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Chapter 50Medical Image Processing Using NovelWavelet Filters Based on Atomic Functions:Optimal Medical Image Compression

Cristina Juarez Landin, Magally Martinez Reyes,Anabelem Soberanes Martin, Rosa Maria Valdovinos Rosas,Jose Luis Sanchez Ramirez, Volodymyr Ponomaryov,and Maria Dolores Torres Soto

Abstract The analysis of different Wavelets including novel Wavelet familiesbased on atomic functions are presented, especially for ultrasound (US) and mam-mography (MG) images compression. This way we are able to determine with whattype of filters Wavelet works better in compression of such images. Key proper-ties: Frequency response, approximation order, projection cosine, and Riesz boundswere determined and compared for the classic Wavelets W9/7 used in standardJPEG2000, Daubechies8, Symlet8, as well as for the complex Kravchenko–RvachevWavelets ψ(t) based on the atomic functions up(t), fup2(t), and eup(t). The com-parison results show significantly better performance of novel Wavelets that isjustified by experiments and in study of key properties.

Keywords Wavelet transform · Compression · Atomic functions · Ultrasoundimages · Mammography images

1 Introduction

Different fields such as astronomy, medical imaging, and computer vision managedata of large volume. Hence, these data should be compressed to optimize the stor-age devices. There exist a lot of approaches in signal compression. Here, we presentWavelet-based techniques for compression procedures focusing on different thresh-old rules. The basic idea behind these techniques is to use Wavelets to transform dataset into a different basis, where the non-important information can be eliminated.Also, we have tested as classical, as novel Wavelet algorithms based on atomicfunctions, which present excellent compression results. Below, decimated Wavelettransforms (WT) and the MAE fidelity criterion are used to evaluate the different

C. Juarez Landin (�)Autonomous University of Mexico State, Hermenegildo Galena No. 3, Col. Ma. Isabel,Valle de Chalco, Mexico State, Mexicoe-mail: [email protected]

H.R. Arabnia and Q.-N. Tran (eds.), Software Tools and Algorithms for BiologicalSystems, Advances in Experimental Medicine and Biology 696,DOI 10.1007/978-1-4419-7046-6 50, c© Springer Science+Business Media, LLC 2011

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[email protected]

498 C. Juarez Landin et al.

compression methods for US and MG images. Investigating the key properties ofthe different Wavelets, we can justify the obtained experimental results in the com-pression of US and MG images.

1.1 Wavelet Transform and Filter Banks

The Discrete Wavelet Transform (DWT) is easy to realize using filter banks [1].DWT can be implemented applying some equations, but it is usually made usingfilter bank techniques. The most popular scheme of the DWT for 2D signal appliesonly two filters for rows and columns, as in the symmetric filter bank.

In tests carried out previously, it was found that better results are obtained whencompressing the ultrasound images with the Symlet Wavelet, and the mammographyimages with the Daubechies Wavelet [2]. Based on this fact, we realize an evaluationto compare the acting of three Wavelet families based on atomic functions with theWavelets that presented better acting. We use the complex Kravchenko–RvachevWavelets ψ(t) based on the atomic functions up(t), fup2(t), and eup(t) [3].

1.2 Compression by Wavelet Threshold

The three main steps of compression using the Wavelet coefficient and thresholdtechnique are as follows:

(a) Calculate the Wavelet coefficient matrix applying WT to the original image(b) Modify (threshold or shrink) the detail coefficients to obtain the reduced number

of coefficients(c) Encode the modified coefficients to obtain the compressed image

In the two-level sub-bands decomposition, the coefficients on the first level aregrouped into the vertical details (LH1), horizontal details (HL1), diagonal details(HH1), and approximations (LL1) sub-bands. The approximations part is then sim-ilarly decomposed in second level sub-bands. The directions reflect the order, inwhich the high-pass (H) and low-pass (L) filters of the WT are applied along thetwo dimensions of the original image.

1.3 Compression by Wavelet Threshold

The thresholding functions [4] determine how the thresholds are applied to data. Themost popular are four thresholds, and a single threshold (±t) is required for hard[δ H

t (w)], soft

[δ S

t (w)], and garrote

[δ G

t (w)]

functions, but for semisoft function[δ SS

t1,t2 (w)], there are require two thresholds (±t1 and ±t2). Hard function does not

50 Medical Image Processing Using Novel Wavelet Filters 499

modify the original data of the Wavelet coefficients; hence, for this reason we useonly hard function in the experiments. The hard function is given below:

S =

{x si |x| > t0 si |x| ≤ t

(1)

where x is the original signal, S is the thresholding signal, and t is threshold.

2 Wavelet Key Properties

The Wavelet decomposition algorithm uses two analysis filters: H(z) (lowpass) andG(z) (highpass). The reconstruction algorithm applies the complementary synthesisfilters: H(z) (lowpass) and G(z) (highpass). These four filters constitute a perfectreconstruction filter bank. In the present case, the system is entirely specified bythe lowpass filters H(z) and H(z), which form a biorthogonal pair. The highpassoperators are obtained by simple shift and modulation and shown below:

G(z) = zH (−z) and G(z) = z−1H(−z) (2)

For all the Wavelet families used, the following properties were obtained to justifythe results.

2.1 Frequency Response

This characteristic allows determining the behavior of the analysis and synthesisfilters in a graphic way to appreciate the differences that there are among differentWavelet families used.

2.2 Approximation Order

This determines the number L of factors (1 + z−1) that divide the transfer functionH(z). The approximation order plays a crucial role in Wavelet theory [5]. It impliesthat the scaling function ϕ(x) reproduces all polynomials of degree lesser or equalto n = L−1; in particular, it satisfies the partition of unity (Σkϕ(x− k) = 1). Theyare also directly responsible for the vanishing moments of the Wavelet analysis:∫

xnψ(x)dx = 0 for n = 0, 1, 2, . . ., L− 1. Finally, the order L also corresponds tothe rate of decay of the projection error as a scale how it goes to zero [6].

The next point concerns the stability of the Wavelet representation and of itsunderlying multi-resolution bases. The crucial mathematical property is the trans-lation of the scaling functions and Wavelets in Riesz bases [7]. Thus, one needs tocharacterize their Riesz bounds and other related quantities.


2.3 Riesz Bounds

The tightest upper and lower bounds, B < 1 and A > 0, of the autocorrelation filterof ϕ(x) are the Riesz bounds of ϕ(x) that are given by:

A,B = inf,supc∈�2

∣∣∣∣∑k∈Z ckϕ(x− k)

∣∣∣∣L2∣

∣∣∣c

∣∣∣∣�2

(3)

The existence of the Riesz bounds ensures that the underlying basis functions are inL2, and that coefficients of transform are linearly independent in the �2 space. TheRiesz basis property expresses equivalence between the L2 norm of the expandedfunctions and the �2 norm of their coefficients in the Wavelet or scaling functionbasis. There is a perfect norm equivalence (Parseval’s relation), if and only if A =B = 1, and in this case the basis is orthonormal.

2.4 Projection Cosine

The (generalized) projection angle θ between the synthesis and analysis subspacesVa and Va is defined as [8]:

cosθ = inff∈Va

∣∣∣∣Pa f

∣∣∣∣L2∣

∣∣∣ f

∣∣∣∣L2

=1

supω∈[0,2π ]

√aϕ(ω) ·aϕ(ω)

(4)

This fundamental quantity is scale independent, it allows comparing the perfor-mance of the biorthogonal projection Pa with that of the optimal least squaressolution Pa for a given approximation space Va. Specifically, we have the follow-ing sharp error bound [9]:

∀ f ∈ L2,∣∣∣∣ f −Pa f

∣∣∣∣L2 ≤

∣∣∣∣ f − Pa f

∣∣∣∣L2 ≤ 1

cosθ∣∣∣∣ f −Pa f

∣∣∣∣L2 (5)

The projection angle θ between the synthesis and analysis subspaces should be 90◦in orthogonal spaces. In other words, the biorthogonal projector Pa will be essen-tially as good as the optimal one (orthogonal projector onto the same space) whenthe value cosθ is close to 1.

2.5 Criteria of Fidelity

To evaluate in objective manner the fidelity of the compressed images, we apply themean absolute error (MAE) and the compression rate (CR) criteria that characterizethe difference between two images [10].


3 Simulation Results

We carried out numerous experiments to compare the performance of the compres-sion algorithm. In these results, classical Wavelet filters are compared with threedifferent families of Wavelet filters based en AFs. We use five decomposition levelsin the compression procedure. Figures 1 and 3 present the obtained results in Ultra-sound and Mammography respectively for MAE criterion. Figures 2 and 4 presentthe obtained results in Ultrasound and Mammography respectively for CR measure.

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Level 1Level 2Level 3Level 4Level 5




Fig. 1 MAE criterion for compressed ultrasound images with (a) Symlet filters, based on AF,(b) up(t), (c) fup2(t), and (d) eup(t) Wavelet filters

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Fig. 2 CR criterion for compressed ultrasound images with (a) Symlet filters, based on AF,(b) up(t), (c) fup2(t), and (d) eup(t) Wavelet filters


0.90

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Fig. 3 MAE criterion for compressed mammography images with (a) Daubechies filters, basedon AF, (b) up(t), (c) fup2(t), and (d) eup(t) Wavelet filters





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Fig. 4 CR criterion for compressed mammography images with (a) Daubechies filters, based onAF, (b) up(t), (c) fup2(t), and (d) eup(t) Wavelet filters

Finally, the Table 1 presents the key properties of the different Wavelets used incompression of the US and MG images.

It is known from statistical theory that the approximation property of estima-tion of random variable can be given in the form of relative error δ = 2(1− r),where r is the correlation coefficient that is equal to projection cosine in this case.Hence, calculations of this error using Table 1 show that Wavelet based on eup(x)


Table 1 Summary of key properties for different Wavelet filtersKey properties for different Wavelet filters

Wavelet 9/7 Daubechies 8 Symlet 8 A.F.W. up(t) A.F.W. fup2(t) A.F.W. eup(t)

Type Dec. Rec. Dec. Rec. Dec. Rec. Dec. Rec. Dec. Rec. Dec. Rec.

Approximationorder

4 4 4 4 4 4

Projectioncosine

0.98387 0.98879 0.98781 0.99176 0.99472 0.99769

Riesz 0.926 0.943 0.833 0.849 0.880 0.896 0.792 0.806 0.713 0.726 0.641 0.653

bounds 1.065 1.084 1.267 1.290 1.273 1.295 1.514 1.542 1.802 1.834 2.145 2.183

can produce relative variance error of 0,00464 (6.8% in RMS value), but at the sametime Wavelet Daubechies 8 gives value of 0,02242 (more than 15% in RMS value),and Wavelet 9/7 value of 0,03234 (more than 18% in RMS value). Hence, Waveletbased on eup(x) function gives about three less a relative error in RMS values thanWavelet 9/7 used in JPEG2000 standard.

4 Conclusions

Numerous tests were carried out in comparing different Wavelets functions tochoose the best one for compression of medical US and MG images. It is observedthat for two modalities of images used in the tests, the Wavelets families based onatomic functions presented smaller levels of MAE, with relationship to their equiv-alent of the Daubechies and Symlet families. The compression level also decreasedlightly but better image quality is conserved.

The Wavelet family based on the atomic function eup(x) present better levelsof MAE for two image modalities. It is observed in the carried out analysis thatDaubechies and Symlet filters are most selective than the Wavelet 9/7 used in stan-dard JPEG2000. The filters of the Wavelet based on the atomic function eup(x) arepresented the best frequency response. One can see that all the used filters have thesame approximation order. This derives mainly in two things, the Wavelet filtershave the same number of coefficients, and therefore, they imply the same computa-tional complexity when being implemented in the compression algorithms, and theconvergence of the error will be of the same order for all the filters.

The existence of the limits Riesz bounds demonstrates that the coefficientsof the analysis and synthesis filters are lineally independent. The projection co-sine measure shows that the families of Wavelets based on atomic functions arenear to optimal ones; this implies that they are “better” orthogonal and “most in-dependent.” Also, the fact that they are lineally independent assures that errorsare hardly presented to the decomposition/reconstruction procedure. Likewise, thedifferent properties of the Wavelet filters found here demonstrates that the filtersof the Wavelets families based on atomic functions can realize the approximation


better in comparison with traditional Wavelets, potentially giving superior qualityof compression, guaranteeing the same level of error.

Acknowledgments This work is supported in part by IPN, UAEM project 2703 and PROMEP.

References

1. Jahne, B., 2004. Practical Handbook on Image Processing for Scientific and Technical Appli-cations. CRC Press, Boca Raton

2. Sanchez, J.L., Ponomaryov, V., 2007. Wavelet compression applying different threshold typesfor ultrasound and mammography images. GESTS Intern. Trans. Comput. Sci. Eng., vol. 39,no. 1, pp. 15–24

3. Gulyaev, Y., Kravchenko, V., Pustovoit, V., 2007. A new class of WA-systems of Kravchenko–Rvachev functions. Dokl. Math., vol. 75, no. 2, pp. 325–332

4. Jan, J., 2006. Medical Image Processing, Reconstruction and Restoration: Concepts and Meth-ods. Taylor & Francis, CRC Press, Boca Raton

5. Vetterli, M., Kovacevic, J., 1994. Wavelets and Subband Coding. Prentice-Hall, EnglewoodCliffs

6. Villemoes, L., 1994. Wavelet analysis of refinement equations. SIAM J. Math. Anal., vol. 25,no. 5, pp. 1433–1460

7. Meyer, Y., 1990. Ondelettes. Hermann, Paris8. Unser, M., Aldroubi, A., 1994. A general sampling theory for nonideal acquisition devices.

IEEE Trans. Signal Process., vol. 42, no. 11, pp. 2915–29259. Strang, G., Nguyen, T.Q., 1996. Wavelets and Filter Banks. Wellesley-Cambridge Press,

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