Image Processing Technique Using Band Splitting in the FresnelTransformed Signal Domain

Satoshi Ito and Yoshifumi Yamada

Faculty of Engineering, Utsunomiya University, Utsunomiya, 321-8585 Japan

SUMMARY

There are two methods for solving the Fresnel trans-form formula using Fourier transforms, one of which usesthe Fourier transform once and the other uses it twice. WhenFresnel transformed signal of an image is calculated usingthe inverse algorithm of the latter Fresnel transform methodand then the Fresnel transform is calculated by the formerFresnel transform method, the image can be scaled by anarbitrary rate. When the image is down scaled, an aliassignal produced in the calculation of the discrete Fresneltransform appears as edge component of the image havinghigh-frequency components of the original image. Theimages corresponding to each hand of the Fresnel trans-formed signal are reconstructed in a different region of thereconstructed image domain like the wavelet-based mul-tiresolution image analysis. In this paper, the authors de-scribe the band splitting effect due to the Fresnel transformsand compare its characteristics with those of a waveletimage analysis. Also, as an image processing application,the authors performed image sharpening processing forenhancing the signal of the high-frequency region whilesuppressing noise. The results confirmed that favorableeffects are obtained which are similar to those obtained forwavelets. © 2003 Wiley Periodicals, Inc. Syst Comp Jpn,34(8): 51–61, 2003; Published online in Wiley InterScience(www.interscience.wiley.com). DOI 10.1002/scj.10290

Key words: wavelet transform; Fresnel transform;band splitting; multiresolution analysis.

1. Introduction

A wavelet transform, which splits the frequency bandof the original image in the image domain while includingposition information, has been widely applied for contrastenhancement of images [1] and noise suppression [2–4].With a wavelet transform, the multiresolution analysis levelof an image can be arbitrarily set and different processingcan be performed for each level or image region. Like thewavelet transform, a multiresolution image analysis can beperformed by using the Fresnel transformed signal band-splitting effect (FREBAS method).

If the FREBAS method is used to compute a down-scaled image, alias signals that are produced when comput-ing the Fresnel transform signal appear as edge images indifferent regions in the reconstructed image domain. Thisoperation is regarded as if the Fresnel transformed signalwere equivalently band-split, and the images were recon-structed from every band.

Conventionally, in a multiresolution analysis basedon Fourier analysis, an enormous amount of calculationswere required to perform operations for reducing the imagematrix by data subsampling [5] or to perform convolutionoperations by using different scale functions [6]. However,while the FREBAS method is based on Fourier transformsand phase modulation processing, it can generate similarimages to multiresolution analysis in a domain regarded asa Fourier transform region of the image without changingthe matrix size of the image, and it can split a band into anarbitrary number of bands according to only the coefficientsused in the Fresnel transform.

© 2003 Wiley Periodicals, Inc.

Systems and Computers in Japan, Vol. 34, No. 8, 2003Translated from Denshi Joho Tsushin Gakkai Ronbunshi, Vol. J85-D-II, No. 2, February 2002, pp. 242–251

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In this paper, we investigate the FREBAS method,which can perform a similar multiresolution analysis as awavelet transform, by comparing its image analysis methodwith the method due to a wavelet transform, and we attemptthe same image processing applications as for the wavelettransform to check the applicability of the FREBAS methodto image processing.

2. FREBAS Method

2.1. Dual solutions of the Fresnel transformformula

There are two methods for solving the Fresnel trans-form formula. One method is to expand the quadratic phaseterm of the Fresnel transform formula and then performFourier transform processing after multiplying by the quad-ratic phase term. (This method is referred to as Method 1.)The other method is to obtain the solution according toinverse filtering. (This is referred to as Method 2.) Theimage Fresnel transform signal band-splitting method pro-posed in this paper uses these two methods to performimage scaling. Therefore, simple explanations of Methods1 and 2, which are components of the image scaling proc-essing, are presented below. To simplify the explanations,the signal is treated as a one-dimensional signal.

If we let o(x) be the image function, c be the coeffi-cient that optically corresponds to the distance parameter,and u(x′) be the Fresnel transformed signal, the relation-ships among these items can be expressed as

2.1.1. Solution according to Method 1

If we expand the exponential term of formula (1) andrearrange the equation, we derive the following formula forthe inverse Fourier transform:

Ωx in formula (3) is a variable that transforms the x domainto the frequency domain. Since the dimension differs fromthat of ωx in the frequency domain of Method 2, which isdescribed later, we denote it as Ωx. Formula (2) takes theform of an inverse Fourier transform. Therefore, we canobtain o(x) as follows by taking the Fourier transform and

then multiplying by the quadratic phase modulation term[7]:

In formula (4), let F represent the Fourier transform. Thisprocessing procedure is illustrated in Fig. 1(d) → (e) → (f).(d) is the Fresnel transformed signal, (e) is the signal withinthe brackets [ ] multiplied by the quadratic phase modula-tion term in formula (4), and (f) is the signal (c/π)F [u(x′)exp(jcx′2)], which is obtained by taking the Fouriertransform of the signal in (e). o(Ωx / 2c) can be obtained bymultiplying expjΩx

2 / (4c) by the signal in (f).

If we let N be the number of data in Method 1 and∆x′ and ∆x1 be the sampling steps in the x′ and x domains,respectively, of formulas (1) and (3), the relationship informula (5) can be derived [8] for the relationship amongthese items from ∆Ωx = 2c∆x1, which is obtained fromformula (3), and from the Nyquist frequency relationship∆Ωx = 2π / (N∆x′), when ∆Ωx is the computation step informula (4):

(1)

(2)

(3)

(4)

(5)

Fig. 1. Scaling of images using the two Fresneltransform pairs.

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2.1.2. Solution according to Method 2

This method solves the convolution integral accord-ing to an inverse filtering technique. If we compute theFourier transform for x′ in formula (1), we obtain [7]

where O(ωx) is the Fourier transform pair of o(x). o(x) canbe obtained by obtaining O(ωx) from formula (6) andcomputing its inverse Fourier transform:

This processing procedure is illustrated in Fig. 1(d) → (c)→ (b) → (a). Signal (b) is obtained by taking the Fouriertransform of the Fresnel transformed signal in (d) to obtainsignal (c) and then multiplying signal (c) by the quadraticphase modulation term and then by the proportion coeffi-cient. The image o(x) in (a) is obtained by taking the inverseFourier transform of signal (b). Since Method 2 takes theFourier transform of a signal with sampling step ∆x′, per-forms phase modulation, and then takes the inverse Fouriertransform to return to the original x′ domain, if we let ∆x2

be the pixel width of the reconstructed image, this value willbe equal to ∆x′ [8].

2.2. Scaling of images using the dual solutionsof the Fresnel transform formula

2.2.1. Transformation to the Fresneltransform formula according to Method 2

We use the reverse algorithm of Method 2 for formula(6) to obtain the Fresnel transformed signal of the image.Let image domain 2 denote the domain in which the originalimage o(x) exists. If we let O(ωx) be the Fourier transformfunction of image o(x) and take the inverse Fourier trans-form after multiplying this O(ωx) by the quadratic phasemodulation term and a constant, we can compute the Fres-nel transformed signal u(x′):

Let up be a discrete Fresnel transformed signal that iscomputed from N image data values, and consider imagereconstruction according to a discrete Fresnel transform byusing a continuous Fourier transform formula thereafter.Since the step ∆x′ of the Fresnel transformed signal in

formula (8) is equal to ∆x2 as explained in Section 2.1.2, ifthe periodicity of the discrete Fourier transform is taken intoconsideration, the discrete Fresnel transformed signal canbe represented by

This transformation process corresponds to Fig. 1(a) → (b)→ (c) → (d).

2.2.2. Reconstruction of scaling imageaccording to Method 1

The Fresnel transformed signal in formula (9) can besolved by using Method 1. The image can be reconstructedby multiplying by the quadratic phase modulation term anda constant, taking the Fourier transform, and multiplyingby expjΩx

2 / (4c), as shown in formula (4). However, since

this quadratic phase modulation using discrete data is mul-tiplied only for the interval N∆x2, we introduce rect(x/N∆x2)as a rectangle function. If we let oscale denote the image thatis obtained by starting from image (a) given in Fig. 1 andundergoing the processing up to (f), this image can berepresented by the following formula. This transformationprocess corresponds to Fig. 1(d) → (e) → (f):

The following shift-invariant relationship betweenthe Fresnel transformed signal and the reconstructed imagewas used in formula (12):

(6)

(7)

(8)

(9)

(11)

(12)

(10)

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If a variable transform is performed according to formula(3) for formula (12), which represents the image that isobtained in the Fourier transform domain when viewedfrom the image function in Fig. 1(a), the following formulais obtained. Let this domain be denoted by image domain1:

From formula (5), the condition that the image scalesof image domains 1 and 2 are equal is expressed as ∆x1 =∆x2, and if we let c denote the distance parameter c at thistime, formula (5) yields

By using this c as a basis and introducing the coefficient Dfor c, c can be expressed again as

By introducing this D value, ∆x1 and ∆x2 can be related asshown in formula (17) from formula (5). Consequently, Dhas the meaning of a scaling parameter.

The convolution integral of the o(x − mN∆x2)exp(−jcx2)and sinc(2cN∆x2x) functions in formula (14) can be trans-formed to the following formula by letting the width of thefield of vision of image domain 2 be X2 = N∆x2:

If the result in formula (18) is used and the image analysisformula in formula (14) is rewritten by denoting the imagecomponent of degree m in formula (14) by o(m, x), formula(20) is obtained:

The expression within the braces ⋅ in formula (18)represents the convolution integral of the function o(x –mX2)exp−jc(x − mX2)2, which assigned a quadratic phasemodulation and a transition of mX2 to the image function,and the sinc function, which has linear phase modulation.The sinc function envelope at this time, which is determinedby the c value when ∆x2 was assigned, does not depend onthe value of m. However, the phase modulation functionexp(j2cmX2x) has a form that is proportional to m. As aresult, the larger the value of m, the greater the phase changewithin the main lobe of the sinc function, and as a result ofthe convolution operation, only the high-frequency compo-nents in the 2cmX2 neighborhood of the image will remain.Figure 2 shows the situation when o(m, x) is calculated.However, for explanatory purposes, the functions for thereal parts of each signal are displayed. o(m, x) is obtainedby convoluting the object function o(x – mX2)exp−jc(x −mX2)2 centered at mX2 with the sinc function having aphase modulation, and that is reconstructed in a region thatdepends on the value of m in image domain 1. At this time,since the scaling image is a function having a period of DX2

in image domain 1, which is clear from formula (14), imagecomponents for which mod(m/D) is equal are reconstructedin the same region in image domain 1 due to aliasing thatis produced when computing the Fresnel transformed sig-nal according to Method 2 (mod denotes the remainderwhen m is divided by D).

With a wavelet transform, image analysis is per-formed while scaling the scaling function. However, withthe FREBAS method, only the modulation function isscaled without the sinc function envelope changing, and aconvolution integral is computed with an image functionfor which quadratic phase modulation was performed. Fig-

(13)

(14)

(15)

(16)

(17)

(18)

(19)

(20)

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ure 3 shows the absolute values of the scaling image accord-ing to formula (14) when the values of D are 4 and 8. Toshow the situation in which m high-degree terms appear,enhancement processing was executed for low-intensitycomponents.

2.3. Band splitting of Fresnel transformedsignal

In this section, we will investigate the image analysisalgorithm in the Fresnel transformed signal domain in Fig.1(d) viewed from the original image and in the Fouriertransformed signal domain in Fig. 1(c). First, to investigatethe image analysis algorithm in the Fresnel transformedsignal domain, we will use the Method 1 Fresnel transfor-mation formula in formula (4) to transform o(m, x) informula (14) to the Fresnel transformed signal u(m, x).However, the image analysis uses formula (12) before thevariable transformation is performed:

Formula (21) is formed by multiplying a rectangle windowfunction with width X2 by the function u(x′ − mX2), whichtransitioned u(x) by mX2. Since the transition amount mX2

depends on m, the function u(x′) can be considered as afunction for which the band is split for each X2 (= N∆x).Figure 4 shows the band-splitting situation.

Next, we take the Fourier transform of the signal informula (21) and investigate it in the Fourier transformdomain viewed from the image:

Formula (22) is formed by multiplying the quadratic phasemodulation term, which is centered at –mX2, by the Fouriertransform function of the image O(ωx) and then convolutingthe resulting function according to sinc(X2ωx). Conse-quently, the signal is averaged and information is lost in aregion in which the phase change is at least 2π within themain lobe width of the sinc function, which is 2π/X2. If welet ± ωb be the range having significant information and letωmax be the Nyquist frequency, then ωb can be obtainedaccording to

(21)

Fig. 2. Image analysis algorithm using the FREBASmethod.

Fig. 3. Examples of down-scaled image according tothe FREBAS method.

(22)

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From formulas (22) and (24), o(m, x) is centered at− mωmax / D in the frequency domain, and the bandwidth isroughly ωmax / D. In other words, since the frequency bandis divided by D in the frequency domain, it is clearlydistributed so that it mostly corresponds to the regionoccupied by o(m, x) in image domain 1. Figure 5 is anexample showing a decomposed sample image and thecorresponding frequency distribution when D = 8. It is clearthat the regions in which the decomposed image exists ineach domain almost correspond to the positions of thebands in the frequency domain.

3. Image Processing Applications

3.1. Signal energy concentration computation

Suppose that the proposed method is used for bandsplitting the image in the Fresnel transform domain andreconstructing the image in each band. Since the image thatappears in Fig. 1(f) at this time represents the edge portionof the image for m > 1, its energy concentration is expectedto be greater than the spectral amplitude that exists in thesame frequency band due to the Fourier transform [signalFig. 1(b)] of the original image. The property that theenergy of the signal is concentrated in a specific portion inthis way is similar to a wavelet-based multiresolution analy-sis. For signals that include noise, wavelet analysis is oftenapplied in image processing, such as image sharpening orSNR improvement processing, by suppressing noise wherenoise is dominant while maintaining or performing suitableprocessing of signals with relatively large amplitudes.Therefore, we decided to compare the energy concentrationin each band of the image analysis domain according to theFREBAS method.

We decided to use several sets of image data here andto compare them by using the FREBAS method, wavelets,and Fourier transformed signals. For the FREBAS method,we set D = 8 for the scaling parameter and performed thecomputations in each rectangular band having a width ofX2 (= N∆x2) in image domain 1, which is the image analysisdomain. The bands are denoted as bandfr0 to bandfr4 fromthe center. For wavelets, we set Daubechies’ N = 6 and letthe image scale with the lowest resolution be level 3, whichmatches the FREBAS method [9], and performed the com-putations in the multiresolution analysis domain of theimage. For wavelets, since a binary decomposition of thefrequency band is performed for the image from the high-frequency components to the low-frequency componentsand the frequency bandwidth differs for each level, thenumber of data values in each region (level) differs fromthe FREBAS method, in which the frequency band is nearlyequally partitioned. Therefore, we calculated the averageamplitude of the signals having amplitudes in the upper 5%

Fig. 4. Band splitting of Fresnel transformed signalaccording to the FREBAS method.

(23)

(24)

Fig. 5. Fresnel transformed band decomposed imageand its frequency distribution according to

the FREBAS method.

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in each region (level) and the energy concentration occu-pied in each region (level). Although the concept of levelsdoes not exist with Fourier transformed signals, we subdi-vided the frequency band by using the frequency bandwidth|ωx| < 2cX2, according to which the band is nearly equallypartitioned in the FREBAS method, and assigned band0 toband4 to the subdivisions, starting from the low-frequencyside. For the image data, we used Lena and mandrill fromthe SIDBA standard images [10] and the Shepp–Loganphantom [11], which is known as an X-ray CT standardnumerical phantom. All image data are 256 × 256 pixel data.Table 1 shows the results; to simplify the comparisons ofthe energy concentrations in the same bands as the Fouriertransformed domain signals, the width of each column is

drawn so that it is roughly proportional to the frequencybandwidth of each level or band. Among these three ana-lytical methods, the Fourier transformed signals have highenergy concentrations in the low-frequency region, and thevalues are low in the middle- and high-frequency regions.However, with both the wavelet analysis and FREBASmethod, the energy concentrations are low in the low-fre-quency region, and these values get larger in the high-fre-quency regions. When we compared the FREBAS methodand wavelet analysis method, we found that the waveletanalysis method had higher energy concentrations in thehigh-frequency regions according to this evaluationmethod. This evaluation showed that the FREBAS methodhas the characteristic that the signal energy is concentrated

Table 1. Average amplitude and energy concentration of the signals with highest amplitudes in the image analysis domain

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in a specific portion as in a wavelet-based multiresolutionanalysis.

3.2. Overview of computational processingprocedure

This section describes the processing procedure usedfor the actual computations.

[Step 1] Fourier transform

Steps 1 to 3 comprise processing for transforming theimage to a Fresnel transformed signal by using the reversealgorithm of Method 2. First, compute the Fourier trans-form of the image data according to an FFT (DFT). [Thisis the processing shown in Fig. 1(a) → (b).]

[Step 2] Quadratic phase modulation processing

Assign the parameter c (or D) and multiply by thequadratic phase modulation and the constant (πD / c~)exp(–jπ/2)expj(Ωx

2 + ωy2)(4c / D) according to formula (8). If

we let (i, j) be the indices of two-dimensional data and makethe quadratic phase modulation coefficient dimensionless,the quadratic phase modulation term can be concisely rep-resented as expj(Dπ / N)(i2 + j2). [This is the processingshown in Fig. 1(b) → (c).]

[Step 3] Inverse Fourier transform

Transform to a Fresnel transformed signal accordingto an inverse FFT (DFT) transform. [This is the processingshown in Fig. 1(c) → (d).]

[Step 4] Quadratic phase modulation processing

Steps 4 and 5 comprise image reconstruction proc-essing according to the algorithm of Method 1. Multiply by(c / πD)exp(−j(c / D)(xg 2 + yg 2)) according to formula (4).This quadratic phase modulation term can be representedas exp−j(π / (DN))(i2 + j2). [This is the processing shownin Fig. 1(d) → (e).]

[Step 5] Fourier transform

Compute the Fourier transform according to an FFT(DFT). If the signal intensity is considered a problem andthe phase is not considered a problem in the subsequentfiltering processing, the phase modulation processingexpj(Ωx

2 + Ωy2) / (4c / D) in formula (4) need not be exe-

cuted. [This is the processing shown in Fig. 1(e) → (f).]

[Step 6] Image processing

Perform processing for the band-split signal.

[Steps 7 to 11] Inverse transform of Steps 1 to 6

Hereafter, perform the inverse transforms of Steps 1to 5 to return to the original image domain 2.

3.3. Image sharpening processing

The FREBAS method has the characteristic that thesignal energy is concentrated in a specific portion of thesignal domain while the frequency band of the image ispartitioned as in a wavelet analysis. Therefore, we executedimage sharpening as an example in which different proc-essing is executed in each band while suppressing noise.For the image data, we added white noise having a Gaussiandistribution to the image data that was used in Section 3.2.For the amount of noise, we assigned noise having a 2%standard deviation relative to 255, which is the maximumamplitude of the image. The high-frequency components ofthe signal are increased because of sharpening processing.Therefore, for the wavelet analysis, we increased the level0 region signal to triple its value. For the FREBAS method,to perform processing that produces nearly the same effectas the processing for increasing the wavelet level 0 signal,we increased the band-4 and band-3 signal to triple its valueand the band-2 signal to double its value. However, weperformed threshold value processing to prevent amplifica-tion of the noise component. The white noise in an imageanalysis that uses wavelets is also white noise in the wavelettransform domain, and its statistical properties do notchange. Also, although the FREBAS method is nonlinearprocessing that includes quadratic phase modulation proc-essing, since an alias is generated and noise is returnedaccording to the discrete Fourier transform, the frequencyspectrum of the white noise can also be regarded as whitenoise in image domain 1, which is the reconstructed imagedomain. We let σn be the standard deviation of the noise andlet 3σn be the threshold value for both methods. Also,although the FREBAS method uses complex operations,we evaluated the real part of the sharpened image data asthe output image.

We took the difference IenhNC − Ienh of the image Ienh,which was obtained by sharpening the original noiselessimage, and the sharpened image IenhNC, which was pro-duced with accompanying threshold processing, and calcu-lated the standard deviation σ(IenhNC − Ienh) of that entireimage. The value of σ(IenhNC − Ienh) is an index representingthe sum of the image degradations due to residual noise andthreshold value processing. We also calculated the standarddeviation σ(IenhN − Ienh) of the difference IenhN − Ienh of thesharpened image of the original noiseless image and IenhN,which is the image that was sharpened without denoising.Table 2 shows the results. The value of IenhN − Ienh indicatesthe degree to which the noise increases. However, sincenearly the same values were obtained by the two methods

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that were compared, the bandwidths of the high-frequencysignal for which enhancement processing was performedfor sharpening can be considered to be nearly equal. Whenwe look at σ(IenhNC − Ienh) for the sharpened image forwhich threshold processing was executed, the results are thesame for Lena, but better results are obtained for the imageprocessed according to the FREBAS method for mandrill.Figure 6 shows magnifications of the mandrill image. (a) isthe noiseless image before processing, (b) and (c) are im-ages for which sharpening was executed without denoisingby using wavelet analysis and the FREBAS method, respec-tively, and (d) and (e) are sharpened images for whichthreshold value processing was executed according to eachof the methods, respectively. Although sharpening wasachieved in the images of both (b) and (c), wavelet analysis

achieved a greater degree of sharpening. For the sharpenedimages for which threshold value processing was executedin (d) and (e), the method using wavelet analysis had betterdenoising characteristics in regions with small image am-plitude variation. As a result, for image data with small-amplitude variation such as the Shepp–Logan phantommodel, the σ(IenhNC − Ienh) value is smaller, and the resultswere extremely good. On the other hand, for images havingvariation in the image amplitude, the error due to thresholdprocessing appears as a large spike of noise in the image.However, for the sharpened image for which thresholdprocessing was executed according to the FREBASmethod, sharpening is performed without generating obvi-ous noise. Therefore, when averaging is performed formandrill with its large image amplitude variation, the FRE-BAS method provides better results compared with idealprocessing results. With Lena, since the region with smallimage amplitude variation is larger than with mandrill, theσ(IenhNC − Ienh) values are equal.

4. Examination

The FREBAS method is an image analysis methodthat is similar to wavelet analysis in that band splitting isperformed for the signal, image analysis is performed indomains where the image is significant, and image position

Table 2. Results of image sharpening using waveletanalysis and the FREBAS method

Fig. 6. Image sharpening processing: (a) original image (mandrill); (b) sharpened image using wavelet analysis withoutdenoising; (c) sharpened image using the FREBAS method without denoising; (d) sharpened image using

wavelet analysis with denoising; (e) sharpened image using the FREBAS method with denoising.

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information is saved. These methods differ in that sincewavelet analysis hierarchically processes the binary de-composition of the frequency band, the upper levels do notvary even when the decomposition level gets deeper. How-ever, with the FREBAS method, the distribution of all bandsvaries according to the number of band subdivisions D. Asa result, although the computation time varies according tothe level for wavelet analysis, the computation time doesnot depend on D for the FREBAS method. One type ofmultiresolution image analysis that is similar to waveletanalysis is the multidimensional median transform [6].Since this includes a convolution operation, the computa-tion time is generally longer. With the FREBAS method,Fourier transform processing is performed six times andquadratic phase modulation processing is performed fourtimes. However, if a fast Fourier transform can be used, theamount of computation will not be as great as for a methodthat uses a convolution integral. In this paper, we used ahigh-speed algorithm called the fast wavelet transform [12]as a discrete wavelet transform. When image sharpeningcomputations were performed according to this wavelettransform processing using Daubechies level 3 on a com-puter with a 600-MHz Pentium III processor, approxi-mately 1.0 second was required. Since this requiredapproximately 1.9 seconds with the FREBAS method, theamount of computation is tolerable for the practical appli-cation of a method that requires more execution time thanthe fast wavelet transform.

Some distinctive characteristics of the FREBASmethod are that different processing can be performed ineach band, in a similar manner as for wavelet image analy-sis, and that the regions in which the image and noise aredominant, respectively, are clearly separated in the imageanalysis domain. Therefore, as an example of image proc-essing, we executed image enhancement processing forenhancing a specific frequency band while suppressingnoise in order to investigate the possibility of performingimage processing similar to wavelet analysis. The resultwas that image sharpening was achieved while suppressingnoise amplification, due to threshold processing. This resultindicated that the FREBAS method can be applied to imageprocessing fields where the main signal components aremanipulated while suppressing noise in a similar manner aswavelet image analysis. Wavelet image analysis is used foran extremely wide range of applications such as imagedenoising using signal degeneracy, image enhancement andoutline extraction using multiresolution analysis, and im-age compression using the property that energy is concen-trated in specific coefficients. Since the FREBAS methodcan perform similar processing as wavelet analysis forperforming multiresolution analysis of an image whilemaintaining local information of the image, the FREBASmethod can be applied in some of the fields where imageprocessing applications use wavelet analysis. In particular,

the FREBAS method can achieve superior image process-ing for noise processing because it does not generate obvi-ous noise in the image when threshold processing isperformed for denoising.

5. Conclusions

When image down-scaling is performed by using thetwo algorithms for solving the discrete Fresnel transformformula, the alias signal that is produced when computingthe Fresnel transformed signal is separated in the recon-structed image domain, and equivalent band splitting of theFresnel transformed signal is performed in the image do-main.

This band splitting of the Fresnel transformed signaldomain can be interpreted as image data being analyzed byconvoluting quadratic phase modulated image data withsinc functions having different modulation indices. TheFREBAS method can perform band splitting in domainswhere the image is significant so that the image can beanalyzed while saving image position information. Also,since it has the characteristic that signal energy is concen-trated in a specific portion, its analysis method is consideredsimilar to a wavelet analysis. However, a difference be-tween the FREBAS method and wavelet analysis is that thecomputation time differs according to the level in waveletanalysis because the band is split with equal spacing in theFresnel transformed signal domain. However, with theFREBAS method, the computation time does not varyaccording to the number of subdivisions.

We used the fact that the FREBAS method has char-acteristics similar to those of wavelet analysis and per-formed image sharpening processing for band-split signals.We performed image analysis for noise-contaminated im-ages by enhancing the high-frequency signals greater thanor equal to a given threshold value and denoising signalsfor data below the threshold value. The result was thatsharpening was achieved while suppressing the magnifica-tion of noise in a similar manner as when wavelet analysisis used. In the future, we plan to apply the FREBAS methodto various types of image processing to improve the imageSNR by using the proposed method.

Acknowledgments. The authors are grateful to As-sistant Professor Yoshitsugu Kamimura of the Departmentof Information Engineering in the Engineering School ofUtsunomiya University, as well to postgraduate MasajiMori and graduates Takashi Isaka and Masaki Iizuka ofUtsunomiya University for their cooperation during thecourse of this research.

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AUTHORS (from left to right)

Satoshi Ito (member) graduated from the Department of Electrical Engineering of Utsunomiya University in 1987,completed his postgraduate program in 1989, and joined Toshiba, Inc. He has been a research associate in the Department ofInformation Engineering of Utsunomiya University since 1994. His research interests are NMR imaging techniques using thesimilarity in equation of NMR signal and wave-front attraction as well as NMR image processing techniques. He received anEncouragement Award from the Japanese Society of Medical Imaging Technology, and is a member of the Japanese Society forMedical and Biological Engineering, the Japan Society for Magnetic Resonance in Medicine, and the Optical Society of Japan.

Yoshifumi Yamada (member) completed the doctoral program at Tohoku University in 1971 and became a researchassociate in the School of Engineering Research. After serving as a lecturer at Hokkaido University and an assistant professorat Utsunomiya University, he has been a professor in the Department of Information Engineering of Utsunomiya Universitysince 1990. His research interests are medical information processing, new types of magnetic resonance imaging technology,and optical image reconstruction of magnetic resonance images. He holds a D.Eng. degree, and is a member of the JapaneseSociety of Medical and Biological Engineering, the Society of Instrument and Control Engineers, the Japan Society for MagneticResonance in Medicine, IEEE, and SMRM.

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