are noteworthy advancements in storage technologies and processing functionalities, but the technological advances are not fully able to efficiently handle the volumes, variation, varieties, values and velocities of the huge amount of data. This calls for the need to explore efficient image compression techniques that can store only the relevant information that is needed for reconstruction of the image. In essence, all image data are merely matrices of pixel values and the inherent redundancies should be exploited in compressing every image.
A Wavelet (little wave) is a function that has high time and frequency concentration about a given point. In comparing wavelets with Fourier Transforms (FT), the latter have the shortcoming of addressing simply the frequency components of the signals. That is, temporal information is usually unavailable. Due to Heisenberg’s uncertainty principle, we may either have poor frequency resolution and better time resolution or high frequency resolution and poor tempo- ral resolution. Wavelet transforms are mostly useful for non-stationary signals and their basis function varies in both frequency and spatial ranges. Wavelet transforms are implemented in such a manner as to obtain high frequency resolutions at the low frequency portions and good temporal resolutions at the high frequency portions of the signals.
Image compression methodologies are widely used in converting data from big and sparse formats into a more compact and dense format. These mathematical algorithms are classified into the following: Lossless compression approaches and the lossy compression techniques. In the lossy compression methods, compression of images generally comes with data loss or with an associated level of deterioration while retaining the crucial attributes of the images. For the lossless algorithms, it is guaranteed that the data being compressed has a high similarity with the initial data given. This contrast is highly crucial since lossless methodologies are not as efficient at compression as their lossy counterparts. Large data like a huge volume of image data that requires storage, pro- cessing and transmission over a network provides a viable example of the advantages of data compressions. This is because the major aim of image compressions is how to obtain the optimum image quality with very small mathematical computation, data transmission and reduced storage costs.
In this work, we examine multi-resolution analysis using wavelet transforms and this is one of the major mathematical and analytical frameworks employed in signal processing and representations, and we study its applications to the area of compressing image data in wireless sensor networks. The proposed approach consists of the applications of wavelet transforms, threshold detections, quantization data encoding and ultimately apply the inverse transforms. The contribution of this work is that it specifically focuses on multi-resolution analysis with wavelet transforms by comparing 3 wavelets at the 5 decomposition levels. Simulation results are provided to demonstrate the effectiveness of the methodology.
The body of knowledge is replete with research studies focusing on the application of Wavelet transform to the field of image compression. Researchers have explored different algorithmic frameworks for image compression such as 3-D wavelet transform, linear quantization, one-dimension address complexity, variable length block-coding, adaptive arithmetic with entropy code and so on. The work in [
In a slightly different perspective, researchers have utilized variable size block coding in representing the cluster of zeros, in conjunction with data decomposition techniques for image compression. A simple but efficient mathematical tool for compressing image data with the aid of a multi-level dyadic wavelet decomposition technique is found in [
It is possible to explore the crucial features of wavelets transform in the compression of still image data forms and this may include the degree to which the quality of the image data has been compromised through the processes of wavelet compression and decompressions [
An exhaustive presentation regarding the usage of wavelet transform for image data compressions is provided in [
This research work will particularly focus on the 1st generation of wavelets. The 1st generation wavelets are the classes of important wavelet invented around the 1980s. The basis function of such wavelets is dyadically scalable with respect to the translation property of one unique mother-wavelet basis function. A major drawback of such wavelets is that they are deployable for infinite or periodic signals but they cannot be optimized in bounded regions or domains. The wavelets are typically employed in detecting self-similarity, compressing images, de- noising signals, and detecting discontinuity [
・ Meyer wavelet,
・ Coiflet Wavelet,
・ Haar Wavelet and
・ Daubechies Wavelet.
The analytics in our study begins from mother wavelets like the Haars, Daubechies and Morlets [
S/N | Wavelet Generations | Differences | Applications |
---|---|---|---|
1 | First Generations | These wavelets can be used for periodic or infinite signals but they cannot be optimized in bounded domains | This wavelet transform is used in self-similarity detection, image compressions, signal de-noising, identification of pure frequencies and detection of discontinuities. Other applications are: Acoustic signal compressions, in fingerprint image compression, image processing, enhancement and restoration. Fractal analysis and de-noising noisy data. |
2 | Second Generations | Fastest for moderately short-filters but one needs to first find the factorizations of the filter banks matrices, but these factors are very well documented for JPEG2000 wavelets. | They are used tremendously for efficient coding in compressions algorithms, computer graphics, geographical data analysis and lossy data compressions. The FBI has used the CDF wavelets in fingerprint compression scans. With these wavelets, compressions ratios of about 20 to 1 could be achieved. They have been applied in multi-resolution analysis, system identifications, and parameter estimations. |
3 | Third Generations | They do not oscillate and do not show aliasing and degrees of shift variance in their magnitudes. They exhibit a 2D attribute of the signal to be transformed and produce redundancies | They are applied in sparse-representation, multi-resolution and useful features characterizations based on the image structures. They are also used in medical profession because they provide intuitive bridges between time and frequency data that could clarify interpretations of complex head-trauma spectra produced with Fourier transform. They are also used in music industry for transcriptions of music since they produced precise results that were not possible earlier with the Fourier transforms. It is capable of capturing short-bursts of repeating and alternating music notes. |
4 | Next Generations | They will be too specific and too constrained because These wavelet transforms are still very much at the research stage and meant for specific applications | The applications will include human-vision characterizations, frequency localizations, feature extractions, analysis of seismic information, analysis of biomedical information, wireless sensor networks [ |
unto scaled and shifted versions of the given mother wavelets. Wavelet decomposition analytics is employed in dividing information of the image data into the approximations and detail sub-signals. Approximation sub-signals usually display the overall trends in the pixels and the associated values as well as the three details sub-signal each from the vertical, diagonal and horizontal detail of the signals. The details could be equated to zero provided they are negligible, without substantial changes in the image data. Hence, when signals are decomposed into their corresponding sub-signals, four images are obtained, the first are the approximate images and the remaining three correspond to the diagonal details, horizontal details and vertical details as
Inverse technique to decomposition operation is called signal reconstruction operation. The overall decompositions and reconstructions are provided in
will be one approximate detail followed by three detailed sub-signals on the diagonal, vertical and horizontal details. The wavelet reconstruction computation should generate approximate images identical to the original image signals.
The remaining sections detailed how we employed wavelet analytics for image data compression. The following provides the general steps utilized in the data compression processes.
The compression methodology employed in this work exploits the wavelets expansion of sensor data and retains the maximal absolute-value-coefficients. Thus, we could choose some global thresholds or certain compression performance-metrics or a relative-squared-norm recovery performance metrics and it only requires the selection of a unit parameter. We load the sensor data and observe the compression performance for a selection of any un-optimized wavelets so as to have a total squared-norm recovery of the loaded sensor data. Finally, global threshold is implemented and the idea is to use multi-resolution analysis with wavelet transforms to compare 3 wavelets at the 5 decomposition levels. This has the potential to help the research community in understanding and identifying what wavelet is suitable for which application?
The compression attributes of any choice of wavelet basis are typically related to the relative scarcity of wavelet domain representation of the signals or data. The main idea associated with compression is hinged upon the fact that regular signal component might be accurately approximated by using the following: Very small amount of the approximation coefficients (at suitably selected level) and few of the detailed coefficient. The analysis is concluded by including the values of perf0 and perfl2 in the MATLAB codes for decompositions.
Decompositions: This entails how to load the data or images, selecting the appropriate wavelets, selecting a decomposition level N, computation of the wavelet decomposition for the data fat that chosen level N.
Threshold the details coefficients: It entails selection of the appropriate threshold and the application of hard thresholding to the details coefficients for every level in 1 to N.
Reconstructions: The computation of the wavelets synthesis is completed by using the originally obtained approximation coefficient for level N as well as the modified detailed coefficients for levels 1 - N.
In order to evaluate the difference between the original and compressed images, we used the following performance metrics: % root-mean-square difference (PRD), the compression ratio (CR) and the quality factor (QF).
Let X represent the original image and Y, the reconstructed image, the PRD is expressed as:
P R D = ∑ N [ X − Y ] 2 ∑ N X 2 × 100 % (1)
CR is the ratio of the uncompressed image size to the compressed image size and it is expressed as:
C R = UncompressedImageSize CompressedImageSize (2)
While the QF is the ratio of the square of the CR to the PRD and it is expressed as:
Q F = C R 2 P R D (3)
The integrity of the decomposition process relies heavily on the choice of wavelet. Thus, we examine the choice, among three wavelets, for the most appropriate mother wavelet for signal compression. The three wavelets are Haar wavelet, Daubechies type 4 and type 8 wavelets (DB4 and DB8). The data is loaded to the algorithmic framework sequentially and the compressions ratio of 50:1 is observed. The L2 norm recovery percent is 99.261% and the Compression-score percent is 97.993%. We then quantified the PRD, CR and QF of the compressed signal. The simulation results are shown in Figures 3-5 from which it is observed that the decomposition level as well as the choice of mother wavelet affects the PRD, CR and QF of the compressed signal. In
The paper investigated image compressions using the 1st generation wavelets transforms. Based on reviewing the recent wavelet-based image compression algorithmic frameworks, various challenges are identified which include: The in-
herent challenges linked with the high-complexity of data-encoding process of current algorithmic frameworks. Many of the modern methodologies need the developer to generate codebooks or lookup table that engulfs more implementation or computation cost. Better data quality and better compression ratios can be obtained by employing some other more compression-efficient wavelet transforms like the standardized CDF 9/7 utilized in fingerprint compressions. In this work, we found that within the first generation wavelets that we used, the Daubechies-4 type of wavelets performed better. In our future work, we will explore other types of wavelets and compare with the ones we used in this research study.
Thanks to our colleagues in the Center of Excellence for Communication Systems Technology Research (CECSTR) and the SECURE Center of Excellence (Systems to Enhance CYBERSECURITY for Universal Research Environment for all their support. Without the help from the Chancellor’s Research Initiative (CRI), this research would not have been possible. We are grateful for that.
Oduola, W.O. and Akujuobi, C.M. (2017) Wavelet Transform for Image Compression Using Multi-Reso- lution Analytics: Application to Wireless Sensors Data. Advances in Pure Mathematics, 7, 430-440. https://doi.org/10.4236/apm.2017.78028