# CHARACTERIZING IMAGE QUALITY: BLIND ESTIMATION OF THE POINT SPREAD FUNCTION FROM A SINGLE IMAGE

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3 Æ 5 Æ Æ ¹ ² ) 9 ² 9 % 6 ² ) explicitely 5 { }~"9=ql ƒq ˆ Š Œ & }~ \$ ƒ \$ Ž using the definition of the Fourier transform of (Castleman, 979). In case we have a sinus-type pattern c7 \$ r \$ r E šs} 9:l the response of the system is a sine-wave with contrast. As the MTF usually falls off for large frequencies, contrast of tiny details is diminished heavily. In our special context we obtain the MTF for the Gaussian shaped PSF,.œ ŸžŠ ƒ which Œ again is a Gaussian, however, with the matrix Š as parameter. Observe that we have ª «ƒ - «ƒ - 7. Contrast Sensitivity Function In order to evaluate the usefulness of the imaging system with a certain PSF or MTF the so called contrast sensitivity function (CSF) is used. The contrast sensitivity function gives the minimum contrast at a periodic edge pattern which can be perceived by a human. In our case we want to apply this notion to edge detectors. Assume we have a periodic pattern of edges characterized by the wavelength and the contrast ±. Further assume the image to be sampled with a pixel size of ²! or and the noise has standard deviation ³. An ideal edge detector would adapt to the wavelength of the pattern and perform an optimal test whether an edge exists or not. For simplicity we assume that the pattern is parallel to one of the two coordinate systems and that the edge detector uses the maximum possible square of size µ. The difference ² between the means and of the two neighboured areas can be determined from the ¹ º l ² pixels in the two areas. It has standard deviation»o ½¼ ¾ À ¾ Á ¾ ÂÁ tãåä Thus in case we perform the test with a significance number Æ and require a minimum probability Ç% for detecting the edge we can detect edges with a minimum height ²È%& The factor 6 % 6&% Ç % Ç%»O depends on the significance level of the test and the required probability of detecting an edge. It is reasonable to fix it; in case we choose a small significance number Æ ) É and a minimum detectability Ç% ) Ë we have 6&% ) ŠÌiÍ. The minimum detectable contrast in a reasonable manner depends on the size of the window and the noise level: The larger the 6&% ÇÉ% ³ ³ ) ³ ) noise standard deviation and the smaller the window the larger the contrast of the edge needs to be in order to be detectable. As we finally want to relate the contrast sensitivity to the 9Î frequency and obtain the contrast sensitivity function CSF"94 ) ² % 6 % ² 9³ It goes linear with the frequency, indicating higher frequency edge patterns require higher contrast..4 Resolving power The resolving 9 power Ï usually is defined as that frequency where the contrast is too small due to the properties of the imaging system to be detectable. As periodic patterns with small wave length will loose contrast heavily they may not be perceivable any more. 49: "94 The MTF has MTF"94Ð maximum value and measures the ratio in contrast, whereas the CSF measures the minimum contrast being detectable. In order to be able to compare the MTF with the CSF we need to normalize the CSF. This easily can be done in case we introduce the signal to noise ratio SNR ÒÑ ³ with Ñ being the contrast. Then the relative contrast sensitivity function reads as rcsf"94 ) CSF"94 Ñ 6%² Ó9 ³ Ñ 6&%Š² Ó9 SNR which immediately can be compared with the MTF. One usually argues, that the resolving power is the frequency where the relative contrast, measured by the MTF, is identical to the minimum relative contrast being 9 detectable (cf. fig. ). Thus the resolving power % is implicitly given by % rcsf"9 % c usable image contrast MTF u resolving power CSF Figure : Relations between the modulation transfer function (MTF), the contrast sensitivity function (CSF) and the resolving power (RP). 9 In the -dimensional case we can explicitely give % E.Ô LambertW Õ Ö SNR ) % u

4 Ñ Ä à Ö Ö The LambertW-function is defined implicitly by (c.f. (Corless et al., 996)) LambertW"ÀÃ exp LambertW4S ) SNR~ Figure 4: Resolving power in linesmm for aerial images with a pixel size of 5 m as a function of SNR (left, ) and of the width of the PSF (right, SNR=) Figure 4 shows the resolving power of our ideal edge detector in linesmm for aerial images as a function of the signal to noise ratio and of the width of the point spread function. The resolving power increases with increasing SNR and reaching 5- linesmm for good SNRs. It decreases with increasing blur, falling below linesmm for # Ø :Ù. These results are reasonable, as they are confirmed by practical experiences with digital aerial images (c.f. (Albertz, 99))..5 Contrast, Gradient and Local Scale We now derive a simple relation between the contrast, the gradient and the local scale, which we will use to determine the local scale at an edge. We assume an edge in an image to be a blurred version of an ideal edge. In case the PSF is a Gaussian,.- 4 c the edge follows erf c Ñ erf Ú s~ LÛ Ù where Ù is the mean intensity and Ñ is the contrast. Following (Fuchs, 998) the contrast can be determined from the standard deviation É of the signal around the edge, ÜÉ. The gradient magnitude of the edge is given by the first of the edge function, which in our case. Thus we have the relation is Ñ, - From this and Ñ Ñ Á E Ý Ñ Á E SÉ we can easily derive The practical procedure determines the variance of the signal from 7Þß yž'"þq Ý à, -ˆá ži, -Šá where the kernel width â is chosen to be large enough to grasp the neighbouring regions. We use a kernel size of â. The gradient magnitude should be estimated robustly from a small neighborhood. We use a Gaussian kernel with for estimating the gradient magnitude..6 Blind estimating the PSF from a single image We are now prepared to develop a procedure for blindly estimating the PSF from a single image. Blind estimation means, we do not assume any test pattern to be available. As the PSF is derived via the sharpness of the edges, and the PSF is the image of an ideal point, a 6 -function, we need to assume that the image contains edges which in the original are very sharp, thus close to ideal step-edges. This can e. g. be assumed for images of buildings or other manmade objects, as the sharpness of the edges in object space is much higher than the resolution of the imaging system can handle. Formally, if the image scale is äã*å, the width æ of the image of the sharp edge would be æ çˆ å and we assume that this value is far beyond what the optics or the sensor can handle. Now, for each edge we obtain a single value è. In case it would be the image of an ideal edge in object space it can be interpreted as an edge with the expected mean frequency E è in the MTF in that direction. Thus we obtain a histogram from all edges with è Éë è Õéê Ðë and è ìž Éë è Õéê Ðë where the direction vector points across the edge. We use two values, as we do not want to distinguish between edges having different sign. In case the edge is already fuzzy in object space, the estimated value è of the edge will be larger, thus the E è will be smaller. Therefore 9 we search for the ellipse which contains all points è and has smallest area. º This ellipse is an estimate for the shape of the ellipse for of the PSF. EXPERIMENTAL RESULTS, thus The following examples want to show the usefulness of the approach. In detail we do the following:. Using an ideal test image (Siemens star) with known sharpness we compare our estimation with given ground truth (cf. fig. 5).. Using the same test image but with noise we check the sensitivity of the method is with respect to noise (cf. fig. 6).. Using real images with known artificial blur we check whether the method works in case the edge distribution is arbitrary (cf. fig 7).

5 ò Ë 4. Using scanned aerial imagery with different sharpness, caused by the scanning procedure, we test whether the method also reacts to natural differences in sharpness (cf. fig. 8). In all cases the minimum resolving power of an ideal edge detector is given. In the case of digital images we refer to a pixel size of 5 m.. Demonstration on synthetic Data Test on noiseless data. The following sequence of gradually blurred images was used to test the proposed method to determine the point spread function and the resolving power with respect to correctness of the implemented algorithm. üþýÿ ö üþýÿ ô üþýÿ ù lmm lmm ólmm ñ ïð ólmm ü ýÿ ï ólmm ô ïð ñ ö ð ù ö ð ô Ü õòlmm ølmm òlmm ólmm Figure 5: Siemens - star at various steps of image sharpness (² ú, SNR Ø Ø ). left: test image, right: histogram of edges, resolving power of optimal edge detector. Ø m, É³ The method gives reasonable results: For each test image, the histogram of edges is a circle with the correct radius E, being the reciprocal width of the point spread function used to generate the image. Test on noisy data. To test the sensitivity of the algorithm with respect to image noise the Siemens star ü ýÿ ö Figure 6: Siemens star with image noise (SNR= * ólmm ) Ë at various steps of ) Ë from fig. 5 was speckled with Gaussian noise, the noise variance being ³. The results in fig. 6 show that the method is quite robust with respect to image noise. Note that the slightly decreasing resolving power of the ideal edge from the first to the last row is caused by the increasing image noise.. Results on real data Real data with artificial blur. The method was also tested on a real image of the MIT building which was gradually blurred by convolution with Gaussian filters of increasing filter width (cf. fig 7). We see that the method seems to yield correct results. In almost each histogram of edges the ellipse containing all points is elongated, indicating anisotropy of the image sharpness for the given image. Aerial image with various sharpness. Finally, the method was applied to digitized versions of an aerial image (cf. fig. 8, top row) scanned three times with a pixel size of 7 m. Various image sharpness has been realized physically by imposing layers of transparencies between the original and the scanner platform, thus exploiting the limited depth of view of the optical system of the scanner. Ë ).

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