KAISER FILTER FOR ANTIALIASING IN DIGITAL PHOTOGRAMMETRY

Transcription

1 KAISER FILTER FOR ANTIALIASING IN DIGITAL PHOTOGRAMMETRY Kourosh Khoshelham Dept. of Land Surveying and Geo-Informatic, The Hong Kong Polytechnic University, Ali Azizi Dept. of Surveying Engineering., University of Tehran, Iran. Abstract Image Aliasing is a problem appearing as artifacts in digitally resampled images, which degrades the quality of the image. In digital rectification and texture mapping pixels from an input image are transformed to pixels of an output image. Discrete nature of a digital image causes Aliasing in transformed image. In this paper source of aliasing and theory of antialiasing is described. The necessity of a precise filter design in antialiasing is discussed and a filter based on Kaiser adjustable window is designed. Different practical antialiasing methods are described as well as interpolation methods, which are conventional in photogrammetry. Selected antialiasing methods are implemented and applied on a close range image. An objective analysis is carried out by applying the inverse transforms to rectified images and deriving some measures to estimate the information loss for each method by comparing original and reconstructed images. Results indicate that interpolation methods are not capable of removing or reducing Aliasing in highly decimating transformations. The output images of interpolation methods therefore suffer from edge corruption and interfusion of small features. Applying kaiser filter with a precise AntiAliasing method results in the least information loss and considerably reduces aliasing at the expense of higher computation load. KEY WORDS: Digital Rectification, Aliasing, Resampling, Kaiser Filter, Antialiasing, Filtering.. Introduction Image Aliasing is a problem appearing as artifacts in digitally resampled images, which degrades the quality of the image. This degradation appears as jagged edges, interfusion and disappearance of small features and moiré patterns. This especially happens in highly decimating transformations where a large number of input pixels in the original image are transformed to a single output pixel in the rectified image. Therefore the sampling interval must be chosen very short in order to transform the entire data from input to output image, which is not feasible in most cases. Investigating the aliasing problem in photogrammetry is very necessary especially when we face the fact that in aerial and close range photogrammetry, digital rectification of oblique images is a very common task. For example in close range projects it is often desirable to have a more visualized output by mapping a texture image, taken from the object, to the final CAD vector model. However taking picture from an appropriate position is not always possible and one has to take an oblique picture and apply a highly decimating transformation to map the texture, which results in aliasing. Antialiasing methods for removing or reducing image aliasing have been more investigated by researchers in the field of computer graphics. In one of the first texture mapping techniques, Catmull (974) considered the aliasing problem, however for the first time it was Crow (977) who presented the idea of Prefiltering for image antialiasing. Prefiltering appeared in many subsequent research works as a basic antialiasing technique (Feibush et. al., 980, Crow, 984, Glassner, 986, Heckbert, 989, Ghazanfarpour, 99). Williams (983) introduced the idea of image pyramids to develop a fast filtering method and Gagnet et al., (982) presented oversampling method with bilinear interpolation for image antialiasing. In photogrammetric context the problem is usually discussed in brief and interpolation methods are suggested as solution. In almost all photogrammetric softwares interpolation methods are the only options for rectifying an image. However performance of the interpolation methods in rectification of oblique images and their ability in removing or reducing the aliasing has not been shown. The situation indicates the necessity of a thorough study on the performance of the interpolation methods and application of antialiasing techniques in digital photogrammetry. The objective of this research is therefore to evaluate the conventional interpolation methods as well as existing antialiasing

2 techniques and design an optimized resampling filter in order to preserve the most possible amount of image information during the rectification process, which will also improve the visual quality of the rectified image by reducing the degradations caused by aliasing. The paper proceeds with a technical description of the resampling theory and what causes the aliasing problem in the next section. Practical limitations for antialiasing methods are discussed in section 3 where windowing method for filter design is introduced. A brief survey of the existing antialiasing methods is brought in section 4. Experimental results of applying selected methods and the filter designed based on kaiser adjustable window on a close range image comes in section 5. The paper ends with conclusion in section Resampling and Aliasing In a digital transformation pixels from an input image are transformed to pixels in an output image. Then for each pixel within this output image, color or gray value of the correspondent pixel in input image should be assigned. Depending on the nature of the geometric errors, affine, projective or polynomial models can be used as a mathematical model for this transformation. There are two main methods to apply a transformation to a digital image: inverse mapping and forward mapping. The algorithm for inverse and forward mapping methods can be as simple as the followings: Inverse mapping: For y For x Compute u(x, y) and v(x, y) Output[x, y] = input[u, v] End End Forward mapping: For v For u Compute x(u, v) and y(u, v) Output[x, y] = input[u, v] End End The problem one may encounter in this procedure is that pixels after transformation necessarily do not have integer coordinates. In other words, the regular grid of texture pixels after transformation appears to be an irregular distribution of pixels. Therefore, pixels after applying transformation, using inverse/forward mapping, fall in space between the input/output pixels. A digital image can be considered as a discrete twodimensional signal. The problem mentioned above is due to the discrete nature of this two-dimensional signal. Therefore, a solution to this problem may be achieved by reconstructing a continuous signal from the original discrete signal. Thus, geometric transformation may be applied to this continuous signal. Finally, sampling the continuous transformed signal produces a discrete output image. Reconstruction step is carried out using a reconstruction filter. It can be shown mathematically that this reconstruction filter is a sinc function. This is because a discrete signal in frequency domain has a set of repeated copies of the main spectrum over the frequency range through which only the central one appears in the spectrum of the continuous correspondent signal. The reconstruction filter consequently must be a box filter in frequency domain, which, keeps the central copy and removes other copies (Fig. ). A box function in frequency domain corresponds to a sinc function in space domain (Fig. 2). However, applying a sinc function with infinite range to the signal makes its practical implementation problematic. After transformation, the continuous transformed signal must be sampled to generate output discrete signal. The sampling function in space domain is a so-called comb function defined by a train of equally spaced impulses. A continuous function can be sampled if it is multiplied by the comb function. In frequency domain, the sampling process may be expressed by convolution of the Fourier transform of the comb function and the continuous signal. The Fourier transform of a comb with impulses that are placed (d) apart, is also a comb but with impulses that are placed (/d) apart. 2

3 Figure : The Reconstruction filter is a box in frequency domain. Figure 2: The Reconstruction filter is a sinc in space domain. Thus, convolution in frequency domain makes the spectrum of a discrete signal appear as the replica of the spectrum of the continuous signal placed /d apart. This means that the selection of an appropriate sampling interval in comb function is crucial. The sampling interval should be short enough so that neighboring copies do not overlap. In this sense Shannon (949) has presented sampling theorem: A continuous signal with frequency range of Wb can be reconstructed precisely from a discrete signal, if sampling frequency, Ws, is at least twice Wb. This sampling frequency is called nyquist frequency. Figure 3A shows the result of sampling with nyquist frequency. Because sampling interval is short enough, the neighboring copies of the signal spectrum do not overlap. From a practical point of view and considering the limitations of computers in memory and speed, in some applications considering a sampling interval so small that satisfies the nyquist criterion is not possible. In digital rectification and texture mapping selection of such a short sampling interval helps us to avoid Aliasing in areas where the transformation is highly decimating. But it will lead to a huge number of redundant pixels in areas where the transformation is not highly decimating, e.g. near the horizon in aerial oblique images. Therefore there is a trade off between sampling interval and computation cost and in practice we have to select a larger sampling interval. This causes the neighboring copies to overlap. It can be seen from figure 3B that high frequencies are added to low frequencies in overlaps. This is the Aliasing problem. As Fig. 3.B shows in considerable parts of the spectrum low frequencies are damaged as well as high frequencies. 3

4 (A) Figure 3: A sampled signal in frequency domain: A) Copies do not overlap (No Aliasing), B) Copies overlap (Aliasing) This results in an output image of a poor quality. The large sampling interval degrades the image quality, which results in moiré patterns (figure 4) and jagged edges (figure 5). (B) Figure 4: The moiré pattern in right image is due to inconsiderate sampling (from Wolberg, 990). Figure 5: Jagged edges due to Aliasing in a checkerboard image. The Prefiltering technique (Crow, 977) for antialiasing is based on filtering the signal before transformation in order to remove some high frequency components of the signal. For such a signal then a larger sampling interval is applicable. Prefiltering simply applies a low pass filter to image before 4

5 sampling. It should be noted that aliasing damages both low frequency and high frequency components of the image, whereas prefiltering sacrifices a few number of high frequency components in order to save more valuable low frequencies. Figure 6A shows the prefilter applied on a signal in frequency domain. For prefiltered signal then neighboring spectrums do not overlap. Hence there is no Aliasing (figure 6B). Considering the significance of prefiltering, the resampling procedure can be modified by adding a prefiltering step. Therefore an ideal resampling procedure consists of four basic steps (Smith, 983): Reconstruction, transformation, prefiltering and sampling (figure 7). (A) (B) Figure 6: A) Prefiltering in frequency domain. B) Prefiltering prevents Aliasing. Figure 7: Ideal resampling (from Wolberg, 990). 5

6 3. Resampling in practice Ideal resampling as described in section 2 cannot be implemented practically. Reconstruction filter is a sinc function, which is infinite in range. This sort of filters are called Infinite Impulse Response (IIR). An ideal low pass filter for prefiltering also has to be an IIR. A filter, for which implementation is possible in practice, should be a Finite Impulse Response (FIR). Thus in practice we have no choice except approximating IIR filters with their associated FIR ones. Box, Triangular and gaussian filters in space domain - known as B-Spline filters - have been used for filtering, but their frequency responses are far from ideal box in frequency domain. Therefore they either cannot reduce Aliasing or result in excessive blurring. One other solution is to truncate the sinc function in a limited range. Figure 8 shows how this operation affects the filter frequency response. The ripples in the filter frequency response are known as Gibbs phenomenon and shows up as a ghost in the filtered image (Strang, 996) Figure 8: Truncated sinc and its frequency response Windowing method for filter design is based on smoothing out the ripples in frequency response. Assume that the frequency response of a window function is convolved with the frequency response of the truncated sinc. This is again like a low pass filtering in order to produce a smoother frequency response for truncated sinc. The resulting filter is then inverse Fourier transform of this smoothed function. In practice however this can be done in space domain simply by multiplying truncated sinc by the window function. There are a number of window functions with fixed or adjustable shape. In this research an adjustable Kaiser window with parameter β = 4 is used for filter design. An L2 norm criterion showed that this filter best fits the ideal low pass filter comparing to other windows (figure 9). Therefore better can be expected from Kaiser filter for antialiasing, as it is more optimized comparing to other filters Figure 9: Filter design using Kaiser window in space and frequency domain 6

7 In order to implement ideal resampling more efficiently, Heckbert (989) combined reconstruction filter and prefilter into a single filter, which is called resampling filter : m ρ ( x, k ) = h( x m( u)) r( u k) du () n u R Where u, x are input and output pixel coordinates respectively, r is the reconstruction filter, h is the prefilter and m is the transformation model. The resampling filter ρ ( x, k) is the weight of an input pixel at location k for an output pixel at location x (Heckbert, 989). 4. Survey of AntiAliasing techniques The conventional resampling techniques in photogrammetry are interpolation methods. The simplest interpolation method is nearest neighbor (point sampling), in which each output pixel is transformed to input original image and the color of the nearest pixel is assigned to the output pixel. Bilinear interpolation and cubic convolution are better interpolators since they fit bilinear and cubic surfaces on 4 and 6 neighboring pixels respectively to interpolate the value of the pixel under inspection. The problem appears when more than 6 input pixels in the original image map to a single output pixel in the rectified image. In this case applying interpolation methods result in information loss since the entire information content of the input image is not transformed to the output image. Antialiasing techniques therefore aim for preserving as much information as possible during the resampling process. In order to select the most suitable technique for use in photogrammetric applications we proceed with a brief review of the existing antialiasing techniques. 4.. Catmull, 974 The oldest implementation of a resampling filter returns back to the Subdivision Patch Renderer (SPR) algorithm presented by Catmull in 974 (Catmull, 974). SPR uses an inverse mapping technique. Catmull considered a pixel in output space as a square with sample point at the center. Algorithm starts with transforming four corners of this square to input texture image. In the input space however these corners define a quadrilateral. Unweighted average of the color of the pixels inside the bounding box of the quadrilateral defines the color of the output pixel. This method is relatively fast in terms of computation cost however filtering is not precise since it calculates an average over a rectangle Blinn and Newell, 976 Considering the fact that simple averaging in SPR method acts as a box filter, it can be modified by a triangular Tent filter. Blin and Newell (976) used a pyramid filter with a square base each side of which is two pixels long, in order to allow overlap between neighboring pixels. The quadrilateral is approximated by a Parallelogram instead of rectangle in this method. In other words the mathematical model is approximated by an affine model in the neighborhood of the pixel. Filter function then determines the weight of each pixel in order to compute a weighted average of the pixels inside the parallelogram. Comparing to the method of Catmull (974), this method yields a fairly slight improvement since the box filter is replaced with tent Feibush et al., 980 In contrast with previous methods, this technique is more relevant with resampling theory. It can be implemented with any filter and uses a Look Up Table (LUT), which highly speeds up the computations (Feibush et al., a980). These smart features enable the method to implement high quality filters with a reasonable speed. The algorithm starts with placing the two-dimensional filter function on the center of an output pixel and calculating its bounding rectangle. This rectangle is transformed into the input texture image and again a bounding rectangle is calculated for the resulting quadrilateral. All pixels inside this rectangle should be transformed to output image where a weighted average is computed for these pixels. Weighted averaging is carried out using a LUT in which filter function values for each pixel with regard to 7

8 its distance from the center of the filter are stored. In other words two-dimensional filtering is reduced to a one-dimensional filtering operation using LUT technique Gagnet et al., 982 This method is similar to the method of Feibush et al., except here selection of the samples is done in output space (Gagnet et al., 982). Filters with circular bases are used with a radius larger than a pixel width. Circular base is then approximated with an ellipse after transformation. A regular grid of samples with intervals smaller than pixel size is constructed in output space and placed on output pixels. Samples interval is proportional to the larger side of a parallelogram bounding the ellipse. For example if this side is three times larger than pixel size then a sampling interval of /3 output pixel size must be selected. Each sample point is then transformed to input texture image and its color is computed using a bilinear interpolation. Finally a weighted average of the samples using a truncated sinc weight function is associated to the pixel based on the sample location with regard to the center of the pixel Trilinear Interpolation The trilinear interpolation technique is a fast and clever algorithm, which makes use of filtered copies of texture image in a pyramidal parametric data structure called MIP Map (Williams, 983). In order to find the color for an output pixel first one or two relevant levels in MIP Map must be selected. Location of the sample in each level then is determined using output pixel coordinates and a color is computed by a bilinear interpolation of four neighboring pixels. After doing this intera-level interpolation, a linear interlevel interpolation should be carried out between the values computed in each of the two levels in order to find the color of output pixel. This technique is one of the fastest existing techniques. For calculation of the color of each point, only 8 MIP Map access, 7 multiplications and 4 summations are required. Constructing the MIP Map levels needs a relatively short time as well Summed Area Table Instead of using image pyramid, Crow (984) suggested to construct a table with the same size as the texture image, namely Summed Area Table (SAT). The value of each pixel in the table is replaced by the summation of the colors of all pixels inside a rectangle defined by that pixel and lower left corner of the input image. Assume an output pixel is transformed to a rectangle in input texture image with coordinates xl, yb for the lower left corner and xr, yt for the upper right corner. The summation of pixel values in this rectangle can be calculated in SAT as: T[xr, yt] T[xr, yb] T[xl, yt] + T[xl, yb] Therefore the algorithm starts with an inverse mapping transformation of output pixels into the texture image. The bounding rectangle is then calculated for the resulting arbitrary quadrilateral similar to the method of Catmull (974). But averaging the values in this rectangle is done in SAT. Thus for highly decimating transformations in which each output pixel transforms to thousands of input pixel, the computation load decreases down to 4 SAT accesses, 3 summations and 2 multiplications Repeated Integration Filtering An interesting extension of the SAT method presented by Perlin (985) and Ferrai et al., (986). This method is based on the rule that if one keeps on constructing SAT for n times, each time on the last created SAT, then an ellipsoid area of texture image with diameters parallel to pixels grid can be filtered by 2(n+) times sampling the SAT. In other words this methods achieves the advantage of a more precise filter (B-Spline) from the nth order instead of a box filter in the first level SAT, which in fact is a B-Spline of the first order Heckbert, 989 Heckbert considered the fact that if reconstruction filter and prefilter have an ellipsoid shape with gaussian cross section and the mathematical model for the transformation can be approximated by an affine model locally, then resampling filter will be an ellipsoid gaussian filter as well. Thus he designed an ellipsoid filter with gaussian cross section called Elliptical Wighted Average (EWA) (Heckbert, 989). For each pixel within this filter a weight value is calculated based on its distance to the center of the filter. 8

9 Weights are stored in a LUT before transformation. Therefore for each output pixel, ellipsoid parameters in input texture space are computed using local affine approximation. Then a so-called Q index is calculated which shows the distance between the pixel and the center of the ellipsoid. Q index is used to access LUT and find the associated weight for the pixel. Finally a weighted average of the colors of the pixels falling inside the filter is calculated and assigned to the output pixel Forward Mapping Antialiasing The method developed by Ghazanfarpour (99) can be considered as a different method since it seems to be the only method, which uses forward mapping. The algorithm can work with any filter function as the cross section of a circular based filter. Algorithm starts with an initialization step, which computes and stores the filter in a LUT. It then transforms the pixels from input texture image to output space and calculates the distance of transformed pixel from neighboring output pixels. Then for each output pixel for which this distance is smaller than the filter radius, a portion of the color of the input pixel is assigned. The amount of this portion is selected based on the weight associated with the distance in the LUT. The summation of these portions for each output pixel determines its color at the end of the transformation. The forward mapping antialiasing technique works with input pixels rather than output pixels, therefore when number of output pixels is less than input pixels; it suffers from an extra computation load. Another issue with this method is the danger of occurring holes. Because it works with input pixels, it is possible that algorithm assigns no value for some output pixels. Those pixels are called holes and appear as white or black spots in the output image. Choosing a filter base with a larger support is a good solution, if the computational load is tolerated. Since holes occur in areas of low decimation, a bilinear interpolation can also be used to fill the holes. For highly decimating transformations, however, this no longer is a problem, for each output pixel is associated with a large number of input pixels. 5. Experimental Evaluation Summed Area Table (Crow, 984) and Forward Mapping Antialiasing (Ghazanfarpour, 99) are selected in addition to interpolation methods for experimental evaluation in this section. Summed Area table is a fast antialiasing method, however the type of filter it applies, box filter, is far from ideal. Forward mapping antialiasing on the other hand imposes higher computation cost, but it performs a precise filtering with an arbitrary filter shape. Therefore this technique provides the opportunity to apply an optimized filter designed based on Kaiser window. In summary these two filters represent two important aspects of antialiasing techniques: efficiency and accuracy. In order to evaluate the accuracy of the antialiasing techniques a close range gray scale image of the size 900x700 is selected (figure 0). A projective transformation model is used to rectify the image by applying each of the selected antialiasing techniques. The rectified images are then reconstructed by applying the inverse transformation. Root mean square error (RMSE) and peak signal to noise ratio (PSNR) measures between original and reconstructed images are computed for analysis of the results. These measures can be used to estimate the information loss in the rectification process and evaluate the accuracy of the corresponding antialiasing technique. A good reconstruction can be achieved by a more accurate antialiasing and results in lower RMSE and higher PSNR values. For 8 bit gray scale images these objective measures are defined as: RMSE m n r o = ( x ij xij ) m. n i = j= 2 2 PSNR = 0 log RMSE Where m and n are image dimensions, x and r ij o x ij are the grey values of a pixel in the reconstructed and original images respectively. Figure shows the rectified images using selected antialiasing techniques and reconstructed images from inverse transformation are given in figure 2. Kaiser filter is used with forward mapping antialiasing technique in figures E and 2E. Table summarizes the accuracy measures and computation time for each method. Computation times are given for comparison only and can be reduced in more optimized algorithms. 9

10 Table : Accuracy and efficiency measures for selected antialiasing techniques. Nearest Neighbor Bilinear Interpolation Cubic Convolution Summed Area Table Forward Mapping Antialiasing RMSE PSNR Computation Time (second) As can be seen from table, the forward mapping antialiasing technique with Kaiser filter shows the highest accuracy with both accuracy measures. Summed area table method also performs well according to measures given in table however it has blurred the entire rectified image as can be seen in figure D. A close examination of figure reveals that in interpolation methods aliasing artifacts appear in spirit of corrupted edges and interfused small features. These artifacts are more in nearest neighbor interpolation and less in bilinear interpolation. In figures A,B,C note that how Aliasing corrupts edges for instant in the three windows at the top of the image. It can be said that interpolation methods are not able to retrieve these corrupted edges, because they use gray scale information of not more than 6 pixels. Figure 0: Original image In forward mapping antialiasing method (figure E and 2E) the size of Kaiser filter is in fact determined by the transformation so that it covers all input pixels that are mapped to a particular output pixel. Therefore this approach enjoys less information loss as compared with aforementioned techniques. The shape of the filter is also designed in such a way that amplifies the edges. This is because of the negative lobe of the Kaiser filter (see figure 9). These advantages result in a clear rectified image with sharp edges as can be seen in figure E. According to table, precise filtering is more costly and the computation load is higher for forward mapping antialiasing technique. Therefore for very large images and when aliasing problem is not very crucial, fast antialiasing methods such as summed area table or trilinear interpolation can be recommended. In fact there is always a compromise between accuracy and computation expense in antialiasing methods. 0

11 Figure : Examples of different AntiAliasing techniques: A) Nearest Neighbour interpolation, B) Bilinear Interpolation, C) Cubic Convolution, D) Crow s Summed Aread table, E) Forward Mapping Antialiasing with Kaiser filter.

12 Figure 2: Reconstructed images from rectified images. A. Nearest Neighbor, B. Bilinear Interpolation, C. Cubic Convolution, D. Summed Area Table, E. Forward mapping antialiasing with Kaiser filter. 2

13 6. Conclusion Although in some cases conventional interpolation methods produce images with reasonable quality, but in case of aliasing applying a precise resampling filter is inevitable. This happens in areas where the transformation is highly decimating. Thus a large number of input pixels transform to one pixel in output space. Nearest Neighbor, Bilinear Interpolation and Cubic Convolution interpolation methods use, 4 and 6 pixels respectively to interpolate the gray level of the output pixel. Therefore in case of aliasing, interpolation methods are not capable of preserving the entire information content of the original image in the rectified image. Precise antialiasing requires design of a FIR low pass filter which best approximates the ideal sinc. The antialiasing technique also has to provide the opportunity of using various filters. Methods of Feibush (980), Heckbert (989) and forward mapping antialiasing (Ghazanfarpour, 99) have such an advantage. Results of this research indicate the higher accuracy of the precise antialiasing techniques with a Kaiser resampling filter. These techniques however impose more computation load and one has to trade off between accuracy and computation expense. For highly decimating transformations, where aliasing is more, a precise filtering is required, which leads to more computation expense. However for very large images fast antialiasing methods such as summed area table can be used with lower accuracy. 7. References Blinn, J. F. and M. E. Newell (976) : Texture and Reflection in Computer Generated Images, CACM, Vol. 9, No. 0. Catmull, E. (974) : A Subdivision Algorithm for Computer Display of Curved Surfaces, PhD thesis, Dept. of CS, U. of Utah, Dec Crow, F. C., (977) : The Aliasing Problem in Computer-Generated Shaded Images, CACM, Vol. 20. Crow, F.C., (984) : Summed-Area Tables for Texture Mapping, Computer Graphics, (SIGGRAPH 84 Proceedings), Vol. 8, No. 3. Feibush, E. A., Levoy, M., R. L. Cook (980) : Synthetic Texturing Using Digital Filters, Computer Graphics, (SIGGRAPH 80 Proceedings), Vol. 4, No. 3. Ferrari, L. A., Sankar, P. V., Sklansky, J., Sidney Leeman (986) : Efficient Two-Dimensional Filters Using B-Spline Approximations, Computer Vision, Graphics, and Image Processing. Gangnet, M., Perny, D., P. Coueignoux (982) : Perspective Mapping of Planar Textures, Eurographics 82, 982 (slightly superior to the version that appeared in Computer Graphics, Vol. 6, No., May 982). GhazanfarPour, D., B. Peroche (99) : A high quality filtering using forward texture mapping,, computers & Graphics, Vol. 5, No. 4. Glassner, A. S., (986) : Adaptive precision in texture mapping, SIGGRAPH 86 proceedings, Vol. 20, No. 4. Heckbert, P. S., (986) : Survey of texture mapping techniques, IEEE Computer Graphics and Applications, Vol. 6, No.. Heckbert, P. S. (989) : Fundamentals of texture mapping and image warping, Master s Thesis, Dept. of Electrical Engineering and computer Science, University of California, Berkeley, California. Perlin, K., (985) : Course Notes, SIGGRAPH 85 State of the Art in Image Synthesis seminar notes, Shannon, C. E., (949) : Communication in the presence of noise, IRE proceedings, Vol. 37, No.. Smith, A. R., (983) : Digital Filtering Tutorial for Computer Graphics, parts and 2, SIGGRAPH 83 Introduction to Computer Animation seminar notes. Strang, G., Nguyen, T., (996) : Wavelets and Filter Banks. Wellesley-Cambridge Press. Williams, L., (983) : Pyramidal Parametrics, Computer Graphics, (SIGGRAPH 83 Proceedings), Vol. 7, No. 3. Wolberg, G., (990) : Digital Image Warping, Los Alamitos, Calif. : IEEE Computer Society Press. 3