Measurement of the Point Spread Function of a Noisy Imaging System.

Transcription

1 Measurement of the Point Sread Function of a Nois Imaging Sstem. Christoher D. Claton, and Richard C. Staunton, * School of Engineering, Universit of Warwick, Coventr, CV4 7AL, UK School of Engineering, Universit of Warwick, Coventr, CV4 7AL, UK * Corresonding author: R.C.Staunton@warwick.ac.uk Citation: C. D. Claton and R. C. Staunton, "Measurement of the oint-sread function of a nois imaging sstem," J. Ot. Soc. Am. A, 9-7 (8) This aer was ublished in the Journal of the Otical Societ of America A, and is made available as an electronic rerint with the ermission of OSA. The aer can be found at the following URL on the OSA website: htt:// Sstematic or multile reroduction or distribution to multile locations via electronic or other means is rohibited and is subject to enalties under law. Abstract: The averaged oint sread function (PSF) estimation of an image acquisition sstem is imortant for man comuter vision alications, including edge detection and deth from defocus. The aer comares several mathematical models of the PSF and resents an imroved measurement technique that enabled sub-iel estimation of -D functions. New methods for noise suression and uneven illumination modelling were incororated. The PSF was comuted from an ensemble of edge sread function measurements. The Generalized Gaussian was shown to be an 8 times better fit to the estimated PSF than the Gaussian and a 4 times better fit than the illbo model. 7 Otical Societ of America OCIS codes:.97,.48,.484, 3.46,.67,.6. INTRODUCTION The research reorted concerns mathematical modelling and ractical measurement of the Point Sread Function (PSF) of focused and defocused image acquisition sstems, such as digital TV cameras. This measure of image blur can be utilised to otimize image rocessing functions such as edge detection [-3] and Deth From Defocus (DFD) deth estimation [4-]. An image acquisition sstem ticall consists of otical comonents (such as lenses and aertures) and electronic comonents (such as the -D CCD arra, anti-aliasing and communication circuits). Each of the comonents in the otical and electronic aths can be considered as satial low-ass filters. Considering the sstem in the terminolog of sstem theor, the transfer functions of each of its comonents can be estimated and then all combined to find the overall transfer function. Alternativel, and the aroach taken in this work, the transfer function for the entire sstem can be measured. Ticall, inut signals are rovided in the form of bar atterns, oint sources or ste edges. For entire sstem measurement, the outut signal is the catured digital image. The Fourier transform of the PSF is the Otical Transfer Function (OTF), and both measures have been widel used to characterize sstems. The lens can be thought of as a -D low-ass filter with a satial cut-off frequenc that is limited b diffraction and aberration effects. The -D arra samles the image and also includes low-ass filtering as the individual sensor elements have a finite area required for low light oeration. Together the elements tile the image lane. As a simle model, charge generated b a hoton at a oint in an element will distribute evenl across the tile []. The samled value is then roortional to the accumulated charge from all the hotons converted in the element during the acquisition hase and since the revious samle was acquired. The CCD is read b a raster scanning rocess []. Here the charge in each of the elements is transferred to vertical columns of shift registers. In turn these shift into a horizontal register and are

2 shifted to an analogue charge-to-voltage converter and then further electronics that rovide, nominall, -D signal rocesses such as low-ass filtering and digitization. The low-ass filtering rovided b the lens is necessar as it acts as an anti-aliasing filter for the image discretization. CCD elements require a relativel large area for the camera to work well at low light levels, and this finite area limits the highfrequenc resonse. PSF measurements can be limited b the Nquist frequenc of the discretization in the acquisition sstem, however an averaging and accumulation rocess was researched to overcome this [,3,4]. This rocess required a knife or ste edge to be imaged, i.e. this is not a general technique for all images. The image was of a blurred ste and several edge-sread functions (ESF) were estimated along the length of the edge. The ESFs were then each registered to a reference oint and accumulated to form a suer-resolution ESF that contained frequenc information above the Nquist limit of the samling grid. The PSF is more useful than the ESF for DFD measurement and image simulation as it can be directl convolved with an inut image to estimate the outut image [,6,9]. The PSF is obtained b differentiating the ESF, however, even low levels of noise in the ESF can result in high levels of noise in the PSF and render it unusable. The source of the noise is in the imaging sstem s sensors and electronics. In this research it was found that both digital and ineensive analogue TV cameras had too oor a signal-to-noise ratio (SNR) for the revious PSF measurement techniques to work reliabl. In this aer we comare research into several methods to both reduce the noise in the ESF estimation and to accuratel model usable PSFs for the acquisition sstem. In Section we review PSF measurement techniques and describe how a suerresolution ESF, and eventuall a PSF is comuted from an ensemble of low-resolution measurements. In Section 3 we eamine the theoretical PSF models that result from consideration of both geometric and diffraction otics. In Section 4 we describe imrovements to the traditional suer-resolution PSF measurement technique that involve:- (i) Comensation for non-uniform illumination within the light bo used to roduce the test images; (ii) A regularized numerical differentiation rocess to limit noise in the comuted PSF; (iii) Models of the ESF that have been develoed and used to comute PSFs that have then been comared with the theoretical models described in Section. Fitting the correct ESF model to the measured data is ke to obtaining accurate PSFs for the sstem. Section resents the eerimental results from both focused and defocused sstems. Secific -D results have been used to demonstrate roblems with noise in the ESF estimation, and bias to the PSF when non-uniform illumination remains uncomensated. Then the results of ESF fitting eeriments have been reorted and discussed. Finall -D PSF lots have been roduced for the most successful fitting methods. Section 6 rovides conclusions. MEASUREMENT TECHNIQUES Here we have roosed PSF measurement of the whole sstem, however methods eist to measure individual PSFs for each comonent. These can then be combined to give the overall PSF. Lasers have been used to measure the PSFs of individual iels within the CCD arra [,6]. These are a function of wavelength, and so a comlete characterization is length and comle. However the PSF for the arra is satiall variant and widel used in image restoration. Classicall, images of sinusoidal gratings have been used for OTF lens measurements. When a normal esf oint samle oint Fig.. Migration of samles onto the ESF focal lane arra (FPA) images the grating the discretization means that the FPA must be moved relative to the image to give minimum and maimum MTF curves [7]. Sinusoidal laser interference atterns have been used b Marchwka and Socker [8], and laser seckle techniques can also be emloed [9]. The main limitation of lasers is the monochromatic light. Satial domain techniques have been used to measure the PSF of the lens. A scanned oint source can be used to obtain a D PSF containing local blur and aberration information u to the Nquist limit of the FPA. A D unit intensit ste, known as a knife-edge can be easil roduced edge

3 eerimentall, using a lightbo. Differentiating the resonse of the lens, the ESF, gives the PSF. The knife-edge technique can be etended to acquisition sstem measurement. If the sstem contains a FPA then undersamling effects cause errors in the PSF estimation due to aliasing. In ractice PSF information beond the Nquist limit of the arra is often required. Reichenbach et al. [3] solved the roblem b using man ESF rofiles to create a suer-resolution image of a D edge. Tzannes and Moone [4] fitted a sum of three Fermi-Dirac functions to the edge to reduce the noise during differentiation to obtain the PSF, and Staunton [] etended the technique to measure the ESF for man differentl angled edges to roduce a D PSF and MTF. At edge angles other than or 9 degrees, a resamling of the data was erformed to obtain the discrete ESFs along normals to the edge. A normal was set and then the closest samles to it rojected on to it from a direction arallel to the edge as shown b the dashed lines in Fig.. The samled values were then not equall saced along the normal, but this r d r v v u u Fig.. Simle model of the otical sstem with the image lane on the left. Magnitude ê µm (a) Magnitude ê µm (b) Fig. 3. Normalized-magnitude PSFs for a 6mm, f4 lens. (a) Focused. (b) Defocused. was irrelevant to the following suer-resolution stage. To understand the suer-resolution stage used in [,3,4], consider an edge oriented close to zero degrees, but not actuall at zero. The edge cuts each iel along its length so that art of the iel is brightl illuminated, and art is dark. The samled value for each iel along the edge is roortional to the averaged illumination throughout the iel. Each samled value along the edge is therefore different. In the same wa each of the man ESFs located at the iels along the edge comrises differentl samled values. These low-resolution ESFs are then registered 3

4 with one another and assembled to form a single high resolution ESF that is resamled onto ticall a. iel grid. The roosed knife-edge technique is simle to erform, but the PSF measurements are averaged along the length of the edge. Such an average is advantageous for shift invariant models used in rocesses such as edge detector testing [], deth from defocus [6], or image simulation, but ma be disadvantageous for rocesses that require models of lens aberrations in addition to sherical aberration, such as ma be required for image restoration []. These ma require sace variant estimation of the PSF. 3 THEORETICAL PSF MODELS A Geometrical Otics Pentland [4] showed that for the simle defocused otical sstem shown in Fig., and assuming geometrical otics, the PSF is a illbo shae with a blur circle radius given b v o u F( v o + u) r =, () fu where v o is the distance between the lens and the CCD, u is the deth of the object, F is the focal length of the lens and f is the f-number, which is defined as f = F/d, where d is the diameter of the aerture. The distance u o is the distance at which an object would aear in focus on the image lane. B Diffraction Aroach The PSF h() of a focused lens that is subject to diffraction effects and with otical aberrations as a function θ(), but neglecting samling due to the FPA, is given b [] as π j ξ jθ ( ξ) λf h( ) = A( ξ) e e dξ () where is a osition vector, A(ξ) is the aerture function, λ is the wavelength of light and F is the focal length of the lens. Out-of-focus blurring can be modelled as a quadratic aberration of the form π θ ( ) = + (3) λ u v F where u is the distance between the object and the lens, and v is the distance between the FPA and the lens. Substituting (3) into (), assuming olchromatic light with equal intensities between wavelengths λ and λ, and the aerture function, A(ξ), to be a circle, radius r, then in D the PSF becomes λ h( ) = e λ F u v F dξ dλ (4) λ + r π j ( ξ ( + ) ξ r Fig. 3(a) shows a PSF eected for a focused sstem where onl diffraction is resent. The olchromatic light is modelled as white light with equal intensit comonents in the range 4 to 7 nm. The PSF looks similar to a Gaussian. Fig. 3(b) shows a PSF for a defocused 6mm lens where the camera is focused at.464m and the oint source is at.8m. The PSF has been flattened out and made to look more like a illbo function. 4 IMPROVEMENTS TO EDGE SPREAD FUNCTION ESTIMATION In this research the knife-edge technique was emloed and this section firstl considers an imrovement to Staunton s [] algorithm that incororates the effect of non-uniform illumination of the lightbo. Without noise the ESF could be differentiated to ield the PSF, but differentiating a nois function amlifies the noise. In this aer models of the ESF are develoed from several PSF models and then comared. A regularized numerical differentiation rocess is roosed. f () - - Fig. 4. ESF with a illbo PSF where σ = (solid) and the ideal ste edge (dashed) A Comensation for Non-Uniform Illumination An ideal brightness ste changes abrutl from one constant brightness level to the other. Eerimentall a light bo was emloed with a knife-edge to aroimate the ste, however, in ractice the brightness of each region was significantl non- 4

5 uniform. This resulted in erroneous PSFs. We roose a new model that retains the abrut transition, but allows each region to have a linear change in intensit as a function of satial osition. As an eamle, shown b the dotted trace in Fig. 4, both the bright and darker areas of this articular light bo increase in intensit towards the knife-edge. However other linear illumination field conditions can also be modelled b this scheme. The modified ste was given b s ) = ( m + c ) u( + ) + ( m + c ) u( ) () ( m + c [ (c + m ( + σ ))( σ ) + g ( ) = 4σ ( + σ )(c + m ( + + σ ))] m + c where is the location of the transition []. An eamle of the ESF for a PSF with a blur circle radius σ = is shown in Fig. 4 where the original ste is shown with a dotted line. Note that there are two shar transitions in the resulting ESF. A illbo PSF would result if the lens assed ever satial frequenc, however, due to diffraction it is known that this is not ossible and a smoother PSF will result. f g () - - Fig.. ESF when the PSF is a Gaussian with σ = (solid line) and unevenl illuminated ideal ste edge (dashed line). C ESF Modelled as a Sum of Fermi-Dirac Functions Tzannes and Moone [4] fitted a sum of three Fermi- Dirac functions to the ESF. Their technique resulted in a smoothed transition across the edge. In general form the sum of N Fermi-Dirac functions for modelling the ESF is where u() is the unit ste function, c and c are the brightness of the uer and lower regions and m and m are the gradients of the brightness. B ESF Assuming a Pillbo PSF For a defocused lens under geometrical otics the PSF is a illbo and given b h ( ) = [ u( + σ ) u( σ )] (6) σ where σ is the radius of the illbo, and hence the blur circle. The ESF assuming a illbo PSF and using () becomes < σ σ σ σ < (7) N ai g FD( ) = + d (8) bi i= + e c i where constants a i have been added to normalize the intensit, b i to set the centre oint, c i to control the gradient and d to account for the non-zero brightness of the lowest level. In order to recover the PSF the ESF must be differentiated, which is given b [] as b N a i i e g = FD ( ) c i hfd ( ) = i= b c + i i e ci (9) However a roblem with this model is that it cannot readil take into account the non-uniform illumination in a wa that allows the ste and the PSF to be searated. The PSFs determined eerimentall were consequentl non-smmetrical, as shown for eamle in Figure. D ESF Assuming a Gaussian PSF The Gaussian PSF is the most frequentl assumed model found in the literature on defocused lenses and this is artl due to its simlicit. A D Gaussian with a standard deviation σ and centered at = is then given b ( ) h g ( ) = e. () πσ σ

6 The ESF assuming a Gaussian PSF and a ste edge with non-uniform illumination is given b [] ( ) ( ) ( ) ) ( ) ( ) ( e ( ) g g = m + m σ + m + c erf + + m c erf σ σ σ σ () where erf( ) is the error function, defined as erf ( ) = π e( t ) dt. () If the ideal ste with non-uniform illumination as shown in Fig. 4 is defocused with a Gaussian (σ =, = ) then the ESF is as shown in Fig.. h G () h G () Fig. 6. Generalized Gaussian PSFs where (left) ( =, σ = ) and (right) ( = 4, σ = ) E ESF Assuming a Generalized Gaussian PSF The Generalized Gaussian function [3] is being roosed here as a model of the PSF of a defocused lens. Along with the mean and the standard deviation σ, the ower of the function is required. The function can take the form of a Gaussian when the ower = and a illbo when =, and thus encomasses both of the frequentl used models of defocus. The Generalized Gaussian is hg ( ) = e (3) Γ σ σ where Г( ) is the Gamma function and reresents the modulus. The term before the eonential ensures the function has unit area. Two Generalized Gaussian functions are resented in Fig. 6, where = (eg for a lens in focus), and =4 (eg defocused). The ESF, assuming a ste edge with non-uniform illumination and a Generalized Gaussian PSF, is given b the convolution of () with (3). A closed form, algebraic solution could not be found so the convolution integral was evaluated numericall. The ESF is given b [] as ξ gg ( ) = e m ξ c dξ σ σ [ ( ) + ] Γ ξ + e [ m ξ ) c dξ σ σ ( + ] Γ (4) Using the PSFs shown in Fig. 6 and the ideal ste with non-uniform illumination the resulting ESFs are shown in Fig. 7. F Regularized Numerical Differentiation In order to recover the PSF from the suer-resolution Edge Sread Function (ESF) the resonse must be differentiated and as the data is discrete, finite-difference aroimations must be emloed. However the ESF is nois and both two and five oint numerical differentiation were found to give oor results. Chartrand [4] considered the roblem of finding the derivative of a function when the underling function is nois and has a discontinuit in the derivative. The solution roosed uses total-variation regularization where the derivative of a 6

7 function () defined on the closed interval [, L] is the minimizer of the function L L Y ( u) = α u ' ( ) d + u( z) dz ( ) d () where u () is the first derivative of the function () and α is a regularization term that weights the first term, a enalt term, against the second term, the data fidelit term. f G () - - The total variation suresses the noise without removing discontinuities in the derivative. The aeal of this aroach is that a illbo PSF has two finite discontinuities and this method ensures that the can be recovered and that additionall noise suression is achievable. The main roblem is the choice of the regularization arameter α as it affects the derivative roduced. f G () - - Fig. 7. The ideal stes (dashed lines) and the ESFs (solid lines) assuming Generalized Gaussian PSFs with (left) ( =, σ = ) and (right) ( = 4, σ = ) Fig. 8. An eamle of the windowed image Amlitude. Amlitude / iels / iels Fig. 9. Five-oint numerical differentiation results for f.8, z=.7m, angle= degrees with ESF shown on the left and the PSF on the right 7

8 EXPERIMENTAL RESULTS, FOCUSED AND DEFOCUSED SYSTEMS A The PSF Recover Algorithm Initiall a knife-edge was setu on a lightbo so that it was angled with a slight offset to a row of iels in the FPA. Its image was windowed ( iels) as shown in Fig. 8. Individual ESFs along the edge therefore contained samles from black to white. This width was sufficient even for defocused lens measurements. The samled ESFs were normalized to remove nonuniform illumination along the direction of the edge. Net the central brightness ositions of the ESFs were estimated using a cubic fit and the ESFs aligned to these. This alignment resulted in the samles being dislaced relative to each other. The suer-resolution edge was created b averaging the iel intensities within sub-iel bins to give a ten times resolution imrovement. Having obtained the mean ESF for a given distance, f-number and knife-edge angle it was necessar to find the PSF. The methods eamined were:- Five-oint numerical differentiation; Regularized numerical differentiation using Chartrand s algorithm; Regularized numerical differentiation using Chartrand s algorithm followed b a fit of the resulting PSF to a Generalized Gaussian function; Fitting the ESF to a sum of Fermi-Dirac functions [4]; Fitting the ESF to a defocused ste assuming even illumination and a Gaussian PSF; Fitting the ESF to a defocused ste where the illumination is assumed to have a linear deendence on osition and a Gaussian PSF; Fitting the ESF to a defocused ste assuming even illumination and a Generalized Gaussian PSF; Fitting the ESF to a defocused ste where the illumination is assumed to have a linear deendence on osition and a Generalized Gaussian PSF. In this section results for a 4mm hotograhic lens fitted to a Basler A63fc color camera are resented when the lightbo was.7m from the camera Amlitude. Amlitude / iels / iels Fig.. The actual ESF (dashed line) and Fermi-Dirac fitted ESF (solid line) results for f/.8, Amlitude B Secific D Results The results from the five-oint numerical differentiation in Fig. 9 show that although the ESF looks fairl smooth, the noise is swaming the underling PSF, thus making this aroach unusable without further rocessing. When the measured ESF was fitted to a sum of Fermi-Dirac functions, as shown in Fig., the ESF aeared to have a good fit, however the PSF neither had smmetr or a single eak. These are roerties eected of a hsical PSF.. / iels Fig.. Regularized numerical differentiation results (right) for α = (dashed), α = (dash-dot) and α = (solid) C Regularized Numerical Differentiation In order to determine the otimum regularization arameter α, a series of simulations were erformed. Both illbo and Gaussian PSFs were used to defocus an ideal ste. Three levels of noise were added with SNRs of (high noise),, and 3 (low noise) db, and then the ESF was differentiated using Chartrand s algorithm 8

9 [4]. The MSE was emloed as a distance measure between the actual PSF and the result of the numerical differentiation. The value: α = was otimum for the SNR = db case, while α = was otimum for and 3 db. These values were emloed in the modelling Table. MSE results for f/.8 as a function of the deth to the light bo Mean Square Error (MSE) / -3 Method.44m.49m.69m.647m.7m Fermi-Dirac Generalized Gaussian without I.C. Generalized Gaussian with I.C. Gaussian without I.C Gaussian with I.C Pillbo without I.C Pillbo withi.c of real ESFs. An eamle is shown in Fig. where we have taken the low noise case because it illustrates a roblem with the method in that it could not take into account non-uniform illumination and roduced nonsmmetric PSFs. The remaining fitting methods roduced PSFs with less noise and better smmetr, and so were rocessed further to give D results. Table. Mean MSE results for all three aertures from best to worst Method Average MSE / -3 Generalized Gaussian with.4 Illumination Correction Generalized Gaussian without 6.93 Illumination Correction Sum of three Fermi-Dirac functions 6.7 Gaussian with Illumination 4. Correction Gaussian without Illumination 6.7 Correction Pillbo with Illumination Correction 7. Pillbo without Illumination 97.6 Correction D Edge Sread Function Fitting Eeriments The ESFs were fitted to the various functions for a range of distances. The results for aerture f.8 are dislaed in Table. It shows the MSE of the fit as an average for all angles tested, which were -8 to +9 degrees in -degree intervals. The lane in-focus was at.44m in front of the lens. The results show that the error assuming a illbo PSF decreases for increasing defocusing, which was eected from the diffraction-based otics theor in Section 3. The MSEs of the fits using Generalized Gaussian, Gaussian and illbo models are lower when the non-uniform illumination was taken into account. The eeriment was reeated for f-numbers of f4 and f.6. In Table, the summarized results for all aertures show that the Generalized Gaussian with illumination correction has resulted in the lowest MSE, thus giving the best fit to the data. The illbo model roduced the worst results with a MSE about 4 times greater than that of the Generalized Gaussian. The MSE of the Gaussian fell almost half wa between the Generalized Gaussian and the illbo. E Results assuming Gaussian and Generalized Gaussian PSFs Images of the knife-edge were obtained in mm increments over a 3cm deth range for angles of -8 to +9 degrees in -degree increments. Each image gave a single mean ESF and that ESF was fitted assuming both Gaussian and Generalized Gaussian PSFs, as derived in Section 4. The PSFs were found to be ver nearl circularl smmetric and so the following results are given for the direction onl. Fig. shows the standard deviation of both the Gaussian and Generalized Gaussian as a function of distance for three different f-numbers tested. The lots aear to be smooth and increase monotonicall, ecet for the Gaussian at the maimum distance tested for f.8. Being more robust with increasing deth, the Generalized Gaussian is considered a better model for use with DFD []. The Generalized Gaussian PSF has two arameters, the standard deviation σ and the ower. When the ower as a function of deth was lotted it was found to be noisier than σ as shown in Fig. 3. F Two-dimensional PSFs Comlete D-PSFs are resented below assuming illbo, Gaussian and Generalized Gaussian PSF models for two deths, corresonding to the furthest and closest ositions tested. The non-uniform illumination imrovement was 9

10 used. Figs. 4 to 6 show the PSFs for a distance, z =.7m, between the camera and the lightbo for an aerture of f/.8. The Gaussian PSF model shown in Fig. 4 is for a defocused lens, and is clearl circularl smmetric. The fit has resulted in a smooth contour lot. The Generalized Gaussian PSF model shown in Fig. aears to be a cross between the Gaussian and a illbo. The fit has 9 9 f/.8 () f/4 () 8 f/.6 () 8 f/.8 () f/4 () f/.6 () σ / iels 4 σ / iels deth / m deth / m Fig. : Standard deviation against deth when fitting (left) a Gaussian PSF and (right) a Generalized Gaussian PSF ower f/.8 () f/4 () f/.6 () deth / m Fig. 3: The ower of the Generalized Gaussian against deth resulted in a contour lot that is less smooth than for the Gaussian, which is robabl due to noise in the ESFs and increased comleit of the function due to it having more arameters than all the other models. The illbo model, Fig. 6, has resulted in a reasonabl circular PSF. Figs. 7 to 9 show the PSFs for z =.44m, the in focus case. Note the change of and ais scales in Figs. 7 to 9. Now all three models have less circularl smmetr and have a maimum sread at aroimatel 4 degrees to the ais. The ower of the Generalized Gaussian is less than two, and so the function is more ointed than that for the Gaussian. G Discussion The goodness-of-fit of the Generalized Gaussian PSF is eemlified b the results of Table 3. Where the nonuniform illumination model was emloed, the fit was between 9 and 6 times better than using a Gaussian PSF. The Gaussian PSF has a faster roll-off when the camera is ver defocused comared to that using the Generalized Gaussian because the ower of the Generalized Gaussian increases with defocus, making it more illbo in shae, as highlighted in Fig..

11 Fig. 4. D PSF assuming a Gaussian model for z =.7m and f.8 where and are in iels Fig. : D PSF assuming a Generalized Gaussian model for z =.7m and f.8 where and are in iels Fig. 6: D PSF assuming a Pillbo model for z =.7m and f.8 where and are in iels

12 Fig, 7: D PSF assuming a Gaussian model for z =.44m and f.8 where and are in iels Fig. 8: D PSF assuming a Generalized Gaussian model for z =.44m and f.8 where and are in iels Fig. 9: D PSF assuming a Pillbo model for z =.44m and f.8 where and are in iels

13 Table 3. The average MSE for each method Average Mean Square Error (MSE) / -3 Method, direction f/.8 f/4 Gaussian, - direction Gaussian, - direction Generalized Gaussian, -direction Generalized Gaussian, -direction h (; f =.8; z=.69m) f/ h (; f =.8; z=.7m). - - Fig. : Comarison between PSFs for the Gaussian (dashed line) and Generalized Gaussian (solid line) 6 CONCLUSION We have reorted imrovements to an easil erformed suer-resolution, but averaged, PSF estimation method that enables it to be used for focused and defocused lenses and with noise generated within the camera. The method involved the initial accumulation of a suer-resolution ESF. Previousl the calculation of the PSF from the ESF could give both nois and distorted results. Distortions were attributed to non-uniform illumination of the knife-edge test object. Where a model of the illumination could be incororated into the comutation of the PSF, smmetrical functions resulted. Previousl noise in the ESF was amlified when it was differentiated to form the PSF. We researched a regularized numerical differentiation that greatl reduced the noise. However changes to the value of the regularization arameter chosen resulted in varing distortions of the PSF. The method was not constrained b an assumtion of an underling model of the ESF or PSF, but in ractice, it was found to be outerformed b methods that did. Several widel used models of the PSF were investigated including the illbo and Gaussian, together with the use of Fermi-Dirac fitting functions. A Generalized Gaussian that incororated both illbo, Gaussian, and a continuum of models in between, through the choice of a arameter, was also used to model the PSF. The corresonding models for the ESF were derived from these as reorted in the aer. The results showed that the MSE of the fit using the Generalized Gaussian erformed best across the range of distances and f-numbers tested and that it was 8 times better than the Gaussian model and 4 times better than the illbo model. Pillbo and Gaussian models are often assumed in DFD work and this research has shown that both are sub-otimum. Finall, D PSFs for various knifeedge angles were combined to form D PSFs. REFERENCES. R.G. White and R.A. Schowengerdt, Effect of oint sread functions on recision ste edge measurement, J. Ot. Soc. Am. A-Otics, (994).. R.C. Staunton, Edge oerator error estimation incororating measurements of CCD TV camera transfer function, IEE P-Vis. Image Sign. 4, 9-3 (998). 3. R.C. Staunton, Detected edge evaluation using measured acquisition sstem arameters, Pattern Recogn. Lett. 6, (). 4. A. Pentland, A new sense for deth of field, IEEE T. Pattern Anal. 9, 3-3 (987).. M. Subbarao and N. Gurumoorth, Deth recover from blurred edges, In Proceedings of IEEE Conference on Comuter Vision and Pattern Recognition (IEEE, 988), J. Ens and P. Lawrence, An investigation of methods for determining deth from focus, IEEE T. Pattern Anal., 97-8 (993). 7. M. Subbarao and G. Sura, Deth from defocus: a satial domain aroach, Int. J. Comut. Vis. 3, 7-94, (994). 8. A.N. Rajagoalan and S. Chaudhuri, A variational aroach to recovering deth from defocused images, IEEE T. Pattern Anal. 9, 8-64 (997). 9. M. Watanabe and S. Naar, Rational Filters for Passive Deth from Defocus, Int. J. Comut. Vis. 7, 3- (998).. Li-Ma and R. C. Staunton, "Integration of multiresolution image segmentation and neural networks for object deth recover", Pattern Recogn. 38, ().. P. Favaro and S. Soatto, A geometric aroach to shae from defocus IEEE T. Pattern Anal. 7, (). 3

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