kaiser filter for antialiasing in digital photogrammetry

KAISER FILTER FOR ANTIALIASING IN DIGITAL

PHOTOGRAMMETRY

Kourosh Khoshelham ([email protected])

The Hong Kong Polytechnic University

Ali Azizi ([email protected])

University of Tehran, Iran

Abstract

Image aliasing is a problem appearing as artefacts in digitally

resampled images, which degrades the quality of the image. In digitalrectification and texture mapping, pixels from an input image aretransformed to pixels of an output image. The discrete nature of a digital

image causes aliasing in the transformed image. In this paper the source ofaliasing and the theory of antialiasing are described. The necessity of aprecise filter design in antialiasing is discussed and a filter based on a Kaiser

adjustable window is designed. Different practical antialiasing methods aredescribed as well as interpolation methods, which are conventional inphotogrammetry. Selected antialiasing methods are implemented and applied

to a close range image. An objective analysis is carried out by applyinginverse transformations to rectified images and deriving some measures toestimate the information loss for each method by comparing original andreconstructed images. Results indicate that interpolation methods are not

capable of removing or reducing aliasing in highly decimating transforma-tions. The output images of interpolation methods therefore suffer from edgecorruption and interfusion of small features. Applying a Kaiser filter with a

precise antialiasing method results in the least information loss andconsiderably reduces aliasing at the expense of higher computation load.

Keywords: digital photogrammetry, rectification, aliasing, resampling,Kaiser filter, antialiasing, filtering

Introduction

Image aliasing is a problem appearing as artefacts in digitally resampled images,which degrades the quality of the image. This degradation appears as jagged edges,interfusion and disappearance of small features and moir�e patterns (Wolberg, 1990).Aliasing especially occurs in highly decimating transformations where a large numberof input pixels in the original image are transformed to a single output pixel in therectified image. Therefore, the sampling interval must be chosen very short in order totransform all the data from the input to the output image, which is not feasible in mostcases. Investigating the aliasing problem in photogrammetry is very necessary

The Photogrammetric Record 19(105): 22–37 (March 2004)

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especially as digital rectification of oblique images is a very common task in aerial andclose range photogrammetry. For example, in close range projects it is often desirableto have a more visual output by mapping a texture image, taken from the object, to thefinal CAD vector model. However, taking the picture from an appropriate position isnot always possible and an oblique picture may have to be taken and a highlydecimating transformation applied to map the texture, which results in aliasing.

Gonzalez and Wintz (1987) provide a good description on spectral characteristicsof sampled images and the sampling process where aliasing occurs. Antialiasingmethods for removing or reducing image aliasing have been more investigated byresearchers in the field of computer graphics. In one of the first texture mappingtechniques, Catmull (1974) considered the aliasing problem, however, it was Crow(1977) who first presented the idea of prefiltering for image antialiasing. Prefilteringappeared in many subsequent research works as a basic antialiasing technique (Feibushet al., 1980; Crow, 1984; Glassner, 1986; Heckbert, 1989; Ghazanfarpour and Peroche,1991). Williams (1983) introduced the idea of image pyramids to develop a fastfiltering method and Gagnet et al. (1982) presented the oversampling method withbilinear interpolation for image antialiasing. In the photogrammetric context theproblem is usually discussed in brief and interpolation methods are suggested as asolution (Wang, 1990; Schenk, 1999; Mikhail et al., 2001). In almost all photogram-metric software, interpolation methods are the only options for rectifying an image.However, performance of the interpolation methods in rectification of oblique imagesand 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 theinterpolation methods and application of antialiasing techniques in digital photogram-metry. While antialiasing techniques traditionally use a gaussian filter as a high-qualityfilter, the major contribution of this work is to design an optimised resampling filter,based on a Kaiser window, and utilise it in an accurate antialiasing technique in order topreserve the greatest possible amount of image information during the rectificationprocess and improve the visual quality of the rectified image by reducing thedegradations caused by aliasing. The objective of this research is therefore to evaluatethe performance of a Kaiser filter in an accurate antialiasing technique comparing it withconventional interpolation methods as well as existing antialiasing techniques.

The paper proceeds with a technical description of the resampling theory andwhat causes the aliasing problem in the next section. Practical limitations forantialiasing methods are discussed in the third section where the windowing methodfor filter design is introduced. A brief survey of the existing antialiasing methods isgiven in the fourth section. Experimental results of applying selected methods and thefilter design based on a Kaiser adjustable window on a close range image is given inthe fifth section. The paper ends with a conclusion in the final section.

Resampling and Aliasing

In a digital transformation, pixels from an input image are transformed to pixels inan output image. Then for each pixel within this output image, the colour or grey valueof the corresponding pixel in the input image should be assigned. Depending on thenature of the geometric errors, affine, projective or polynomial models can be used as amathematical model for this transformation. There are two main methods to apply atransformation to a digital image: inverse mapping and forward mapping. The

The Photogrammetric Record

� 2004 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd. 23

algorithm for inverse and forward mapping methods can be as simple as the following:

Inverse mapping:

For yFor x

Compute uðx; yÞ and vðx; yÞOutput ½x; y� ¼ input ½u; v�

EndEnd

Forward mapping:

For vFor u

Compute xðu; vÞ and yðu; vÞOutput ½x; y� ¼ input ½u; v�

EndEnd

where [u, v] are pixel coordinates in the input image and [x, y] are pixel coordinatesin the output image. The problem that may be encountered in this procedure is thatpixels after transformation do not necessarily have integer coordinates. In otherwords, the regular grid of texture pixels after transformation appears to be anirregular distribution of pixels. Therefore, pixels after applying transformation, usinginverse/forward mapping, fall in the space between the input/output pixels. A digitalimage can be considered as a discrete two-dimensional signal. The problemmentioned above is due to the discrete nature of this two-dimensional signal.Therefore, a solution to this problem may be achieved by reconstructing acontinuous signal from the original discrete signal. Thus, a geometric transformationmay be applied to this continuous signal. Finally, sampling the continuoustransformed signal produces a discrete output image.

The reconstruction step is carried out using a reconstruction filter, which is a boxfunction in the frequency domain that corresponds to a sinc function in the spacedomain. The sampling function in the space domain is a so-called comb functiondefined by a train of equally spaced impulses. A continuous function can be sampled ifit is multiplied by the comb function. In the frequency domain, the sampling processmay be expressed by convolution of the Fourier transform of the comb function andthe continuous signal. The Fourier transform of a comb with impulses that are placed dapart, is also a comb but with impulses that are placed 1/d apart. Thus, convolution inthe frequency domain makes the spectrum of a discrete signal appear as the replica ofthe spectrum of the continuous signal placed 1/d apart. This means that the selection ofan appropriate sampling interval in the comb function is crucial. The sampling intervalshould be short enough so that neighbouring copies do not overlap. In this senseShannon (1949) has presented the sampling theorem: a continuous signal withfrequency range of Wb can be reconstructed precisely from a discrete signal, ifsampling frequency, Ws, is at least twice Wb. This sampling frequency is called theNyquist frequency. More details on the sampling theorem and the Fourier transformcan be found in Baher (2001).

Khoshelham and Azizi. Kaiser filter for antialiasing in digital photogrammetry

24 � 2004 The Remote Sensing and Photogrammetry Society and Blackwell Publishing Ltd.

Fig. 1(a) shows the result of sampling with the Nyquist frequency. Because thesampling interval is short enough, the neighbouring copies of the signal spectrum donot overlap. From a practical point of view and considering the memory and speedlimitations of computers, in some applications considering a sampling interval so smallthat satisfies the Nyquist criterion is not possible. In digital rectification and texturemapping, selection of such a short sampling interval helps us to avoid aliasing in areaswhere the transformation is highly decimating. But it will lead to a huge number ofredundant pixels in areas where the transformation is not highly decimating, forexample, near the horizon in aerial oblique images. Therefore, there is a trade-offbetween sampling interval and computation cost and in practice a larger samplinginterval has to be selected. This causes the neighbouring copies to overlap. It can beseen from Fig. 1(b) that high frequencies are added to low frequencies in overlaps.This is the aliasing problem. As Fig. 1(b) shows, in considerable parts of the spectrumlow frequencies are damaged as well as high frequencies. This results in an outputimage of poor quality. Fig. 2 shows a warped checkerboard image where selection oflarge sampling intervals has caused jagged edges as a result of aliasing.

The prefiltering technique (Crow, 1977) for antialiasing is based on filtering thesignal before transformation in order to remove some high-frequency components ofthe signal. For such a signal then a larger sampling interval is applicable. Prefiltering

40

30

20

10

00 50

(a) (b)

100 150

40

30

20

10

00 10 20 30 40 50 60

Fig. 1. A sampled signal in the frequency domain:(a) copies do not overlap (no aliasing); (b) copies overlap (aliasing).

Fig. 2. Jagged edges due to aliasing in a checkerboard image.



simply applies a low-pass filter to the image before sampling. It should be noted thataliasing damages both low-frequency and high-frequency components of the image,whereas prefiltering sacrifices a few of the high-frequency components in order to savemore valuable low frequencies. Fig. 3(a) shows the prefilter applied on a signal in thefrequency domain. For a prefiltered signal, neighbouring spectra do not overlap, hencethere is no aliasing (Fig. 3(b)).

Considering the significance of prefiltering, the resampling procedure can bemodified by adding a prefiltering step. Therefore, an ideal resampling procedureconsists of four basic steps (Smith, 1983): reconstruction, transformation, prefilteringand sampling (Fig. 4).

Resampling in Practice

Ideal resampling as described in the second section cannot be implementedpractically. The reconstruction filter is a sinc function, which is infinite in range. This

40

30

20

10

00 20 40

40

30

20

10

0

(a) (b)

40

30

20

10

00 10 20 30 40 50 600 20 40

Fig. 3. (a) Prefiltering in the frequency domain. (b) Prefiltering prevents aliasing.

Discrete Input Discrete Output

Sample

Reconstructed Input

Reconstruct

Warped Input Continuous Output

Warp

u

f (u)

fc (u) gc (x) g ′c (x)

g (x)

x

Prefilter

x xu

Fig. 4. Ideal resampling (from Heckbert (1989) with permission).



sort of filter is called an infinite impulse response (IIR). An ideal low-pass filter forprefiltering also has to be an IIR. A filter, for which implementation is possible inpractice, should be a finite impulse response (FIR). Thus in practice there is no choiceexcept approximating IIR filters with their associated FIR ones. Box, triangular andgaussian filters in the space domain—also known as B-spline filters—have beentraditionally used for filtering, but their frequency responses are far from the ideal boxin the frequency domain. Therefore, they either cannot reduce aliasing or result inexcessive blurring. One other solution is to truncate the sinc function in a limitedrange. Fig. 5 shows how this operation affects the filter frequency response. Theripples in the filter frequency response are known as Gibbs phenomenon that shows upas a ghost in the filtered image (Strang and Nguyen, 1996).

The windowing method for filter design is based on smoothing out the ripples inthe frequency response (Antoniou, 1993). Assume that the frequency response of awindow function is convolved with the frequency response of the truncated sinc. Thisis again like a low-pass filtering in order to produce a smoother frequency response forthe truncated sinc. The resulting filter is then an inverse Fourier transform of thissmoothed function. In practice, however, this can be done in the space domain simplyby multiplying the truncated sinc by the window function. There are a number ofwindow functions with fixed or adjustable shape. In this research an adjustable Kaiserwindow (Kaiser, 1974) with the parameter b ¼ 4 is used for filter design. An L2 normcriterion is used to measure the similarity of the filter with the ideal box, which isdefined as

E2 ¼1

2p

Zp

�p

ðHðwÞ � IðwÞÞ2dw ð1Þ

where IðwÞ and HðwÞ are frequency responses of the ideal filter and the designedfilter, respectively, and E2 measures the difference between the two. The L2 normmeasure was derived for a number of common window functions and results aresummarised in Table I. The filter designed based on the Kaiser window turns out tobest fit the ideal low-pass filter compared with other windows (Fig. 6). Therefore,

1

1

2

3

0

010 20 30 40 10 20 30 40

0.5

0.5

1.5

2.5

-0.5

Fig. 5. Truncated sinc and its frequency response.

Table I. L2 norm measures for some common types of window functions.

Window function Triangular Blackman Hamming Hanning Kaiser (b ¼ 4)

E2 (� 104) 20Æ851 20Æ611 15Æ881 16Æ482 14Æ030



better antialiasing can be expected from the Kaiser filter as it is more optimisedcompared with other filters.

In order to implement ideal resampling more efficiently, Heckbert (1989)combined the reconstruction filter and prefilter into a single filter, which is called theresampling filter:

qðx; kÞ ¼ZRn

hðx� mðuÞÞrðu� kÞ ¶m¶u

��du ð2Þ

where u and x are input and output pixel coordinates, respectively, r is thereconstruction filter, h is the prefilter and m is the transformation model. Theresampling filter qðx; kÞ is the weight of an input pixel at location k for an outputpixel at location x (Heckbert, 1989).

Survey of Antialiasing Techniques

The conventional resampling techniques in photogrammetry are interpolationmethods. The simplest interpolation method is nearest neighbour (point sampling), inwhich each output pixel is transformed to the input original image and the colour of thenearest pixel is assigned to the output pixel. Bilinear interpolation and cubicconvolution are better interpolators since they fit bilinear and cubic surfaces onto 4 and16 neighbouring pixels, respectively, to interpolate the value of the pixel underinspection (Lancaster, 1986). The problem appears when more than 16 input pixels inthe original image map to a single output pixel in the rectified image. In this caseapplying interpolation methods results in information loss since the entire informationcontent of the input image is not transformed to the output image. Antialiasingtechniques therefore aim to preserve as much information as possible during theresampling process (Heckbert, 1986). In order to select the most suitable technique foruse in photogrammetric applications a brief review of the existing antialiasingtechniques is given in the following sections.

Catmull (1974)

The oldest implementation of a resampling filter dates back to the subdivisionpatch renderer (SPR) algorithm presented by Catmull in 1974 (Catmull, 1974). TheSPR uses an inverse mapping technique. Catmull considered a pixel in output space as

1

1

2

3

0

010 20 30 40 10 20 30 40

0.5

0.5

1.5

2.5

-0.5

Fig. 6. Filter design using a Kaiser window in the space and frequency domain.



a square with the sample point at the centre. The algorithm starts with transformingfour corners of this square to the input texture image. In the input space, however,these corners define a quadrilateral. The unweighted average of the colour of the pixelsinside the bounding box of the quadrilateral defines the colour of the output pixel. Thismethod is relatively fast in terms of computation cost, however, filtering is not precisesince it calculates an average over a rectangle.

Blinn and Newell (1976)

Considering the fact that simple averaging in the SPR method acts as a box filter,Blinn and Newell (1976) modified it by a triangular tent filter. They used a pyramidfilter with a square base each side of which is 2 pixels long, in order to allow overlapbetween neighbouring pixels. The quadrilateral is approximated by a parallelograminstead of a rectangle in this method. In other words, the mathematical model isapproximated by an affine model in the neighbourhood of the pixel. The filter functionthen determines the weight of each pixel in order to compute a weighted average of thepixels inside the parallelogram. Comparing with the method of Catmull (1974), thismethod yields a fairly slight improvement since the box filter is replaced with a tent.

Feibush et al. (1980)

In contrast with previous methods, this technique is more relevant withresampling theory. It can be implemented with any filter and uses a look-up table(LUT), which greatly speeds up the computations (Feibush et al., 1980). These smartfeatures enable the method to implement high-quality filters with a reasonable speed.The algorithm starts with placing the two-dimensional filter function at the centre of anoutput pixel and calculating its bounding rectangle. This rectangle is transformed intothe input texture image and again a bounding rectangle is calculated for the resultingquadrilateral. All pixels inside this rectangle should be transformed to the output imagewhere a weighted average is computed for these pixels. Weighted averaging is carriedout using a LUT in which filter function values for each pixel with regard to itsdistance from the centre of the filter are stored. In other words two-dimensionalfiltering is reduced to a one-dimensional filtering operation using the LUT technique.

Gagnet et al. (1982)

This method is similar to the method of Feibush et al., except here selection of thesamples is done in output space (Gagnet et al., 1982). Filters with circular bases areused with a radius larger than 1 pixel width. The circular base is then approximatedwith an ellipse after transformation. A regular grid of samples with intervals smallerthan 1 pixel is constructed in output space and placed on output pixels. The sampleinterval is proportional to the larger side of a parallelogram bounding the ellipse. Forexample, if this side is three times larger than the pixel size then a sampling interval of1/3 output pixel size must be selected. Each sample point is then transformed to theinput texture image and its colour is computed using a bilinear interpolation. Finally, aweighted average of the samples using a truncated sinc weight function is associated tothe pixel based on the sample location with regard to the centre of the pixel.



Trilinear Interpolation

The trilinear interpolation technique is a fast and clever algorithm, which makesuse of filtered copies of the texture image in a pyramidal parametric data structurecalled an MIP Map (Williams, 1983). In order to find the colour of an output pixel firstone or two relevant levels in the MIP Map must be selected. Location of the sample ineach level then is determined using output pixel coordinates and a colour is computedby a bilinear interpolation of four neighbouring pixels. After doing this intra-levelinterpolation, a linear inter-level interpolation should be carried out between the valuescomputed in each of the two levels in order to find the colour of the output pixel. Thistechnique is one of the fastest existing techniques. For calculation of the colour of eachpoint, only 8 MIP Map access, 7 multiplications and 14 summations are required.Constructing the MIP Map levels needs a relatively short time as well.

Summed-area Table

Instead of using an image pyramid, Crow (1984) suggested constructing a tablewith the same size as the texture image, namely a summed-area table (SAT). The valueof each pixel in the table is replaced by the summation of the colours of all pixelsinside a rectangle defined by that pixel and lower left corner of the input image.Assume an output pixel is transformed to a rectangle in the input texture image withcoordinates ½xl; yb� for the lower left corner and ½xr; yt� for the upper right corner. Thesummation of pixel values in this rectangle can be calculated in a SAT as

T ½xr; yt� � T ½xr; yb� � T ½xl; yt� þ T ½xl; yb�:

Therefore, the algorithm starts with an inverse mapping transformation of outputpixels into the texture image. The bounding rectangle is then calculated for theresulting arbitrary quadrilateral similar to the method of Catmull (1974). But averagingthe values in this rectangle is done in a SAT. Thus for highly decimating trans-formations in which each output pixel transforms to thousands of input pixels, thecomputation load decreases down to four SAT accesses, three summations and twomultiplications.

Repeated Integration Filtering

An interesting extension of the SAT method is presented by Perlin (1985) andFerrari et al. (1986). This method is based on the rule that if one keeps on constructinga SAT for n times, each time on the last created SAT, then an ellipsoid area of textureimage with diameters parallel to the pixel grid can be filtered by 2ðn þ 1Þ timessampling the SAT. In other words this method achieves the advantage of a more precisefilter (B-spline) from the nth order instead of a box filter in the first level SAT, which infact is a B-spline of the first order.

Heckbert (1989)

Heckbert considered the fact that if the reconstruction filter and prefilter have anellipsoid shape with gaussian cross section and the mathematical model for the



transformation can be approximated by an affine model locally, then the resamplingfilter will be an ellipsoid gaussian filter as well. Thus he designed an ellipsoid filterwith gaussian cross section called elliptical weighted average (EWA) (Heckbert, 1989).For each pixel within this filter a weight value is calculated based on its distance to thecentre of the filter. Weights are stored in a LUT before transformation. Therefore, foreach output pixel, ellipsoid parameters in input texture space are computed using localaffine approximation. Then a so-called Q index is calculated which shows the distancebetween the pixel and the centre of the ellipsoid. The Q index is used to access theLUT and find the associated weight for the pixel. Finally, a weighted average of thecolours of the pixels falling inside the filter is calculated and assigned to the outputpixel.

Forward Mapping Antialiasing

The method developed by Ghazanfarpour and Peroche (1991) can be consideredas a different method since it seems to be the only method which uses forwardmapping. The algorithm can work with any filter function as the cross section of acircular-based filter. The algorithm starts with an initialisation step, which computesand stores the filter in a LUT. It then transforms the pixels from input texture image tooutput space and calculates the distance of the transformed pixel from neighbouringoutput pixels. Then for each output pixel for which this distance is smaller than thefilter radius, a portion of the colour of the input pixel is assigned. The amount of thisportion is selected based on the weight associated with the distance in the LUT. Thesummation of these portions for each output pixel determines its colour at the end ofthe transformation. The forward mapping antialiasing technique works with inputpixels rather than output pixels, therefore when the number of output pixels is less thanthe input pixels it suffers from an extra computation load. Another issue with thismethod is the danger of occurring holes. Because it works with input pixels, it ispossible that the algorithm assigns no value to some output pixels. Those pixels arecalled holes and appear as white or black spots in the output image. Choosing a filterbase 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 usedto fill the holes. For highly decimating transformations, however, this no longer is aproblem, for each output pixel is associated with a large number of input pixels.

Experimental Evaluation

SAT (Crow, 1984) and forward mapping antialiasing (Ghazanfarpour andPeroche, 1991) are selected in addition to interpolation methods for experimentalevaluation in this section. SAT is a fast antialiasing method, however, the type of filterit applies, box filter, is far from ideal. Forward mapping antialiasing on the other handimposes higher computation cost, but it performs a precise filtering with an arbitraryfilter shape. Therefore, this technique provides the opportunity to apply the optimisedfilter design based on a Kaiser window. In summary these two techniques representtwo important aspects of antialiasing methods: efficiency and accuracy.

In order to evaluate the accuracy of the antialiasing techniques a close range greyscale image of size 900�700 is selected (Fig. 7). A projective transformation modelis 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) measuresbetween original and reconstructed images are computed for analysis of the results.These measures can be used to estimate the information loss in the rectification processand evaluate the accuracy of the corresponding antialiasing technique. A goodreconstruction can be achieved by a more accurate antialiasing and results in lowerRMSE and higher PSNR values. For 8-bit grey scale images these objective measuresare defined as

RMSE ¼ 1

mn

Xmi¼1

Xnj¼1

ðxrij � xoijÞ2

!1=2

; PSNR ¼ 10 log10

2552

RMSE

where m and n are image dimensions, and xrij and xoij are the grey values of a pixel inthe reconstructed and original images, respectively.

Fig. 8 shows the rectified images using selected antialiasing techniques andreconstructed images from inverse transformation are given in Fig. 9. The Kaiser filteris used with the forward mapping antialiasing technique in Figs. 8(e) and 9(e). Table IIsummarises the accuracy measures and computation time for each method. Compu-tation times are given for comparison only and can be reduced in more optimisedimplementations.

As can be seen from Table II, the forward mapping antialiasing technique withKaiser filter shows the highest accuracy with both accuracy measures. The SATmethod also performs well according to measures given in Table II, however, it hasblurred the entire rectified image as can be seen in Fig. 8(d). A close examination ofFig. 8 reveals that in interpolation methods aliasing artefacts appear in the form ofcorrupted edges and interfused small features. These artefacts are more prevalent innearest neighbour interpolation and less in bilinear interpolation. In Fig. 8(a)–(c) notehow aliasing corrupts edges for instance in the three windows at the top of the image. Itcan be said that interpolation methods are not able to retrieve these corrupted edges,because they use grey scale information of not more than 16 pixels. In the forwardmapping antialiasing method (Figs. 8(e) and 9(e)) the size of the Kaiser filter is in factdetermined by the transformation so that it covers all input pixels that are mapped to aparticular output pixel. Therefore, this approach enjoys less information loss ascompared with the aforementioned techniques. The shape of the filter is also designedin such a way that amplifies the edges. This is because of the negative lobes of the

Fig. 7. Original image.



Kaiser filter (see Fig. 6). These advantages result in a clear rectified image with sharpedges as can be seen in Fig. 8(e).

According to Table II, precise filtering is more costly and the computation load ishigher for the forward mapping antialiasing technique. Therefore, for very largeimages and when the aliasing problem is not very crucial, fast antialiasing methodssuch as SAT or trilinear interpolation can be recommended. In fact there is always acompromise between accuracy and computation expense in antialiasing methods.

Conclusion

Although in many cases conventional interpolation methods produce images withreasonable quality, in the case of aliasing applying a precise resampling filter is

Fig. 8. Results of different antialiasing techniques: (a) nearest neighbour interpolation; (b) bilinearinterpolation; (c) cubic convolution; (d) Crow’s summed-area table; (e) forward mapping antialiasing with

Kaiser filter.



Fig. 9. Reconstructed images from rectified images: (a) nearest neighbour; (b) bilinear interpolation;(c) cubic convolution; (d) summed-area table; (e) forward mapping antialiasing with Kaiser filter.

Table II. Accuracy and efficiency measures for selected antialiasing techniques.

Nearestneighbour

Bilinearinterpolation

Cubicconvolution

Summed-areatable

Forwardmapping antialiasing

RMSE 41Æ613 37Æ353 42Æ171 32Æ050 28Æ481PSNR 31Æ938 32Æ408 31Æ881 33Æ072 33Æ585Computation time (s) 4Æ079 5Æ718 16Æ328 123Æ922 587Æ750



inevitable. This happens in areas where the transformation is highly decimating. Thus alarge number of input pixels transform to one pixel in output space. Nearest neighbour,bilinear interpolation and cubic convolution interpolation methods use 1, 4 and 16pixels, respectively, to interpolate the grey level of the output pixel. Therefore, in thecase of aliasing, interpolation methods are not capable of preserving the entireinformation content of the original image in the rectified image. Precise antialiasingrequires the design of a FIR low-pass filter which best approximates the ideal box. Theantialiasing technique also has to provide the opportunity of using various filters. Themethods of Feibush et al. (1980), Heckbert (1989) and forward mapping antialiasing(Ghazanfarpour and Peroche, 1991) have such an advantage. Results of this researchindicate the higher accuracy of the precise antialiasing techniques with a Kaiserresampling filter. These techniques, however, impose more computation load and onehas to trade off between accuracy and computational expense. For highly decimatingtransformations, where aliasing is more critical, a precise filtering is required, whichleads to more computational expense. However, for very large images fast antialiasingmethods such as SAT can be used with lower accuracy.

Acknowledgements

The work described in this paper was supported by the Deputy of Research,University of Tehran and theHongKongPolytechnicUniversity (ProjectNo.G-W122).

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California. 340 pages.

R�esum�e

Le ph�enomene d’aliasing qui introduit des artefacts dans le r�e-�echantillonnage num�erique d’images provoque des d�egradations dans laqualit�e de ces images. Lorsque l’on opere une rectification num�erique et quel’on integre la texture dans la cartographie, on transforme les pixels del’image d’entr�ee en pixels de l’image de sortie. La nature discrete de l’imagenum�erique entraıne ce ph�enomene d’aliasing dans l’image transform�ee. Onanalyse dans cet article les causes de ce ph�enomene et les moyens th�eoriquesde l’�eliminer. Il faut disposer d’un filtre anti-aliasing pr�ecis et a cet effet on aconcu un filtre bas�e sur une fenetre r�eglable de Kaiser. On d�ecrit �egalementdiff�erentes m�ethodes pratiques d’anti-aliasing ainsi que des m�ethodesd’interpolation, qui sont courantes en photogramm�etrie. On a s�electionn�equelques m�ethodes d’anti-aliasing pour les mettre en œuvre sur une image dephotogramm�etrie a courte distance. Une analyse objective a �et�e effectu�ee enappliquant les transformations inverses aux images rectifi�ees et end�eterminant par des mesures la perte d’information de chaque m�ethode encomparant les images de d�epart a celles reconstruites. Les r�esultats montrentque les m�ethodes d’interpolation ne sont pas capables d’�eliminer ni meme der�eduire cet aliasing lorsqu’il s’agit de transformations tres profondes. Lesimages issues des m�ethodes d’interpolation souffrent notamment d’une�erosion des contours et d’une confusion des petits d�etails.

L’emploi d’un filtre de Kaiser et d’une m�ethode d’anti-aliasing depr�ecision r�eduit au minimum possible la perte d’informations et leph�enomene d’aliasing au prix d’une charge sup�erieure de calculs.

Zusammenfassung

Durch Aliasing, d.h. Bandbereichsuberlagerungen, werden bei der

Interpolation in digitalen Bildern Artefakte erzeugt, die die Bildqualitatverschlechtern. Bei der Digitalen Entzerrung und bei der Texturextraktionwerden Bildelemente eines Eingangsbildes in Bildelemente eines Ausgangs-

bildes transformiert. Durch die diskrete Natur der digitalen Bildern trittAliasing in dem transformierten Bild auf. In diesem Beitrag werden Ursachen



fur das Aliasing und die Theorie zum Antialiasing vorgestellt. DieNotwendigkeit fur ein prazises Design von Filtern fur das Antialiasing wird

diskutiert und ein Filter, der auf ein Kaiser adaptives Fenster beruht, wirdvorgestellt. Verschiedene Verfahren zum Antialiasing, die in der Praxiseingesetzt werden, werden beschrieben, dazu auch Interpolationen aus derklassischen Photogrammetrie. Ausgewahlte Antialiasing Methoden werden

implementiert und auf Nahbereichsaufnahmen angewandt. Bei der objektivenAnalyse, die auf die Anwendung von Rucktransformationen der entzerrtenBilder beruht, werden einige Maße abgeleitet, die es erlauben, den

Informationsverlust jeder Methode durch den Vergleich zwischen Original-bildern und rekonstruierten Bildern abzuschatzen. Die Ergebnisse weisendarauf hin, dass Interpolationsverfahren nicht fahig sind, in stark

unterteilenden Transformationen das Aliasing zu entfernen oder entscheidendzu reduzieren. Die Ausgabebilder der Interpolationsmethoden zeigen deshalbveranderte Kanten und Vermischung kleiner Bildmerkmale. Die Anwendungeines Kaiser Filters mit einer genauen Antialiasing Methode weist den

kleinsten Informationsverlust auf und kann Aliasing entscheidend reduzieren,allerdings auf Kosten eines hoheren Rechenaufwandes.

Resumen

El pixelado de bordes es un problema que se manifiesta en las imagenesremuestreadas digitalmente como un artefacto que degrada la calidad de laimagen. En la rectificacion digital y generacion de texturas, los pıxel de la

imagen de entrada se transforman en los pıxel de una imagen de salida. Lanaturaleza discreta de una imagen digital da lugar al pixelado de bordes enla imagen transformada. En este artıculo se describe el origen del pixelado y

la teorıa de suavizado de bordes. Se examina la necesidad de disenar unfiltro preciso en el suavizado y se disena un filtro basado en la ventanaajustable de Kaiser. Se describen varios m�etodos practicos de suavizado y deinterpolacion de uso comun en fotogrametrıa. Se aplican varios m�etodos desuavizado de bordes especıficos a una imagen de fotogrametrıa terrestre. Serealiza un analisis objetivo aplicando transformaciones inversas a lasimagenes rectificadas y se extraen algunas medidas para estimar la p�erdidade informacion en cada m�etodo, comparando las imagenes original yreconstruida. Los resultados indican que los m�etodos de interpolacion no soncapaces de eliminar o reducir el pixelado de bordes en las transformaciones

con una elevada reduccion de resolucion. Las imagenes resultantes de losm�etodos de interpolacion experimentan, por lo tanto, la corrupcion de losbordes y la fusion entre objetos pequenos. La aplicacion del filtro de Kaiser

con un m�etodo de suavizado de bordes preciso da lugar a una menor p�erdidade informacion y reduce considerablemente el pixelado a cambio de unmayor tiempo de calculo.



kaiser filter for antialiasing in digital photogrammetry

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