An Adaptive Irregular Sampling Method for ... - Semantic Scholar
An Adaptive Irregular Sampling Method for Progressive Transmission G. Ramponi and S. Carrato DEEI, University of Trieste, Italy e-mail: framponi,[email protected]
Abstract
An adaptive nonuniform sampling technique for alias-free image representation is presented. It is both fast, thanks to the use of pseudo-Poisson disks (PPDs), and e ective, because a higher sample density is used in areas which are rich in details; these zones of the image are located using a skewness-based operator, which is both simple and accurate, and gradient information. Experimental results on real-world images are presented, which show that this technique provides images free of aliasing and with good detail content. As an example of application, a progressive image coding scheme based on the proposed technique is also presented.
1 Introduction
Non-standard sampling techniques are widely used in the computer graphics literature to avoid unpleasant e ects in the reproduction of synthetic images. In particular, irregular sampling grids are used in which the sampling points are placed in an aperiodic fashion on the image. Indeed, one of the visually most annoying disturbances of standard sampling is aliasing, due to the presence of above-Nyquist frequency components in a signal sampled on a rectangular grid. Starting from the consideration that the eye has a limited number of receptors but does not introduce aliasing, sampling techniques have been devised which reproduce the spatial organization of such receptors, resembling a two-dimensional Poisson stochastic distribution. For example, good results can be obtained just by jittering the locations of a periodic grid: aliasing is attenuated or eliminated while, in exchange, noise is introduced. Even if the overall energy content of the error is the same, aliasing generates patterns that can be more easily perceived than noise. The jittered grid is an irregular but uniform distribution of samples. In some cases non-uniform grids are selected, in which the density of the samples varies as a function of the image content. Turning again to the eye as a model, we observe indeed that in the human eye the fovea exists, where the density of receptors is higher and which is made to coincide with the xation point of our gaze. This xation point in turn tends to follow the most detailed portion of the scene which is looked at. We can obtain a similar e ect by locally modifying the density of the sampling in the detailed areas of a picture.
In [1], these methods are ported closer to the image processing eld. The concept of locally band-limited image is introduced, and an approach is presented for the representation of these images by spatially-varying systems; stochastic estimation of parameters characterizing the local bandwidth is discussed, to generate an optimal sampling scheme. In [2], a farthest-point sampling strategy is developed, which permits to progressively add sample points by placing them in the middle of the least-known area. The image is then reconstructed using a simple Four-Nearest-Neighbors (4NN) interpolator, which forms a weighted average of the available samples closest to the desired pixel. An adaptive version of the sampling algorithm is also proposed in [2]; in this case, the sample density is increased in detailed areas, exploiting a nonstationary image model in which the distance between two points is modi ed if the local variance is high. The obtained results are very satisfactory, and clearly show the disappearance of the aliasing and the better reproduction of the image details in the adaptive method. It should be mentioned however that, while in the basic algorithm the sampling map can be recreated autonomously by the reconstruction system, in the adaptive version it is necessary to know the location of the samples to be able to reconstruct the image. The method we propose di ers from and improves on the technique in [2] in two ways: a much simpler (even if possibly less powerful) random technique is used to generate the uniform but irregular sampling grid, and a di erent, more e ective method is used to determine the areas in which signi cant details are present. Under the latter viewpoint, the method we propose has also the important advantage of being reproducible by the reconstruction system autonomously, without the necessity of sending a map of the sampling points. The result is a technique which allows for progressive transmission of an image: rst, a lowresolution image is constructed using a small amount of samples taken on the random grid and a slightly modi ed 4NN interpolator; then, a better quality result is achieved by a re nement step which exploits the further samples located in detail areas. In the following, we rst describe the criteria used to locate the samples needed for the interpolation.
2.2 Adaptive sample location using skewness and gradient
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3
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Figure 1: Spectrum of a random PPD distribution. Then we show a possible image transmission scheme which, notwithstanding the irregularity and adaptivity of the sampling grid, does not require the transmission of the samples positions. Finally, experimental results on a real-world image are presented.
2 Algorithm description
As already mentioned, in the proposed method we determine the sample locations following a two-pass scheme. First, an irregular uniform sampling grid is considered; then more samples, close to the image edges, are added in order to improve the quality of the interpolation.
2.1 Irregular uniform sample location using PPD
We call Pseudo Poisson Disk (PPD) the method used to generate the uniform random distribution. It is a simpli ed version of the conventional Poisson Disk technique [2]: the latter, in which the distance between each sampling point and its neighbors is kept larger than a prede ned value (the disk diameter), is computationally very heavy. In our case, we simply select a random coordinate pair in an image and, at the same time, we de ne a forbidden pseudocircular area around the selected point; the mechanism is repeated until the desired number of samples is reached. Of course, the diameter of the forbidden area must be set as an inverse function of the sampling density. With a correct choice of this parameter, our PPD method shares the main characteristic of the Poisson Disk distribution, i.e. its capability of moving the energy of the noise towards high frequencies, where the sensitivity of the eye is lower. In this way the visual aspect of the resulting image is better, compared to other random distributions such as conventional Poisson or the jittered grid. Fig. 1 shows the spectrum of a PPD distribution used for sampling a 256 256 image, plotted as a 1-D function by averaging along concentric circles; it can be seen that most of the noise energy is located at frequencies above =2, and the plot is similar to the one shown in [2] for the Poisson Disk distribution.
The most important aspect to be considered is, however, the method with which signi cant areas of the scene are selected where the sample density should increase. For this task, we use as local activity information the skewness and the gradient of the luminance, estimated in a neighborhood of each point of the image. The underlying idea is that, for a better image reconstruction, more points are needed in proximity of edges; however, if conventional interpolators are used, samples exactly on the edges give no useful information, because they tend to blur the reconstructed edge. The local skewness has already proved to be able to characterize the details of an image [3]; in particular, it possesses the property of being quite insensitive to noise, since common noise distribution are symmetric. In this case, the algorithm we propose for creating the re nement sampling map can be described as follows: We measure the absolute value of the local skewness as: 1 (x(i ? k; j ? l) ? (i; j ))3 ; 3 (i; j ) = N
X M
where (i; j ) is the sample mean of the data in a mask M, composed of N points. Only the pixels in which j3 j exceeds a certain threshold are selected for the re nement step mentioned above. As it can be easily observed looking at the luminance distributions in Fig. 2, 3 is zero when the processing mask in which it is measured is located in a uniform zone and it is nonzero on the edge of an object, unless the edge itself lies exactly on the center of the mask. If conventional interpolation schemes are used, this behaviour makes the skewness a very good indicator of the usefulness of a sample to improve the quality of the reconstructed image. Indeed, when the sample belongs to a nely textured or detail area, 3 6= 0 indicating the need of the corresponding luminance value for a better reconstruction; on the other hand, we have 3 = 0, and we avoid to select the sample in the following cases: { if the sample belongs to a uniform area; in this case indeed the 4NN method does not need this sample to achieve perfect reconstruction; { if the sample belongs to an area in which the luminance changes linearly, where again the 4NN method yields an exact result even without this sample; { if the sample exactly corresponds to the edge of an object (the ex point of a sigmoid model of the edge such as the one used in [4]); in this case the contribution of this sample would be disruptive, because the reconstruction would spread its value in the neighborhood, introducing blurring.
3 = 0
3 < 0
3 = 0
3 > 0
3 = 0
Figure 2: Position of the edge of an object with respect to the mask M and corresponding local luminance histograms.
Figure 3: Detail of the image \Lenna" (left) together with the resulting sampling grid (right).
To localise with further precision the samples use-
ful for reconstruction, we also employ an estimate of the local value of the luminance gradient. In fact, points exactly along the edges, which as already explained have to be avoided, are generally characterised by large values of the gradient. Consequently, it is advisable to remove from the re nement grid those samples the gradient of which is higher than a prede ned threshold. In order to keep the number of samples as low as possible, we do not allow samples of the re ned grid to be too close one to each other or to samples of the original grid. This e ect is obtained by de ning a small circular forbidden zone, analogous to the one used for the PPD sampling, around each already chosen sample. To give an idea of the sample positions which are obtained with our technique, a detail of the image \Lenna" together with the resulting adaptive sampling grid is shown in Fig. 3.
2.3 Image interpolation
Various algorithms for image interpolation from scattered grids have appeared in the literature [5, 6]. In general, they achieve better performances at the cost of a huge increase in complexity. We limit our attention here to the already mentioned 4NN technique,
suggested in [2], which has proved to be both simple and e ective. According to this approach, each unknown pixel value is obtained as the weighted average of its four closest available samples, where the weights are larger for those pixels which are closer to the point to be interpolated. Generally, the weights are chosen as the inverse of the distance d of the respective pixels from the point to be interpolated. However, other choices are possible, such as modifying the weighting function according to the distance of the farthest sample used [7]. The basic 1=d approach yields very good interpolation in smooth areas, but tends to blur image edges because pixels which are rather far still contribute to the output with weights which are not neglectable. Consequently, in order to improve the interpolation quality along the edges, it is advisable to weight even more heavily the closer points with respect to the farthest ones. In our approach, this can be easily done by using as a weight the value 1=d for points with low skewness (i.e., far from the edges) and 1=d2 for the other ones.
3 An image transmission scheme
In this section we present a possible application of the proposed adaptive interpolation scheme: a sampling and reconstruction system wich can be used to code images following a progressive transmission approach. Figure 4 shows the block structure of the algorithm. The input image, IN, is sampled using the coarse, uniform, irregular grid generated by the PPDs. Using our 4NN interpolator, these samples permit to reconstruct an alias-free, low resolution image, OUT(low res.). On the other hand, the original image is further sampled according to the re nement grid generated exploiting the activity measure; at the reconstruction side, these extra samples may be added to the previous ones to produce, again using the 4NN interpolator, the high resolution version of the image, OUT(high res.). It has to be observed that no sample position has to be transmitted to the decoder. In fact, the PPD grid is built using a pseudo-random number generator, so that it suces for the decoder to use the same seed; moreover, and more importantly, when computing 3 and the gradient we use the image obtained by recon-
IN
PPD sampler
OUT (low res.)
4NN filter 4NN filter
PPD map generator
Activity measure
PPD map generator Merge map
Activity measure
map Merge
Refining sampler
4NN filter
OUT (high res.)
Figure 4: Block structure of a sampling/reconstruction system. structing the PPD-sampled data instead of the original ones. The price to pay is of course a slightly less accurate edge detection, because we compute skewness and gradient of an image which is somehow noisy with respect to the original one, and with less sharp edges. In particular, in order to avoid the detection of false contours due to the coarse interpolation, a smoothing lter is applied to the low resolution image before the edge detection process.
4 Experimental results
Figure 5 reports some preliminary results of the proposed sampling and reconstruction system, obtained from a 256 256 version of the test image \Lenna". In particular, we compare the performance of our algorithm with respect to what is obtained using a regular grid with the same number of samples. The regular grid is formed by 85 85 = 7225 points; for our algorithm, the coarse grid has 4700 samples, while 2525 more samples are used in the re nement grid. The diameter of the PPD is 5 pixels, while the forbidden distance between any two samples of the re nement grid is 1 pixel. For each pixel, the skewness has been computed on a 3 3 mask, while the gradient estimate is obtained applying the Sobel operator on the low resolution image previously smoothed by a simple 5 5 low-pass lter of approximately Gaussian shape. It is apparent that the jaggedness of the contours obtained from sampling on a regular grid disappears with the skewness-re ned random sampling. Also the PSNR with respect to the original image is signi cantly improved (from 25.27 to 26.25 dB).
5 Conclusions
We presented an algorithm for adaptive nonuniform sampling of still images. Due to the irregularity of the grid, subsampled images are free from aliasing, while good image quality is obtained by increasing the sample density in the areas of the images which are rich in details. Localisation of active areas is performed using both a skewness operator, which is fast, accurate, and robust to noise, and the local gradient. We also proposed a sampling/reconstruction scheme, where a low resolution image is transmitted
using an irregular uniform grid, and high resolution information may be added by considering further samples located in proximity of the edges of the image. A pecularity of this approach is that we locate these samples using the low resolution image, instead of the original one, so that the transmission of the samples position is not required.
Acknowledgements
This work has been partially supported by the European ESPRIT LTR Project # 20229 \NOBLESSE". The authors would like to thank R. Tomassoni for his help in the analysis and validation of the algorithm.
References
[1] Y.Y. Zeevi and E. Shlomot, \Nonuniform Sampling and Antialiasing in Image Representation," IEEE Trans. on Signal Processing, vol.41, no.3, March 1993, pp.1223-1236. [2] Y. Eldar, M. Lindenbaum, M. Porat, and Y.Y. Zeevi, \The Farthest Point Strategy for Progressive Image Sampling," IEEE Trans. on Image Processing, vol.6, no.9, Sept. 1997, pp.1305-1315. [3] G. Ramponi and S. Carrato, \Performance of the Skewness{of{Gaussian (SoG) Edge Extractor," Proc. Seventh European Signal Processing Conf., EUSIPCO-94, Edinburgh, Scotland, Sept. 13-16, 1994. [4] M. Petrou and J. Kittler, \Optimal Edge Detectors for Ramp Edges," IEEE Trans. Pattern Anal. Mach. Intell., vol.PAMI-13, no.5, pp.483491, 1991. [5] V. Caselles, J-M. Morel, C. Sbert, \An axiomatic approach to image interpolation," IEEE Trans. on Image Processing, Vol. 7, No. 3, March 1998. [6] D. J. Battle, R. P. Harrison, M. Hedley, \Maximum entropy image reconstruction from sparsely sampled coherent eld data," IEEE Trans. on Image Processing, Vol. 6, No. 8, Aug. 1997. [7] H. Le Floch, C. Labit, "Irregular image subsampling and reconstruction by adaptive sampling," Proc. ICIP'96, Lausanne, CH, 1996.
a
b
c
d
Figure 5: Subsampling by a factor of 9 of a 256x256 version of \Lenna", and reconstruction results using 4NN's. Uniform sampling grid (7225 points) (a) and reconstructed image (b); map of the adaptive grid (4700 PPD plus 2525 edge points) (c) and reconstructed image (d).
Abstract
An adaptive nonuniform sampling technique for alias-free image representation is presented. It is both fast, thanks to the use of pseudo-Poisson disks (PPDs), and e ective, because a higher sample density is used in areas which are rich in details; these zones of the image are located using a skewness-based operator, which is both simple and accurate, and gradient information. Experimental results on real-world images are presented, which show that this technique provides images free of aliasing and with good detail content. As an example of application, a progressive image coding scheme based on the proposed technique is also presented.
1 Introduction
Non-standard sampling techniques are widely used in the computer graphics literature to avoid unpleasant e ects in the reproduction of synthetic images. In particular, irregular sampling grids are used in which the sampling points are placed in an aperiodic fashion on the image. Indeed, one of the visually most annoying disturbances of standard sampling is aliasing, due to the presence of above-Nyquist frequency components in a signal sampled on a rectangular grid. Starting from the consideration that the eye has a limited number of receptors but does not introduce aliasing, sampling techniques have been devised which reproduce the spatial organization of such receptors, resembling a two-dimensional Poisson stochastic distribution. For example, good results can be obtained just by jittering the locations of a periodic grid: aliasing is attenuated or eliminated while, in exchange, noise is introduced. Even if the overall energy content of the error is the same, aliasing generates patterns that can be more easily perceived than noise. The jittered grid is an irregular but uniform distribution of samples. In some cases non-uniform grids are selected, in which the density of the samples varies as a function of the image content. Turning again to the eye as a model, we observe indeed that in the human eye the fovea exists, where the density of receptors is higher and which is made to coincide with the xation point of our gaze. This xation point in turn tends to follow the most detailed portion of the scene which is looked at. We can obtain a similar e ect by locally modifying the density of the sampling in the detailed areas of a picture.
In [1], these methods are ported closer to the image processing eld. The concept of locally band-limited image is introduced, and an approach is presented for the representation of these images by spatially-varying systems; stochastic estimation of parameters characterizing the local bandwidth is discussed, to generate an optimal sampling scheme. In [2], a farthest-point sampling strategy is developed, which permits to progressively add sample points by placing them in the middle of the least-known area. The image is then reconstructed using a simple Four-Nearest-Neighbors (4NN) interpolator, which forms a weighted average of the available samples closest to the desired pixel. An adaptive version of the sampling algorithm is also proposed in [2]; in this case, the sample density is increased in detailed areas, exploiting a nonstationary image model in which the distance between two points is modi ed if the local variance is high. The obtained results are very satisfactory, and clearly show the disappearance of the aliasing and the better reproduction of the image details in the adaptive method. It should be mentioned however that, while in the basic algorithm the sampling map can be recreated autonomously by the reconstruction system, in the adaptive version it is necessary to know the location of the samples to be able to reconstruct the image. The method we propose di ers from and improves on the technique in [2] in two ways: a much simpler (even if possibly less powerful) random technique is used to generate the uniform but irregular sampling grid, and a di erent, more e ective method is used to determine the areas in which signi cant details are present. Under the latter viewpoint, the method we propose has also the important advantage of being reproducible by the reconstruction system autonomously, without the necessity of sending a map of the sampling points. The result is a technique which allows for progressive transmission of an image: rst, a lowresolution image is constructed using a small amount of samples taken on the random grid and a slightly modi ed 4NN interpolator; then, a better quality result is achieved by a re nement step which exploits the further samples located in detail areas. In the following, we rst describe the criteria used to locate the samples needed for the interpolation.
2.2 Adaptive sample location using skewness and gradient
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
−3
−2
−1
0
1
2
3
Figure 1: Spectrum of a random PPD distribution. Then we show a possible image transmission scheme which, notwithstanding the irregularity and adaptivity of the sampling grid, does not require the transmission of the samples positions. Finally, experimental results on a real-world image are presented.
2 Algorithm description
As already mentioned, in the proposed method we determine the sample locations following a two-pass scheme. First, an irregular uniform sampling grid is considered; then more samples, close to the image edges, are added in order to improve the quality of the interpolation.
2.1 Irregular uniform sample location using PPD
We call Pseudo Poisson Disk (PPD) the method used to generate the uniform random distribution. It is a simpli ed version of the conventional Poisson Disk technique [2]: the latter, in which the distance between each sampling point and its neighbors is kept larger than a prede ned value (the disk diameter), is computationally very heavy. In our case, we simply select a random coordinate pair in an image and, at the same time, we de ne a forbidden pseudocircular area around the selected point; the mechanism is repeated until the desired number of samples is reached. Of course, the diameter of the forbidden area must be set as an inverse function of the sampling density. With a correct choice of this parameter, our PPD method shares the main characteristic of the Poisson Disk distribution, i.e. its capability of moving the energy of the noise towards high frequencies, where the sensitivity of the eye is lower. In this way the visual aspect of the resulting image is better, compared to other random distributions such as conventional Poisson or the jittered grid. Fig. 1 shows the spectrum of a PPD distribution used for sampling a 256 256 image, plotted as a 1-D function by averaging along concentric circles; it can be seen that most of the noise energy is located at frequencies above =2, and the plot is similar to the one shown in [2] for the Poisson Disk distribution.
The most important aspect to be considered is, however, the method with which signi cant areas of the scene are selected where the sample density should increase. For this task, we use as local activity information the skewness and the gradient of the luminance, estimated in a neighborhood of each point of the image. The underlying idea is that, for a better image reconstruction, more points are needed in proximity of edges; however, if conventional interpolators are used, samples exactly on the edges give no useful information, because they tend to blur the reconstructed edge. The local skewness has already proved to be able to characterize the details of an image [3]; in particular, it possesses the property of being quite insensitive to noise, since common noise distribution are symmetric. In this case, the algorithm we propose for creating the re nement sampling map can be described as follows: We measure the absolute value of the local skewness as: 1 (x(i ? k; j ? l) ? (i; j ))3 ; 3 (i; j ) = N
X M
where (i; j ) is the sample mean of the data in a mask M, composed of N points. Only the pixels in which j3 j exceeds a certain threshold are selected for the re nement step mentioned above. As it can be easily observed looking at the luminance distributions in Fig. 2, 3 is zero when the processing mask in which it is measured is located in a uniform zone and it is nonzero on the edge of an object, unless the edge itself lies exactly on the center of the mask. If conventional interpolation schemes are used, this behaviour makes the skewness a very good indicator of the usefulness of a sample to improve the quality of the reconstructed image. Indeed, when the sample belongs to a nely textured or detail area, 3 6= 0 indicating the need of the corresponding luminance value for a better reconstruction; on the other hand, we have 3 = 0, and we avoid to select the sample in the following cases: { if the sample belongs to a uniform area; in this case indeed the 4NN method does not need this sample to achieve perfect reconstruction; { if the sample belongs to an area in which the luminance changes linearly, where again the 4NN method yields an exact result even without this sample; { if the sample exactly corresponds to the edge of an object (the ex point of a sigmoid model of the edge such as the one used in [4]); in this case the contribution of this sample would be disruptive, because the reconstruction would spread its value in the neighborhood, introducing blurring.
3 = 0
3 < 0
3 = 0
3 > 0
3 = 0
Figure 2: Position of the edge of an object with respect to the mask M and corresponding local luminance histograms.
Figure 3: Detail of the image \Lenna" (left) together with the resulting sampling grid (right).
To localise with further precision the samples use-
ful for reconstruction, we also employ an estimate of the local value of the luminance gradient. In fact, points exactly along the edges, which as already explained have to be avoided, are generally characterised by large values of the gradient. Consequently, it is advisable to remove from the re nement grid those samples the gradient of which is higher than a prede ned threshold. In order to keep the number of samples as low as possible, we do not allow samples of the re ned grid to be too close one to each other or to samples of the original grid. This e ect is obtained by de ning a small circular forbidden zone, analogous to the one used for the PPD sampling, around each already chosen sample. To give an idea of the sample positions which are obtained with our technique, a detail of the image \Lenna" together with the resulting adaptive sampling grid is shown in Fig. 3.
2.3 Image interpolation
Various algorithms for image interpolation from scattered grids have appeared in the literature [5, 6]. In general, they achieve better performances at the cost of a huge increase in complexity. We limit our attention here to the already mentioned 4NN technique,
suggested in [2], which has proved to be both simple and e ective. According to this approach, each unknown pixel value is obtained as the weighted average of its four closest available samples, where the weights are larger for those pixels which are closer to the point to be interpolated. Generally, the weights are chosen as the inverse of the distance d of the respective pixels from the point to be interpolated. However, other choices are possible, such as modifying the weighting function according to the distance of the farthest sample used [7]. The basic 1=d approach yields very good interpolation in smooth areas, but tends to blur image edges because pixels which are rather far still contribute to the output with weights which are not neglectable. Consequently, in order to improve the interpolation quality along the edges, it is advisable to weight even more heavily the closer points with respect to the farthest ones. In our approach, this can be easily done by using as a weight the value 1=d for points with low skewness (i.e., far from the edges) and 1=d2 for the other ones.
3 An image transmission scheme
In this section we present a possible application of the proposed adaptive interpolation scheme: a sampling and reconstruction system wich can be used to code images following a progressive transmission approach. Figure 4 shows the block structure of the algorithm. The input image, IN, is sampled using the coarse, uniform, irregular grid generated by the PPDs. Using our 4NN interpolator, these samples permit to reconstruct an alias-free, low resolution image, OUT(low res.). On the other hand, the original image is further sampled according to the re nement grid generated exploiting the activity measure; at the reconstruction side, these extra samples may be added to the previous ones to produce, again using the 4NN interpolator, the high resolution version of the image, OUT(high res.). It has to be observed that no sample position has to be transmitted to the decoder. In fact, the PPD grid is built using a pseudo-random number generator, so that it suces for the decoder to use the same seed; moreover, and more importantly, when computing 3 and the gradient we use the image obtained by recon-
IN
PPD sampler
OUT (low res.)
4NN filter 4NN filter
PPD map generator
Activity measure
PPD map generator Merge map
Activity measure
map Merge
Refining sampler
4NN filter
OUT (high res.)
Figure 4: Block structure of a sampling/reconstruction system. structing the PPD-sampled data instead of the original ones. The price to pay is of course a slightly less accurate edge detection, because we compute skewness and gradient of an image which is somehow noisy with respect to the original one, and with less sharp edges. In particular, in order to avoid the detection of false contours due to the coarse interpolation, a smoothing lter is applied to the low resolution image before the edge detection process.
4 Experimental results
Figure 5 reports some preliminary results of the proposed sampling and reconstruction system, obtained from a 256 256 version of the test image \Lenna". In particular, we compare the performance of our algorithm with respect to what is obtained using a regular grid with the same number of samples. The regular grid is formed by 85 85 = 7225 points; for our algorithm, the coarse grid has 4700 samples, while 2525 more samples are used in the re nement grid. The diameter of the PPD is 5 pixels, while the forbidden distance between any two samples of the re nement grid is 1 pixel. For each pixel, the skewness has been computed on a 3 3 mask, while the gradient estimate is obtained applying the Sobel operator on the low resolution image previously smoothed by a simple 5 5 low-pass lter of approximately Gaussian shape. It is apparent that the jaggedness of the contours obtained from sampling on a regular grid disappears with the skewness-re ned random sampling. Also the PSNR with respect to the original image is signi cantly improved (from 25.27 to 26.25 dB).
5 Conclusions
We presented an algorithm for adaptive nonuniform sampling of still images. Due to the irregularity of the grid, subsampled images are free from aliasing, while good image quality is obtained by increasing the sample density in the areas of the images which are rich in details. Localisation of active areas is performed using both a skewness operator, which is fast, accurate, and robust to noise, and the local gradient. We also proposed a sampling/reconstruction scheme, where a low resolution image is transmitted
using an irregular uniform grid, and high resolution information may be added by considering further samples located in proximity of the edges of the image. A pecularity of this approach is that we locate these samples using the low resolution image, instead of the original one, so that the transmission of the samples position is not required.
Acknowledgements
This work has been partially supported by the European ESPRIT LTR Project # 20229 \NOBLESSE". The authors would like to thank R. Tomassoni for his help in the analysis and validation of the algorithm.
References
[1] Y.Y. Zeevi and E. Shlomot, \Nonuniform Sampling and Antialiasing in Image Representation," IEEE Trans. on Signal Processing, vol.41, no.3, March 1993, pp.1223-1236. [2] Y. Eldar, M. Lindenbaum, M. Porat, and Y.Y. Zeevi, \The Farthest Point Strategy for Progressive Image Sampling," IEEE Trans. on Image Processing, vol.6, no.9, Sept. 1997, pp.1305-1315. [3] G. Ramponi and S. Carrato, \Performance of the Skewness{of{Gaussian (SoG) Edge Extractor," Proc. Seventh European Signal Processing Conf., EUSIPCO-94, Edinburgh, Scotland, Sept. 13-16, 1994. [4] M. Petrou and J. Kittler, \Optimal Edge Detectors for Ramp Edges," IEEE Trans. Pattern Anal. Mach. Intell., vol.PAMI-13, no.5, pp.483491, 1991. [5] V. Caselles, J-M. Morel, C. Sbert, \An axiomatic approach to image interpolation," IEEE Trans. on Image Processing, Vol. 7, No. 3, March 1998. [6] D. J. Battle, R. P. Harrison, M. Hedley, \Maximum entropy image reconstruction from sparsely sampled coherent eld data," IEEE Trans. on Image Processing, Vol. 6, No. 8, Aug. 1997. [7] H. Le Floch, C. Labit, "Irregular image subsampling and reconstruction by adaptive sampling," Proc. ICIP'96, Lausanne, CH, 1996.
a
b
c
d
Figure 5: Subsampling by a factor of 9 of a 256x256 version of \Lenna", and reconstruction results using 4NN's. Uniform sampling grid (7225 points) (a) and reconstructed image (b); map of the adaptive grid (4700 PPD plus 2525 edge points) (c) and reconstructed image (d).
