The modulation transfer function (MTF) is the mostcommon tool for the characterization of optical ima-ging systems [1,2]. It is attractive due to its simpli-city in the evaluation of the performance of imagingsystems. The MTF has been measured over the yearsby using templates such as the U.S. Air Force (AF)resolution chart, for instance [3], for assessing thespatial frequency response of imaging systems. TheAF resolution chart consists of a large set of smallpatches of Ronchi rulings (a set of equally spacedbars and spaces) of increasing density [4]. Thepatches come in groups of two, at orthogonal orienta-tions to each other, for simultaneous testing of theradial and sagittal response of imaging systems interms of the spatial frequency response and the cor-responding contrast. Calculation of the MTF usingsuch bar targets can be found in [5–8]. Another kindof template used for such a purpose is the chirpedRonchi grating, which consists of one-dimensional(1D) bars and spaces, with progressive increasingdensity (from here the “chirped” notation), thus pro-
viding a wide range of spatial frequencies. Thesetemplates clearly identify at what spatial frequencythe imaging system fails (cannot reproduce certainspatial frequencies) or renders poor contrast images.Recently, a novel method for the calculation of theMTF, using random targets [9], has been described.The two templates mentioned above, are 1D struc-tures that allow themeasurement of what one shouldcall 1D MTF. The question that one should be con-cerned with is whether the 1D MTF is sufficient fordetermining the properties (specifications) of an ima-ging system. This question received much attentionrecently from manufacturers of bar code detectionsystems who realized that the common 2D bar codes(PDF 417, Maxicode, and Data matrix code) requireimaging systems with higher contrast capabilities inthe acquisition stage (when such contrast is mea-sured in terms of 1D MTF values), in comparisonto those used for 1D target (bar code) detection. Itis thus obvious that the definition of the MTF shouldbe revisited.
2. One-Dimensional versus Two-DimensionalModulation Transfer Function
The MTF of imaging systems is provided by the auto-correlation of the pupil function [1]. Thus, for 1D
pupils (i.e., slit function), the in-focus MTF has theshape of a triangle [Fig. 1(a)], while for two-dimensional (2D) pupils (i.e., circular function), thein-focus MTF has a shape that strongly resembles atriangle but ends with a curving slope at highfrequencies [Fig. 1(b)].
One should be warned at this point, that eventhough the MTF has been evaluated for circularapertures, it is still essentially a 1DMTF, albeit withcircular symmetry and, as such, it should rather becalled a “1D circular MTF.” For instance, it is com-mon practice to use spoke targets as the objectfunction for characterizing imaging systems and toobserve how the out-of-focus image of such objects ex-hibits the inability to image certain spatial frequen-cies or exhibits contrast reversal for a certain rangeof spatial frequencies. This is beautifully demon-strated in Fig. 2, where the image of a spoke targetis shown in either an in-focus condition [Fig. 2(a)] oran out-of-focus condition [Fig. 2(b)].
An examination of the 1D MTF corresponding to aslit pupil and the 1D circular MTF curve, often erro-neously called 2D MTF, corresponding to a circularpupil, are shown in Fig. 3 for an out-of-focus condi-tion of ψ ¼ 5. ψ is a common measure used to quan-tize the out-of-focus condition [1] and measures thequadratic phase added to the pupil function whenthe imaging system is not in focus. ψ is the maximumphase error of the spherical wave front at the edge ofa circular aperture that has a diameterD, for an ima-ging system that uses a lens with a focal length of f :
ψ ¼ πD2
4λ
�1
dobjþ 1dimg
−
1f
�: ð1Þ
Imaging is considered “diffraction limited” as longas ψ < 1.
The physical explanation for contrast reductionand inversion is due to the fact that in an out-of-focuscondition, the point spread function is wide, and thusit captures information from the addressed area aswell as from its adjacent features. As such, when
Fig. 1. Modulation transfer function curves of (a) 1D slit pupiland (b) 2D circular pupil, in focus.
Fig. 2. Spoke target images (a) in focus and (b) out of focus.
the center of the point spread function hits a darkregion, it acquires light also from adjacent regionsthat may be bright, thus providing a somehowbrighter output. Likewise, when it hits a bright re-gion, the total acquired intensity is dimmer, becausethe expanded point spread function covers adjacentregions, from which no energy is reflected, if they aredark. Contrast reversal occurs when the energy col-lected from a bright region is lower than that ac-quired from a dark region in view of the above. Itis obvious that this effect is catastrophic in the caseof bar code reading, because it prevents proper decod-ing and identification.
The severity of this effect is enhanced when decod-ing 2D bar codes. An example of such a bar code(Fig. 4) shows that bright regions may be surroundedby dark regions, and likewise dark regions may besurrounded by bright regions, not only along the hor-izontal direction as one encounters with 1D barcodes, but also in the vertical direction as well. Thisobservation readily explains why, for 2D bar code im-
agers, one gets a signal with a lower contrast thanthat obtainable from detecting 1D bar codes of thesame density.
We now suggest adopting new kinds of templatesthat are truly 2D. Two such templates are shownin Fig. 5 in two representations: Cartesian [Fig. 5(a)]and polar [Fig. 5(b)]. The Cartesian checkerboardpattern is the “generic pattern” that allows the deter-mination of the spatial frequency response of a new2D resolution chart: instead of the two twin patches,one rotated by 90 ° with respect to the other, onewould have a single set of checkerboard patternsof increasing density. Likewise, the polar templateis the checkerboard equivalent pattern of the spoketarget and contains a whole range of spatial frequen-cies with increasing density toward the center.
Because the 1D USAF 1951 target has patchesconsisting of three bars and two spaces, we proposethat the checkerboard patterns of the 2D resolutionchart should consist of 5 × 5 squares, starting withlarge structures and then progressively smaller ones.The spatial frequencies of the 2D target match thespatial frequencies of the 1D target. In Fig. 6, thetwo charts, with only two groups are shown.
There is no counterpart representation for the ro-sette pattern (1D or 2D), because we have there awhole set of spatial frequencies in each pattern. Ofcourse, we may be able to use a set of rosettes, each
Fig. 3. Modulation transfer function curves of (a) 1D pupil and (b)2D pupil, for the out-of-focus condition, ψ ¼ 5.
Fig. 4. Typical 2D bar code.
Fig. 5. Suggested 2D templates: (a) Cartesian and (b) polar.
having a different starting spatial frequency, ifdesired.
The contrast obtainable with a polar 2D template(“checkerboardlike” spoke target) for an in-focuscondition and a circular point spread function wascarried out by computer simulations according tothe formula:
contrast ¼ Imax − Imin
Imax þ Imin; ð2Þ
where Imax is the maximum value and Imin is theminimum value of the sampled signal encounteredalong the chosen path.
The contrast obtained with an imaging system ex-hibiting a point spread function corresponding toψ ¼ 1, when used with a 1D circular template (“con-ventional” spoke target with 50 spokes) was 90%[Fig. 7(a)], while that obtained with a 2D polar check-erboard template that had the same density, wasonly 55% [Fig. 7(b)]. One thus readily sees that the“true” contrast level provided by the 2D checker-boardlike template is much lower than that obtainedwith a “conventional circular” 1D template for thesame spatial frequency.
In Fig. 8 we present computer simulation resultsobtained with a 2D Cartesian resolution chart(Fig. 6) for an in-focus condition, as well as an out-of-focus condition. For comparison, the images ofthe AF resolution template are shown for the samecondition.
One can easily notice contrast reversals as well asa loss of certain frequencies, in particular for ψ ¼ 5.
In Fig. 9 we present (via computer simulations)views of the 2D polar resolution chart for an in-focuscondition, as well as an out-of-focus condition.
For comparison, the images of the 1D polar resolu-tion chart are shown for the same condition.
In Fig. 9, one notices how the blur is more pro-nounced at high frequencies (toward the center) andhow the response is “nil” (gray region with no visiblestructure) for some range of spatial frequencies.Notice that the blur circle occurs at lower spatial
Fig. 6. (a) Standard 1D resolution chart versus (b) 2D suggestedresolution chart.
Fig. 7. (upper) 1D and (lower) 2D spoke templates. The red circle indicates the spatial frequency chosen for comparison. Signal traces atthis selected spatial frequency (right).
Fig. 9. Images of the 1D spoke target 2D (left) and the polar re-solution chart (right) for different out-of-focus conditions: (a) ψ ¼ 0(in-focus), (b) ψ ¼ 2, and (c) ψ ¼ 5.
Fig. 10. (a) 1D spoke template and (b) 2D spoke template, atfocus, Ψ ¼ 0. The RGB circles represent three selected spatialfrequencies.
Fig. 8. Images of the 1D resolution target (left) and the 2DResolution target (right) for different out-of-focus conditions: (a)ψ ¼ 0 (in focus), (b) ψ ¼ 2, and (c) ψ ¼ 5.
Fig. 11. (Color online) Signal traces along the three circles drawnin Fig. 10 represent the response at three selected spatial frequen-cies. The RGB traces correspond to the 1D template, and the blacktraces correspond to the 2D template.
frequencies (i.e., larger radius) for the 2D chart, incomparison to its counterpart, the 1D (circular) ro-sette. At every distance, the radial step size, mea-sured at the middle of the zone, is approximatelyequal to the azimuthal one. The calculation of theradial step follows from
radial step ¼�2S − π2Sþ π
�N; ð3Þ
where S is the number of spokes (16 in this particularfigure) and N is the ordinary number of the step,measured from the edge, which is 1. One can noticecontrast reversals, in particular for ψ ¼ 5 here, too.An experiment with the new 2D target has beentaken.
A commercial RGB (red-green-blue) camera (uEYE1225), equipped with a complementary metal-oxidesemiconductor detector with a resolution of 752 ×480 pixels and a pixel dimension of 6 μm was uti-lized. It was equipped with a lens having an effectivefocal length of 16 mm [Computar M1614W], aper-ture size of 1 mm and a field of view of 45 ° alongthe sensor diagonal. The target was incoherentlyilluminated by ambient room light. The nominal dis-
tance has been set at 23 cm, The contrast level atthree different spatial frequencies has been evalu-ated for the nominal focal distance as well as foran out-of-focus condition, ψ ¼ 5 (e26 cm) for both1D and 2D templates. It so happens that for sucha defocus condition, the green curve in the 2D imagehits a region where the contrast level is 0. The bluecurve corresponds to a region that provides contrastreversal for the 2D template but no contrast reversalfor the 1D one.
Both cases were simulated using MATLAB toolsand presented in Figs. 12 and 15. In Fig. 10, imagesof the 1D (right) and 2D (left) polar templates, for ψ ¼0 case are shown. The RGB circles represent thethree selected spatial frequencies.
Fig. 12. (Color online) Modulation transfer function of an opticalsystem in focus is drawn (magenta). The dashed curve representsthe corresponding experimentally measured values. The RGB dotsrepresent the contrast at the selected test frequencies.
Fig. 13. (Color online) (a) 1D spoke template and (b) the 2D spoketemplate, out of focus, Ψ ¼ 5. The RGB circles represent three se-lected spatial frequencies.
Fig. 14. (Color online) Signal traces along the three circles drawnin Fig. 13 represent the response at three selected spatial frequen-cies. The RGB traces correspond to the 1D template, and the blacktraces correspond to the 2D template. Note the almost lost contrastfor green and phase reversal for blue.
In Fig. 11, the signal traces along the RGB circlesare shown. The signals are almost identical for the1D and 2D templates.
In Fig. 12, the MTF of an optical system in focus isdrawn (magenta). The dashed curve represents thecorresponding experimentally measured values.The RGB dots represent the contrast at the selectedtest frequencies.
In Fig. 13, images of the 1D (a) and 2D (b) polartemplates for the ψ ¼ 5 case are shown. The RGB cir-cles represent the three selected spatial frequencies.
In Fig. 14, the signal traces along the RGB circlesare shown. One can easily see that there is a contrastreversal in the blue circle of the 2D template.
In Fig. 15, the 2D optical transfer function of anoptical system out of focus, for the ψ ¼ 5 case, isdrawn (magenta). The dashed curve represents thecorresponding experimentally measured values.The RGB dots represent the contrast at the selectedtest frequencies.
Because the 2D templates exhibit smaller levels ofcontrast than do the 1D templates, both for the Car-tesian and the Polar case, contrast reduction andcontrast reversals can be seen at lower frequencies,hence at larger radii.
4. Conclusions
Using the AF chart for contrast calculation and reso-lution measurements is not sufficient when dealingwith 2D sensors, due to the fact the point spreadfunction is affected by the entire surrounding area.As a result, one should, rather, introduce and use2D checkerboard resolution charts. Those charts(Cartesian or polar) provide different 2D spatial fre-quencies, and the contrast displayed when imagingthose charts is lower than that obtained with 1Dtemplates.
The suggested 2D checkerboard charts include thesame range of frequencies available with 1D charts.These templates should be used if one desires to findout the true spatial frequency response of imagingsystems.
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Fig. 15. (Color online) Optical transfer function of an optical sys-tem out of focus, for the Ψ ¼ 5 case, is drawn (magenta). Thedashed curve represents the corresponding experimentally mea-sured values. The RGB dots represent the contrast at the selectedtest frequencies.
