description of the transfer function of an optical system with wavelet transforms

Description of the transfer function of an opticalsystem with wavelet transforms

Liying Tan, Jing Ma, and Qi Wang

According to the wavefront filtering idea of wavelet optics, the transfer function of an optical systemis described with a wavelet scale function. In the transfer function described with a wavelet scalefunction, different scale parameters a,c and shift parameters b,d correspond to different subtransferfunctions, which correspond to different situations of the optical system. According to the request ofthe optical system, by adjusting all these scale parameters, not only can we obtain the optical imagesunder different conditions, but we can also obtain the singular points under this scale parameter;hence a more ideal output can be obtained by such processing. The transfer function described witha wavelet scale function can be adjusted according to the request of the optical system, which makesthe described transfer function self-adjustable. According to all types of disturbing effects to thesystem, by adjusting the scale and shift parameters, the practical form of the transfer function of anoptical system can be confirmed, which satisfies the request of the self-adjustability of the opticalimaging system. The result of our analysis shows that describing the transfer function of an opticalsystem with a wavelet scale function is not only feasible but also satisfies the request of the self-adjustability of the optical imaging system, and different optical systems can be described by differentwavelet scale parameters. This work breaks from the formal additional describing mode of thetransfer function of an optical system and makes description of the transfer function of an opticalsystem convenient. © 2006 Optical Society of America

OCIS codes: 100.7410, 110.4850.

1. Introduction

In the past few years, researchers have tried toimplement wavelet transforms with optical sys-tems. Szu et al. proposed implementing aone-dimensional wavelet transform by a two-dimensional correlator.1,2 Wang et al. realized aHarr wavelet transform by using the multiplex an-gle method with the volume holographic techniqueand used it in image feature extraction.3 Further-more, there are also projects to realize wavelettransforms with interferometers. Cho et al. dis-cussed the photoelectric difference-of-Gaussianwavelet transform system.4 Qiu accomplished aGabor-type matrix algebra and fast computations ofdual and tight Gabor wavelets.5 Soon et al. im-

proved feature extractions by using a joint wavelettransform correlator.6 Shen et al. discussed the re-lation between monochromatic electromagneticwavelets and the Huygens principle.7 These studieshave established the experimental foundations ofwavelet optics theory. In this paper, according tothe wavefront filtering idea of wavelet optics, weconsider an optical system to be a filter,8,9 and thelight wave is filtered when it passes though theoptical system. After being filtered, the weights ofthe optical field are redistributed, and we import ascale wavelet function to describe the point-spreadfunction of the optical system. Different optical sys-tems can be described by different scale waveletfunctions. In the transfer function described withthe wavelet scale function, different scale parame-ters a,c and shift parameters b,d correspond to dif-ferent subtransfer functions that correspond todifferent situations of the optical system. Accordingto the request of the optical system, by adjusting allthese scale parameters, not only can we obtain theoptical images under different conditions, but wecan also obtain the singular points under this scaleparameter; hence a more ideal output can be ob-tained by such processing. The transfer function

The authors are with the National Key Laboratory of TunableLaser Technology, Harbin Institute of Technology, 92 West DazhiStreet, Harbin 150001, China. L. Tan’s e-mail address is [email protected].

Received 7 March 2005; revised 29 August 2005; accepted 24October 2005; posted 8 December 2005 (Doc. ID 59946).

0003-6935/06/143275-08$15.00/0© 2006 Optical Society of America

10 May 2006 � Vol. 45, No. 14 � APPLIED OPTICS 3275

described with the wavelet scale function can beadjusted according to the request of the optical sys-tem, which makes the described transfer functionself-adjustable. According to all types of disturbingeffects to the system, adjusting the scale and shiftparameters allows the practical form of the transferfunction of an optical system to be confirmed, andthis satisfies the request of the self-adjustability ofthe optical imaging system. The result of our anal-ysis shows that describing the transfer function ofan optical system with the wavelet scale function isnot only feasible but also satisfies the request of theself-adjustability of the optical imaging system, anddifferent optical systems to be described by differ-ent wavelet scale parameters. This work breaksfrom the formal additional describing mode of thetransfer function of an optical system and makesdescription of the transfer function of an opticalsystem convenient.

2. Point-Spread Function

As for an ideal optical imaging system, each pointon the object plane corresponds to a point on theimage plane. The quality of an image will deterio-rate if a point on the object plane cannot be imagedcorrectly to a point on the image plane. It is notedthat a point object of an optical system is describedby a point-spread function. Commonly, the concreteform of the point-spread function is decided by theoptical system, and different optical systems havedifferent point-spread functions. According to thewavefront filtering idea of wavelet optics, we canconsider that the wavefront of a light wave is fil-tered when it passes through an optical system, andthe distribution of the light field on the image planeis the redistribution of the light field after filtering.The light amplitude on the object plane is U1�x1, y1�and that on the image plane is Ui�xi, yi�. The propa-gation circumstance after filtering may be consid-ered as the weight of each point is redistributed.Here we introduce the wavelet scale function ��x, y�as the weight function. That is,

Ui�xi, yi� ��

�

��xi, yi; x0, y0�U0�x0, y0�dx0dy0,

where ��xi, yi; x0, y0� is the pulse response function ofan optical system. So the scale wavelet function��x, y� can be used to describe the point-spread func-tion of an optical system, and the scale wavelet func-tion satisfies the normalizing condition

��x, y�dxdy � 1. (1)

Hence we need to find the wavelet scale function todescribe the point-spread function according to thepractical conditions of the optical systems. Accordingto the characteristics of scale functions, a unique co-

efficient sequence �hn; n � Z� � l2�Z� must exist,which satisfies

��x, y� � �2 �n�Z

hn��2x � n, 2y � n�, (2)

where l2�Z� means that l2�Z� � ��xn; n � Z�;�n�Z �xn�2 � ��, n is an integer number, and Z is theset of total integer numbers. Commonly, Eq. (2) iscalled the scale equation. In fact, the calculating for-mula of this coefficient sequence is

hn � �2�R

��x, y�� 2x � n, 2y � n�dxdy, (3)

where R is the field of real numbers and �� is thecomplex conjugate quantity of the scale wavelet func-tion ��x, y�. If we perform an operation of the Fouriertransform to both sides of Eq. (2), we obtain

�(fx, fy) � �k��

� h(k)

�2�fx

2,fy

2exp(�j�k�2)

� �fx

2,fy

2 �k��

� h(k)

�2exp(�j�k�2)

� Hfx

2,fy

2�fx

2,fy

2, (4)

where ��fx, fy� is the Fourier transform of the scalewavelet function ��x, y�, and fx, fy is the spatial fre-quency of direction x and y. In Eq. (4)

Hfx

2,fy

2� �k��

� h�k��2

exp��j�k�2� or

H�fx, fy� � �k��

� h�k��2

exp��j�k�. (5)

The coefficient h(k) is the scale coefficient, which rep-resents the characteristics of the point-spread func-tion (the pulse response function) of the opticalsystem.

3. Description of the Transfer Function of an OpticalSystem with Wavelet Transforms

A common optical imaging system is not made of asingle lens but is made of two or more lenses. Thereare negative lenses as well as positive lenses. Tomake the description of the transfer function of anoptical system with wavelet transforms simple, wecan consider the optical image system as a black box.The object wave enters the black box through theentrance pupil and goes out of the black box throughthe emergent pupil. The wavefront of the object lightwave is filtered by the black box. After the light wavepasses the optical system, it can be considered to befiltered by the optical system; that is, the wavefront ofthe light wave is filtered. As for the lens, we can

3276 APPLIED OPTICS � Vol. 45, No. 14 � 10 May 2006

consider that light distribution U0�x0, y0� on the objectplane consists of countless small-area elementsU0�x0, y0�dx0dy0, and each area element can be re-garded as a weighted wavelet function. Concerning alens or an image system, we need to know the lightfield distribution on the image plane clearly after thelight field distribution of any small-area elements onthe object plane passes through the optical system. Sowe can consider that the wavefront of the light waveis filtered by the optical system when passingthrough, and, after being filtered, the weight of theoptical field is redistributed. We still apply the scalewavelet function to describe the weight factor in Eq.(2).

We suppose that the input light field is U1�x, y�, andthe point-spread function of the system is describedby a scale wavelet function. After passing through thelens, the light field distribution U(x, y) at point (x, y)(the output of the system) is

U�x, y� ��

�

U1�x1, y1��x � x1, y � y1�dx1dy1.

(6)

In Eq. (6), U1�x, y� is the unlimited light field dis-tribution function. Also,

U�x, y� � U1�x, y� � ��x, y�. (7)

After Fourier transforming Eq. (7), we obtain

U�fx, fy� � U1�fx, fy��fx, fy�

� U1�fx, fy��fx

2,fy

2Hfx

2,fy

2. (8)

So after an input light field passes through an opticalsystem, the output is equivalent to the convolutionproduct of input U1�x, y� and the point-spread func-tion ��x, y� [or transfer function H�fx, fy�]. When thescale factor a in the scale wavelet function �a,b�x, y� ischanged, �a,b�x, y� will function as a band filter, whichallows a series of bandpasses with different centralfrequencies. So we can treat an optical system as aband filter system, and it has several properties asfollows:

(1) The output is depicted by the wavelet trans-form of the input U1�x, y� [namely, the functionU1�x, y� is filtered by a series of band filters]. Thebandwidth and central frequency of the band filterare decided by the scale factor a. The shift factor b isthe spatial coordinate parameter of the output afterfiltering.

(2) The bandwidth and the central frequency ofthe band filter change with the scale factor a. Whena decreases, the central frequency increases and thebandwidth widens. On the contrary, when a in-creases, the central frequency is minimized and thebandwidth becomes narrow.

(3) Therefore the series of band filters formedwith the changing of the scale factor a are all constantQ filters. Q is the quality factor of the optical system,which is equal to the ratio of the central spatial fre-quency and the spatial-frequency bandwidth of theoptical system.

So analyzing the local characteristic of the distri-bution of the light field is valuable when a transferfunction of an optical system is described with awavelet transform. When the change rate of the sig-nal is slow, it implies that the signal mainly consistsof low frequency. At this time, the optical systemdescribed with the wavelet transform corresponds tothe condition when a is large. In contrast, when thechange rate of the signal is fast, it implies that thesignal mainly consists of high frequency. Here theoptical system corresponds to the condition when a issmall. So while the scale factor a changes from largeto small, the filtering range of the optical systemchanges from low frequency to high frequency. Thischaracteristic reflects some zooming characteristicsof the optical system.

In Eq. (7) the pulse response function can be givenin the form of the wavelet scale function. The Gauss-ian linear frequency-modulated complex wavelet isintroduced as the point-spread function

�a,b;c,d�x, y� � exp��12b � x

a 2�exp��12d � y

a 2� exp�j2 fxb � fyd � C�fxb

2 � fyd2��.

Then the relationship between the input and outputis

U2�x2, x2� ��

�

U1�x1, y1��x1, y1; x2, y2�dx1dy1

��

�

U1�x1, y1�exp��12x2 � x1

�x 2� exp��

12y2 � y1

�y 2�exp�j2 fxx2 � fyy2

� C�fxx22 � fyy2

2��dx1dy1. (9)

Function U2�x2, x2� is the output. Then the Fouriertransform to ��x1, y1; x2, y2� in Eq. (9) and the com-mon form of the transfer function of the system can beobtained:

��fx, fy� ��

��

exp��12b � x

a 2�exp��12d � y

c 2� exp�j 4�fxb � fyd�� 2C�fxb

2 � fyd2��dxdy. (10)

Generally, the transfer function of the system ischanged with time by the effect of all types of per-


turbation. To satisfy the request of real-time pro-cessing of the self-adjustable optical imagingsystem, these changes must be estimated in realtime, and the instantaneous transfer function afterthese changes should be confirmed in real time.

According to the scale and shift characters of awavelet, if the scale and shift parameters arechanged properly, the adjusting of the transferfunction can be realized and different outputs canbe achieved. The effects of the changes of the scaleparameters a,c and shift parameters b,d to thetransfer function of the system are further analyzedin Subsections 3.A–3.C. The analysis shows the ef-fects of adjusting the transfer function of the systemby changing scale parameters a,c and shift param-eters b,d and gives the corresponding simulationresults.

A. Analysis of the Effects of Scale Parameters a,c to theTransfer Function of the System When Shift Parametersb,d are Fixed

In the example of the effects of adjusting the transferfunction of the system by changing scale parametersa,c, the scale parameters a and c are selected to beinfinitude. The transfer function is obtained in thefollowing form:

��fx, fy� ��

��

exp�j 4�fxb � fyd�

� 2C�fxb2 � fyd

2��dxdy. (11)

It can be seen that the transfer function of theself-adjustable optical imaging system is the transferfunction described by the Fourier transform in thiscondition except for the quadratic phase factor, so theoutputs have little difference. The outputs underthese two conditions are given in Fig. 1.

From the results it can be seen that differentscale parameters a,c correspond to different sub-transfer functions, which correspond to differentconditions of the optical system, so that the outputimages are different. According to the request of theoptical system, by adjusting the scale parametersa,c, the requested output images can be obtained.These show the adjustability of the transfer func-tion under the wavelet description. The transferfunction under the wavelet description can be ad-justed according to the request of the system, whichmakes the described transfer function be self-adjustable.

Here the output image of the system under thewavelet description when the scale parameters a andc are selected to be infinitude is consistent with theoutput image under the Fourier description.

B. Analysis of the Effects of Shift Parameters b,d to theTransfer Function of the System When Scale Parametersa,c are Fixed

In the example of the effects of adjusting the transferfunction of the system by changing shift parametersb,d, the shift parameters are selected to be b � d� 0. The transfer function is obtained in the following

Fig. 1. (A) Input image. (B) Transfer function when scale parameters a and c are selected to be infinitude. (C) Output image of the systemunder a Fourier description. (D) Output image of the system under the wavelet description (a � c � �).


form:

��fx, fy� ��

��

exp��12x2

a2 �y2

c2�dxdy. (12)

It can be seen that the transfer function of theself-adjustable optical imaging system equals anadjustable Gaussian window in this condition;that is, it is equal to a Gaussian transform of theimage. The output under this condition is given inFig. 2.

In Fig. 2(B) it can be seen that the output imageunder the wavelet description �b � d � 0� containsthe intensity singular points of the input image. Itcan be considered as the profile of the input image.This well satisfies the character of the Gaussiantransform.

From the results it can be seen that the differentshift parameters b,d correspond to different sub-transfer functions, which correspond to different con-ditions of the optical system, so that the outputimages are also different. According to the request ofthe optical system, by adjusting the shift parametersb,d, the intensity singular points of the input imagecan be obtained. Consequently, the ideal image canbe obtained. We note that the described transfer func-tion is self-adjustable.

C. Analysis of the Effects of the Changes of Both theScale Parameters a,c and the Shift Parameters b,d tothe Transfer Function of the System

In the transfer function of Eq. (11), the core functionof the wavelet transform is considered to be composedby a three core function. The core function

exp��12b � x

a 2�exp��12d � y

c 2�corresponds to a Gaussian wavelet function. It isequal to a Gaussian wavelet window of the opticalimaging system. And exp j2�fxb � fyd�� is a Fourier-transform core function. Here exp j2C�fxb

2 � fyd2��

is linear demodulation core function, and it expressesthe quadratic phase factor. It reflects the modulationeffects of the lens and the different propagating dis-tances on light propagation to the phase. Where C isa constant, it is decided by a characteristic of theoptical system.

From Eq. (11), by adjusting the scale and shiftparameters, the practical form of the transfer func-tion of an optical system can be confirmed accordingto all types of perturbation effects to the system, sothat all types of different outputs can be obtained.This satisfies the request of the self-adjustability ofthe optical imaging system. The output and transferfunction in terms of two types of parameters aregiven in Fig. 3.

It is emphasized again that different scale param-eters a,c and shift parameters b,d correspond todifferent subtransfer functions, which correspondto different conditions of the optical system, so thatdifferent output images can be obtained. Accordingto the request of the optical system, by adjusting theshift parameters b,d, the image’s singular pointsunder this scale parameter can be obtained, and byadjusting the scale parameters a,c, different sizeimages can be obtained. It is shown that, by adjust-ing all these parameters, not only can we obtain theoptical images under different conditions, but wecan also obtain the singular points under this con-dition, which makes the described transfer functionself-adjustable. From the preceding discussion, itcan be seen that, according to the request of theoptical system, the transfer function under thewavelet description is flexible and adjustable,which satisfies the request of the optical imagingsystem well beyond the Fourier transform. Espe-cially in a photoelectric mix processing system, byadjusting the software, a satisfactory image or out-put can be obtained.

The above results show that describing the transferfunction of an optical system with the wavelet scalefunction is not only feasible but also satisfies therequest of the self-adjustability of the optical imagingsystem. Because the wavelet transform is more effec-tive than the Fourier transform in dealing with some

Fig. 2. (A) Transfer function when shift parameters b � d � 0. (B) Output image of the system under the wavelet description (b � d �0).


abrupt changes, describing an optical transfer func-tion with the wavelet transform is more suitable forpractical situations of an optical system than theFourier transform.

4. Chromatic Light Illumination

Under the conditions of chromatic light illumination,the form of the wavelet function will be different.Because the amplitude and phase of each object pointchange randomly with time, the statistical relation-ship among the distribution of the light field at eachpoint on the object plane will directly influence thestatistical relationship among the correspondingweight functions on the image plane. This will affectthe final calculation result, which is determined bythe average light intensity of the image with respectto time. To simplify the process, we consider only twosituations as follows:

(1) The changing mode of the light field at eachpoint on the object plane with respect to time is con-sistent. In this condition, although each two points onthe object plane have a different fixed phase differ-ence, the respective changing mode of the absolutephase with respect to time is the same. This situationis equivalent to illumination with spatial coherentlight.

(2) The changing mode of each point’s light fieldon the object plane with time is not the same, and thelight field has a statistical independence between thepoints. This situation is illumination with spatial in-coherent light.

Here the distribution of the light field on the imageplane is

Ui(xi, yi, t) ��

�

Ug(x�0, y�0, t)�(xi � x�0, yi

� y�0)dx�0dy�0. (13)

Here ��xi � x�0, yi � y�0� is also a wavelet scale func-tion. Equation (13) represents the point-spread func-tion at point �xi, yi� on the image plane. The lightintensity distribution of the image with respect totime is

Ii�xi, yi� � �Ui�xi, yi; t�Ui*�xi, yi; t��. (14)

We define

F�� p, q�, (15)

where ��p, q� is the Fourier transform of the weightfunction (wavelet basis function) at point �xi, yi� onthe image plane, which is the transfer function of theoptical system.

According to the practical condition, the pulse re-sponse function is described by the wavelet functionin terms of the Gaussian linear frequency-modulatedcomplex wavelet. Then the output light intensity dis-tribution is

Fig. 3. (A) Transfer function when the parameters are a � c � b � d � 2.3. (B) Output image of the system with the wavelet descriptiona � c � b � d � 2.3. (C) Transfer function when the shift parameters are a � c � b � d � 2.4. (D) Output image of the system with thewavelet description a � c � b � d � 2.4.


I�x2, y2� � ��

�

�U1�x1, y1��2

��x1, y; x2, y2��2dx1dy1

��

�

�U1�x1, y1��2�exp��12x2 � x1

�x 2� exp��

12y2 � y1

�y 2�exp j2�fxx2 � fyy2��

exp j2C�fxx22 � fyy2

2��2

dx1dy1

� exp j4C�fxx22 � fyy2

2��

��

�

�U1�x1, y1��2

exp��x2 � x1

�x 2� exp��y2 � y1

�y 2� exp j4�fxx2 � fyy2��dx1dy1. (16)

Compared with the light intensity distribution de-scribed by the Fourier transform,

I�x2, y2� ��

�

�U1�x1, y1��2

�h�x2 � x1, y2 � y1��2dx1dy1

��

�

�U1�x1, y1��2

exp j4�fxx2 � fyy2��dx1dy1. (17)

We can see that Eq. (16) has an excess term

exp��x2 � x1

�x 2�exp��y2 � y1

�y 2�more than the Fourier description of the image in-tensity distribution, which corresponds to a Gaussianwavelet function. It represents a Gaussian waveletwindow of the optical imaging system.

Under the conditions of illuminating with coherentlight, the analysis method is the same. According tothe scale and shift property of a wavelet, if the scaleand shift parameters are changed properly, the ad-justing of the output intensity distribution can berealized and a different output can be achieved. Theeffects of the changes of the scale parameters a,c andshift parameters b,d to the output intensity distribu-tion of the system are further analyzed in Subsec-tions 4.A–4.C.

A. Analysis of the Effects of Scale Parameters a,c to theOutput Intensity Distribution of the System When the ShiftParameters are Fixed

In the example of the effects of adjusting the transferfunction of the system by changing scale parametersa,c, the scale parameters a and c are also selected tobe infinitude. The output intensity distribution is ob-tained in the following form:

��fx, fy� ��

��

exp�j 4�fxb � fyd�

� 2C�fxb2 � fyd

2��dxdy. (18)

It can be seen that the output intensity distribution ofthe self-adjustable optical imaging system is equiva-lent to the output intensity distribution described bythe Fourier transform in this condition. It can be seenthat different scale parameters a,c correspond to adifferent output intensity distribution. According tothe request of the optical system, by adjusting thescale parameters a,c, the requested output intensitydistribution can be obtained. These show the adjust-ability of the transfer function under the waveletdescription. The transfer function under the waveletdescription can be adjusted according to the requestof the system, which makes the described transferfunction self-adjustable.

B. Analysis of the Effects of Shift Parameters b,d to theOutput Intensity Distribution of the System When theScale Parameters are Fixed

In the example of the effects of adjusting the outputintensity distribution of the system by changing shiftparameters b,d, the shift parameters are also se-lected to be b � d � 0. The output intensity distri-bution is obtained in the following form:

��fx, fy� ��

��

exp��12x2

a2 �y2

c2�dxdy. (19)

It can be seen that the output intensity distributionof the self-adjustable optical imaging system is equalto an adjustable Gaussian window in this condition.The different shift parameters b,d correspond to dif-ferent output intensity distributions. According tothe request of the optical system, by adjusting theshift parameters b,d, the requested output intensitydistribution can be obtained. According to the specialrequest of the optical system, the intensity singularpoints can be found and the ideal image can be ob-tained. This makes the described transfer functionself-adjustable.

C. Analysis of the Effects of the Changes of the ScaleParameters a,c and Shift Parameters b,d to the OutputIntensity Distribution of the System

From Eq. (20), by adjusting the scale and shift pa-rameters in the output intensity distribution of an


optical system according to all types of perturbationeffects, all types of output can be given. The descrip-tion of the transfer function with wavelet transformssatisfies the request of the self-adjustability of theoptical imaging system in the condition of illuminat-ing with coherent light. Different scale parametersa,c and shift parameters b,d correspond to differentoutput intensity distributions.

The above analysis in the condition of illuminatingwith coherent light can also be used in the opticalsystem with all types of perturbation effects. Thetransfer function and output intensity distributiondescribed with wavelet transforms has adjustabilityand flexibility, which satisfies the request of the op-tical imaging system well beyond the Fourier trans-form. Especially in the photoelectric mix processingsystem, by adjusting the software, a satisfactory im-age and output can be obtained.

5. Conclusions

We can see from the discussions here that an opticalsystem can be considered as a filter according to thewavefront filtering idea of wavelet optics. The lightwave is filtered by the optical system, and the weightsof the light field are redistributed after being filtered.For example, the plane wave changes into a conver-gent spherical wave or a divergent spherical waveand so on. We have presented the point-spread func-tion of an optical system with the wavelet scale func-tion. Thus we can describe the transfer function of anoptical system with the wavelet scale function. Dif-ferent scale parameters a,c and shift parameters b,dcorrespond to different subtransfer functions, whichcorrespond to different conditions of the optical sys-tem. According to the request of the optical system, byadjusting all these parameters, not only can we ob-tain the optical images under different conditions,but we can also obtain the singular points under thiscondition. According to all types of perturbation ef-

fects to the system, by adjusting the scale and shiftparameters allows, the practical form of the transferfunction of an optical system to be confirmed. Theresult of our analysis shows that describing the trans-fer function of an optical system with a wavelet scalefunction is not only feasible but also satisfies therequest of the self-adjustability of the optical imagingsystem. This work breaks with the formal additionaldescribing mode of the transfer function of an opticalsystem which makes description of the transfer func-tion of an optical system convenient.

This research was supported by the National Sci-ence Foundation of China (10374023).

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description of the transfer function of an optical system with wavelet transforms

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