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Subpixel Edge Localization and the Interpolation of Still Images

Kris Jensen and Dimitris Anastassiou

Abstract-In this paper, w e present .I nonlinear interpolation scheme for \till image rewlution enhancement. The algorithm is based on a source model emphasihg the visual integrity of detected edges and incorporates a novel edge fitting operator that ha\ been developed for thi5 application. A \mall neighborhood about each pixel in the low-resolution image i s hrst mapped to a best-fit continuous space 5tep edge. The bilesel approximation serbes as a local template on M hich the higher resolution sampling grid can then be superimpowd (where disputed values in region4 of local window o\erlap are averaged to smooth errow). The result is an image of increased resolution with noticeably Pharper edges and, in all tried ca5es. lower mean-squared reconstruction error than that produced b! linear techniques.

1. Ih I 'KODI( I ' ION

MAGE interpolation is required for multiresolution pyra- I niidal coding (e.g. I 11. 121. and 181). iinproved definition television (IDTV) receive1 design. and other more general nceds for picture reso1utioi.i enhancernent, such as still photo- graph zooming. Sampling theory dictates that the Nyquist limit specities the maximum frequency that can be retained when thc sampling density of a siyial is increased by postprocessing since no inforniation is prcB\ent regarding the missing wper- Nvquist region 01' the frequency spectrum. This imposes a fundamental limit on the frequency content of a signal ;I\ ailable for interpolation ;mce i t tias been bandlimited. The umpl ing theorem can. ho\*ever. lie defeated if a rnodcl of the original signal is devisi.d from which the miqsing higher frequency information can be predicted. based on the lower frequency content.

Throughout the past. m..in) linear. and several nonlinear, iniage interpolation techniques h;ive been proposed. With thc rapid increase in available computing power. coupled with great stride\ in imase feature analysis. model-based, often highly nonlinear. intei polativc techniques have become a v i ~ b l c alternative to classic linear methods and have received increasing attention recently. The most prevalent of image feittures iminifesting broact-\pectrum characteristics that can be analyzed over manq scde.; ut' resolution are edges, and this observation. coupled M ith the importance of edges t o the

early stages of human visual proces4ng. explains why most of the newer model-based interpolation algorithms tend to focus on them. By their very nature. edges contain a great amount of high-frequency information. During the image digitization process (or the subsampling of an already digiti& image). inany of the higher spatial frequencies are lost by necessity to avoid aliasing. The interpolation problem is to predict and replace the lost higher spatial frequencies based on existing low-frequency ini'ormation. which indicates the presence of edges.

Several examples of model-baaed approaches to spatial image interpolation can be found in [ I O ] . 171, [Y] , and [ 6 ] . Each of these papers utilizes the concept of an edge in a different fashion to enhance interpolation results. As in 161. the approach adopted here shall be that of edge fitting within small overlapping windows of image data to localize existing edges to subpixel accuraq. The bilevel result of each local step-edge fit is then used ab a template on which a higher resolution sampling lattice c;ui be superimposed. Before detailing the algorithm's specitics. we turn to a brief discussion of the technique we have adopted to solve the problem of edge fitting within a 3 x 3 window of image data.

11. A ShlAl I \b'I?;DOb' EDGE.-FITTING OPEKATUK

An ideal 2-D continuou.; space 5tep edge is a function of four parameters that takes the following form:

where .I' and represent the coordinates on a Cartesian plane, and . c w . s ( ~ ) + y s / i / ( H ) = p specities a straight line separating the two regions of' constant intensity value (see Fig. 1 ).

The problem of' edge fitting is to find the best continuous space step edge approximation to a given window of discrete image data within which such an edge is determined to be presenl. The technique proposed by Hueckel 141, [SI has found widespread use for this application; however, i t was designed for implementation over window s i x s significantly larger than 3 x 3 arid is quite complex. We have developed an accurate algorithm of IOM implementation complexity that is well suited for application over small window sizes. The operator is initially derived over continuous space after which an approximation is introduced for application in the digital domain.

Authorized licensed use limited to: Columbia University. Downloaded on June 02,2010 at 23:37:26 UTC from IEEE Xplore. Restrictions apply.

286 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 4, NO. 3, MARCH 1995

Fig. 1. Continuous space step edge defined by the four parameters A, B. p , and 6'.

B A

/

Fig. 2. in a polar coordinate system at a fixed distance R > p from the origin.

Ideal step edge as a function of the angular coordinate Q when viewed

A . Continuous Space Theory

We first consider the case for 0 = 0 and later generalize the theory for arbitrary 0 through rotation. When viewed at a fixed distance R > p from the origin, an ideal edge becomes a I-D periodic function of the angular coordinate 4. Defining Po = { A , B , p , O = 0) and

o =cos-'(;),

the ideal vertical edge representation becomes

A i f - a L $ < a B if otherwise

as depic!ed in Fig. 2. Working with a truncated orthonormal basis system defined

over 4, the projection of S(q5,Po) on the subspace results in a series of equations from which the values o, A , B and even B (for a rotated edge) are obtained. The truncated basis set of

Fig. 3. Image values on the sampling lattice are assumed constant within a small circular area about their centers (indicated by grey shading). The analysis ring (over which the continuous function is to be interpolated) is superimposed, and the locations of its corresponding sample points are each marked by an x.

a a a

MO

-a -p -a -a a

MI M2

0 -p 0 P 0 -P M3 M4

Resultant operator masks. Only two weightings are required. Fig. 4.

interest is


JEYSEh 9ND ANASTASSIOI;: SLIRI’IXEL EDGE I.OC.AI.IZ.4TlON AND THE INTERPOLATION OF STILL IMAGES

X 0 X 0 - X 0 X 0 X 0 X 0 X 0 X-Q X Q X 0 0 0 0 Q 0 0 0 0 0 0 Q 0 0 0 0 Q 0 0 Q 01 X O X Q X O X O X Q X O X Q X O X O X O ~ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Q 0 0 Q 0 X O X O X Q X O X O X O X O X Q X O X Q O O O O O O Q O O O O O Q O O Q Q O O Q ~ X O X O X O X Q X O X O X O X O X O X O

0 0 0 0 0 Q 0 Q 0 0 0 0 0 0 0 Q Q 0 0 0 X O X O X O X Q K O X O X O X Q X O X Q 0 0 0 Q 0 0 0 0 0 Q 0 0 0 0 Q Q 0 Q Q 0 X O X Q X O X O X O X Q X Q X Q X O X O O O Q Q O Q Q O O Q Q O O O Q Q Q O O O X O X O X Q X Q X Q X O X O X Q X Q X Q 0 0 0 Q Q Q 0 0 0 Q 0 0 0 0 Q Q-90 0 01

Fig 5 lnterpoldtion lattice Lou -re\oIution pixel \ample locations are rep- r cmted hy “X ’’ These pixel\ are to be dehlurred, and the higher resolution \ainple\. which dre denoted “0.” ‘ire to be interpolated

Fig. 6. as\ociated step-edge approximation i \ noted.

Curved bounday observed over a \mall area of analysis. The

a

0.2 c

Fig 8. Edge profile lor - = I O .

I l

I I I 1 I

( C )

f C

Fly. 7 . Failure of edge analy\is o\er 2 x Z pixel window vzes: (a) Continuous space step edge whose location wc wish to determine: Ih) resultant digitization of this edge over a 3 x 3 pixel area a\ i t could appear in a sampled image; (c1-(f) portray the set of four overlapping 2 x 2 windows within which edge detection would be applied. Notice that within each of these smaller windowss. an incorrect edge would be detected passing through the center of the respective region of analhsis. Localization of the original edge to continuous space accuracy hecome\ impossible in this situation.

Defining the projection\

ln[P) =< S(0.P) B71(o) >=

thc related set of \pectral coefficients for vertical edge

I I I I I

Fig. 9. Linear ver\u\ nonlinear up-\ampling: (a) Original low-resolution window; (b) hest fitting ctep-edge approximation; (c) result of bilinear interpolation. where aleraging o f neighboring intensities is employed for interpolation of the needed 16 pixel values. The original nine pixel values remain unchanged in this process and ilre each marked with an x; (d ) result of nonlinear interpolation utilizing the detected discontinuity. In all cases. the spatial location of the hest fitting edge has heen superimposed for purposcs of visual compariwn.

becomes

where ‘ r ! E {1 . :3 } . and

(Po) = 0 (7)



-1 0 0 0

-0

Fig. 10. Original image (256 x 256 pixels).

0 0 0 0 - cos(8) sin(6') 0 0

-sin($) cos($) O 0 0 0 cos(26') sin(28) 0 O -sin(%') cos(26')-

for n E {2!4}. Making the substitution sin(2a) = 2 sin(a) cos(a), we find

Thus, from (1) and (8)

p = R [ a ] -

The step height of the edge S = ( A - B ) is found as

Gal (PO) 6 = 2 sin( a)

from which the projection ao(P0) is used to obtain

and

(9)

(10)

When 0 # 0, a rotation of the coordinate axis by 0 yields the following relation between the rotated and nonrotated

projection vectors:

Given the projections a,( P), the equivalent nonrotated projections a,(Po) are obtained from (13). Equation (7) implies that

, a 2 P o ) = 0 (14)

from which (13) yields

(15) a2(P) cos($) - ul(P) sin($) = 0.

Equation (15) gives the edge orientation


FI:. 1 1 . High-rewlution edge prediction 1256 x 256 pixels) from the low-resolution image data.

B . D i x w t c Appi~o.~-i~~zcitioti

When working with digital images. approximations to the above integral inner products must be introduced in order to obtain the desired measurements u , , . AI with other techniques. thc approach adopted herr is that of fitting a continuous function about the sample points and measuring the desired quantities I / , ! directly from the continuous space interpolation. A 1-D function is interpolxted about the data around a ring centered at the 3 x 3 window's middle pixel. Choosing the ring's radius

- I + v?

R - 7 (17!

to minimize the average distancc and allow each of the outermost eight samples to lie equidistant from the ring. each pixel is approximated as powxsing a constant value within a distance of (z 0.207 or roughly one fifth of a sample uni t ) about i t \ center. With this as\uniption, we have access to sample values that lie directly o n the ring of interest (see Fig. 3 ) .

Given a 3 x 3 windou ot image data [ I . [ / / / . / I ] . a minimal mean-squared error t i t to l l - [ r / / . / I (on the ring) using the truncated basis set of intere\t is found in the form

I

, = o

where the fitting coefticients A, directly correspond the spectral coefficients U , desired.

Over an observed set of discrete data II . [d] . ~i E 12. the coefficients A, yielding a minimal mean-squared error tit to the orthogonal basis set Lz, (truncated to the first A. bases) over S 2 is obtained by minimizing the error

i

L:' = 1 ( I r - [ . i ] -~ - p , B , ( , . ) ?. (19) & E ! ! ,=[I

Taking the derivative over A,. I:' is found to be minimized at

Hence. given a 3 x 3 window of image data W, the A, are found as

I i

A, =< w.Ii.Ii >= I I . [ / r / . / , ] . 21 , [ r r / . / / ] (21) ,,, = 1 ,,= I

where the precomputed (and fixed) discrete masks Mi (corresponding to the 11, ! are depicted in Fig. 4. The weightings / v and ;j are



Fig. 12. Subsampled image as interpolated back to original resolution (256 x 256 pixels) through use of bilinear interpolation.

and

p = - J;; 4 '

The masks MI and M2 used to determine 8 are of the form found in gradient approaches. Furthermore

(24)

yields the isotropic weighting for edge orientation estimation as proposed in [3].

-=Jz P Q

C . Operator Implementation

data W[m, n], the edge fitting process proceeds as follows: When applying the operator to a 3 x 3 window of image

1) Compute A i (i E {0,1,2,3,4}) from (21). 2) Estimate the edge orientation from (16) as

8 = tan-' (k) . (25)

3) Find the distance to the edge from (9) and (13) as

] * (26) A3 cos(28) + A4 sin(28)

.=R[ A1 cos(8) + A2 sin(8)

4) Calculate from (1). 5 ) Compute the step height of the edge from (10) and (13)

(27)

as J;F[AI cos(8) + A2 sin(S)]

2 sin( o) 6=

6) Find the values on either side of the edge B and A from (1 1) and (12), respectively, as

and

A = S + B . (29)

Equations (1 3) and (7) further imply that for an ideal edge

(30) q (P ) cos(28) - ~ ( P ) s i n ( 2 8 ) = 0.

By normalizing with the effective input magnitude (where the window of interest is here denoted We to indicate that the center pixel is not included), the quantity

(31) A4 cos(28) - A3 sin(28) E =

< W",W" >; in which we have divided by the square root of

3 3

< We, We > = W[m, n]W[m, n], (32)

for the purposes of normalization can be thresholded and used as an error measure to gauge the quality of fit and determine whether or not an edge should actually be modeled within W[m,n]. The value of the image window's central pixel should also be checked for consistency with the predicted edge.


F-12. 1.7. Suhwnpled image as interpolated hack to original resolution (2.56 x 256 pixels) through tihe ot the new technique

Alternately, the error between the predicted step edge and the actual pixel values within the 3 x 3 neighborhood could also be computed as a viable distortion criterion; however, this approach has been discarded. owing to the large amount of computational overhead th;tt would be 5 0 introduced.

111. IMPROVED I M A G E INTERPOLATION

The algorithm we introduce ha\ been implemented to increase the resolution of a \ t i l l image by a factor of two in each spatial dimension (see Fig. 5 ) . resulting in a quadru- pling of the original pixel density. Iteration of the process results in a 2"--fold Looin in each dimension, where N is the number of interpolation applications: however, the algorithm can be modified to achieve a zoom of arbitrary proportion. Furthermore, inlerpolation is not limited to any one upsampling lattice. but with modification, i t can be applied to the interpolation of arbitrary lattice structures, with potential extension to the temporal domain (deinterlacing would be one such application).

A The Loc,ul Inluge Model

Within the scope of this investigation. an image is modeled as consisting of two primary structural components: edges and texture. Edge fitting is performed over 3 x 3 windows of pixels (for reasons to be explained shortly), and thus, texture, for the sake of simplicity, is considered to be any structural information existing over a 3 x 3 window that cannot be modeled as an edge within the framework specified in the last section. The

purpose of this algorithm is to increase interpolation accuracy over these edge-based structures. Although no formal model is adopted for other textural characteristics, information that is not directly modeled will be preserved through the use of simpler local interpolative techniques.

Our edge-texture model is based on the simplifying assumption that within a sufficiently small neighborhood about a pixel lying near a boundary between two objects, the objects can be separated by a perfectly straight line into two regions, each with nearly constant intensity value, i.e., sufficiently high magnification of even a curved boundary will reveal a straight line separating two regions of relative homogeneity (see Fig. 6).

Thus, the first step in the interpolation process maps a small neighborhood about each low-resolution pixel to a best fitting continuous space step edge. Should an edge be determined to be present within the window of analysis, i t is taken as being representative of the true underlying image structure within that region. and attempts will be made to accurately increase its resolution during interpolation. The step-edge approximation serves as a local template on which a higher resolution sampling grid will later be superimposed (intensity values in regions of local window overlap will be averaged to smooth the errors resulting from such a rigid step-edge fitting).

B. Scule (f Anul?.sis

Owing to the fact that subpixel accuracy is desired in the localization of detected edges when reconstructing images over



Fig. 14. Local zoom of bilinearly interpolated image.

a higher resolution sampling lattice, a 3 x 3 window has been selected as the local neighborhood size most appropriate for our analysis. Over a 2 x 2 pixel area, horizontal and vertical edges not passing through the window’s center cannot be localized to the desired degree of accuracy. Such edges over this limited range of inspection manifest themselves as edges of the same orientation, yet with a different step height, passing through their respective windows’ centers. Even though the interpolation neighborhoods overlap, all adjacent 2 x 2 windows will experience the same failure, resulting in an inability to accurately localize these edges in continuous space for interpolation. See Fig. 7 for an example of this phenomenon.

Although edges captured in the periphery of a given 3 x 3 window can also fail to be localized in a similar fashion, the greater size of the local templates provides for an overlap such that neighboring windows have the opportunity to localize these edges to continuous space accuracy. The larger number of window overlaps present in this case provides compensation in the event that errors are made in a small number of the local interpolations. In addition to complexity concems, larger neighborhoods were discounted for two other reasons: First, they would average together too much information in the bilevel approximation, blurring detail in the higher resolution reproduction, and second, they would impart a far-too-jagged interpolation template to curved boundaries, force fitting straight lines on a scale that may be larger than appropriate.

C . Edge Sharpening

Boundary estimation in this fashion provides, for each low- resolution pixel neighborhood, the location (in continuous space) of the best fitting edge, as well as the two associated intensity values ( A and B ) lying on each of its two sides. The values of A and B can be determined by a variety of methods. The direct results of the operator described in the previous section could be applied; however, for greater accuracy, one could also average the intensities of those pixels that lie on either side of the detected edge.

A higher resolution sampling grid is now superimposed over the resulting region of the edge fit. The values of the higher resolution pixels are approximated as being one of A , B (for those pixels lying entirely on one side of the edge), or a weighted average of A and B (for those pixels intersected by the resultant line of the edge fit). In the latter case, the weighted average is taken as a function of the distance from the center of the pixel that is intersected by the line and the line itself.

We assume that a pixel’s relevance to an edge drops off with its distance d from the edge and to this end, weight the disputed intensity values as cr(d)A+[l-cr(d)]B for a pixel whose center lies on the side of intensity A and a(d)B + [I - a ( d ) ] A for a pixel whose center lies on the side of intensity B. To allow for varying degrees of edge sharpening, based on the image contents and the results desired, the weighting function used

L


2't i

l'or the experimental implementation was

which i \ a sigmoidal curvc' whose value is for d = 0 and climbs t o a value of I with increasing rl . For typical images.

adjusted to achieve any dt.\ired degree of sharpening in the interpolated edges. The edse protile for = IO is shown in Fig. X.

Fig. 9 depicts the up-sampling process as applied to one idzali7ed. pixel-centered 3 x 3 neighborhood. Note that as implemented here. the 3 1 3 region is tirst mapped to its continuous space edge approximation and then resampled to a 5 x 5 neighborhood with the pixel values assigned as described previously. An example of'the bilinear interpolator's result is included for comparison. The blurring of the edge mmifests itself. and the potential advantage of the nonlinear method is readily seeti. The area o f each original pixel could also have been split into t'our quadrants to achieve the same resolution enhancement. Like most surface-fitting techniques. bilinear interpolation leaves the original pixel values unchanged. wherea\ the nonlinear technique alters thcse values during the averaging of neighboring window rcwlts, deblurring detected edges i n the low-resolution image throughout interpolation.

..- . --. I O was found to yield good results: of course, y may be

Of course. in many windows encountered. the fitting technique is not appropriate. specifically in those areas where there are no edges satisfying the conditions of the model. Finely textured areas. regions of constant luminance. and pixel-wide lines do not conform to the contines of a pure step-edge source model. In such windows. the edge-fitting operator retums a "no tit" response (we thresholded the detection criterion E z 0.05 and required that the values of the windows' central pixels lie between the calculated parameters A and I?). and a simpler interpolation scheme is be employed. In our work, bilinear interpolation is adopted as the default process: Missing pixel values are taken simply as an average of neighboring intensities. In this way. although no higher frequency information is extracted from the lower resolution samples, those underlying inten5ity fluctuations not modeled are at least captured in the interpolation process. Higher order (possibly texturally specific) interpolative techniques could also be employed i n these situations.

1V. E X P E K I M E ~ T A L FINDINGS

To gauge the efficacy of the algorithm. several gray-scale still images were liltered and subsampled t o one-quarter scale and then interpolated back to their original resolution. The par- ticular strengths anti weaknesses of any interpolation scheme can. in this way. be judged hy comparing the up-sampled result to an original wurce image. Filtering of the still images



Fig. 16. Local zoom of original image.

prior to subsampling is required to avoid aliasing so that they are appropriately represented at the lower resolution scale. Although perfect filtering was not found to be crucial to the operator’s performance, a fair degree was found to be required. An FIR filter of size 7 x 7 pixels was used for the following comparisons.

The original 256 x 256 pixel test image “Lenna” is shown in Fig. 10. Fig. 11 shows the reconstructed 256 x 256 pixel high-resolution edge map as predicted from the 128 x 128 pixel subsampled image (edges possessing step magnitudes of less than 10 units, on a scale of 256, are not displayed here for reasons to be discussed later). The high-resolution edge pixels are displayed if they fall within a distance slightly greater than one sample radius of a detected edge. The intensities of these line segments have been graded based on how many of the overlapping detectors agreed on their presence (the whitest of the edges being those most agreed upon). Owing to the fact that this edge map was predicted from the decimated image data alone, some of the textural information present in the original image, but not in its subsampled counterpart, is clearly lost (examine, for example, the texture of the hat).

Figs. 12 and 13 depict the low-resolution image restored to its original scale (256 x 256 pixels) through the use of both bilinear interpolation and the nonlinear algorithm, respectively. Figs. 14 and 15 show details of these interpolated images (the images were zoomed through the use of pixel replication) and may be compared with the original image segment of Fig. 16.

Although, in both interpolated images, some high-frequency textural information is lost, in the nonlinearly interpolated version, it is visually apparent that a certain degree of higher spatial frequency information was indeed replaced, specifically, in those regions possessing distinct edges. This results in not only a more visually pleasing zoom but, in all tried cases, a lower mean-squared error reproduction as well.

The greatest errors incurred during interpolation result from attempts to reproduce those areas that originally possessed high frequencies. With the proposed algorithm, certain high- frequency image components around detected edges are predicted and replaced, yielding a lower error reproduction in such areas. The decreases in observed error energy are due specifically to increased interpolation accuracy in those regions of edge-like luminance discontinuity. Mathematical comparisons of our algorithm to both bilinear and cubic B-spline interpolations have yielded higher SNR reconstructions in all cases tried. Of course, the exact degree of error reduction will vary from image to image and is dependent on the proportion of image pixels falling near detectable edge boundaries.

Since the edge operator responds to edges independent of their step magnitudes, in many cases, slight variations in texture over the small windows of analysis are detected as edges and are processed in the manner previously described. This can result in a small loss of textural reproduction in those areas as local edge fitting may be an inappropriate course of action in such cases. Although we did observe a


JEUSEN 4 2 D ANASTASSIOU- SUBPIXEL EDGE LOCALIZ.ATTION AND THE INTERPOLATION OF STILL IVAGES 295

“noisy” pattem of edges dctected in regions where one would orherwise expect to see little edge activity, the misclassified pixels were found to lie far enough apart that each window was only operated on by one edge detector. Thus, in these cases, there existed ample opportunity for correction, due to the a\ eraged results of surrounding operators. However, to further compensate for low-level edge noise. only edges possessing step magnitudes greater than 10 luminance values (on a scale o f0 to 255) were considered to be of importance. Although in theory this does preserve some of the underlying texture in the interpolations, the visual quality of interpolated images when such a threshold is incorporated is indistinguishable from those cases when it is not. The RMS error in reproduction was also found to be virtually identical in such situations.

Another important aspect of any interpolator’s performance is its ability to operate uell in the presence of noise. We found that the nonlinear algorithm. by virtue of its smoothing tendencies near edges, actually decreases noise levels in its interpolations. Edge 1ocaljLation remains robust due to the averaged results of several overlapping edge detectors, and most edges remain sharp. even at relatively high levels of noise saturation. In all tried cases, the images resulting from interpolation with the new algorithm retained sharper edges and lower reconstruction errors than those obtained with the bilinear approach.

v. CONC12UDING REMARKS

In this paper, a model-based algorithm for application to the spatial interpolation of still images was presented. The technique revolves around the detection and localization of edges present in the lower resolution pictures to impart high- frequency detail information to the interpolations. Existing edges can often be restored to a high degree of accuracy once having been localized in space. An edge fitting operator for this purpose has been incorporated. Simulation results have been included, in which the benetits of the edge model are readily seen when compared with a standard linear interpolative technique. Although the algorithm does not model the diversity of textural characteristics found within most images, the inclusion of an edge-texture source assumption can be used to enhance the quality of an image in the interpolation process. Possible applications include improved pyramidal coding, sampled photographic image enhancement, and IDTV receiver design.

I II

171

REFERENCES

D. Anastassiou. ”Generalited three-dimensional pyramid coding for HDTV using nonlinear inttrpolation.” in Proc Picrure Coding Symp. (Cambndge. MA), 1990. P. I . Burt and E. H. Adelson. “The Laplacian pyramid as a compact image code, fEEE Trans C‘rimniun. col. COM-31, no. 4, pp. 532-540, Apr. 1983.

W. Frei and C. Chen, ”Fast boundary detection: A generalization and a new algorithm.” IEEE Trans. Compur.. vol. C-26, no. 10. pp. 988-998, Oct. 1977. M. H. Hueckel, “ A local visual operator which recognizes edges and lines.” J . Assocc C o m p r . Mach , vol. 20, no. 4, pp. 634-647. Oct. 1973. -, “Erratum for A local visual operator which recognizes edges and lines,” vol. 21, no. 4. p. 350, Apr. 1974. K. Jensen and D. Anastassiou, “Spatial resolution enhancement of images using non-linear interpolation,” in Proc. IEEE f n t . Conf: Acou.rt , Speech. and Signa/ Procrssinq (Albuquerque, NM), 1990. D. M. Martinez and J. S. Lim. “Spatial interpolation of interlaced television pictures.” in Proc. lEEE fn t . Conf. Acousr Sprec~h. SigfiaI Processing (Scotland). Apr. 1989, pp. 1886-1889. K. M. Uz, M. Vetterli, and D. I. LeGall, “Interpolative multiresolution coding of advanced television with coinpatible subchannels.” fEEE Trans. Circuirs S v s t Video Te~hnoh~q? , vol. I , no. I , pp. 8699 , Mar. 1991. D. De Vleeschauwer and I. Bruyland, “Nonlinear interpolators in compatible HDTV image coding,” in Signal Processing of HDTV (L. Chiariglione, Ed.). Y. Wang and S. K . Mitra, “Motion/pattem adaptive interpolation of interlaced video sequences.” in P roc IEEE fnr. Conf Acousr.. Speedi. Signal Proresing (Toronto, Canada). May 1991, pp. 2829-2832.

Amsterdam: Elsevier (North Holland), 1988.

Kris Jensen has bom in Mornstown, NJ, in 1963 He received the B S degrees in mathematic\. physics. and electrical engineering in 1986 following a five-year combined program between Allegheny College and Columbia University He received the M S and Ph D degree5 in electrical engineenng tram Columbia University in 1988 and 1992, respectively

He joined the engineenng staff of the Box Hill Systems Corporation, New York, NY. where he is currently involved in the design and implementation

of large-scale network data management 5 ) stems

Dimitris Anastassiou was bom in Athens, Greece, in 1952 He received the Diplomd in electncdl engineenng tram the National Technical University of Athens, Greece. in 1974 and the M S and Ph D degrees in electncdl engineenng from the University of California Berkeley, in 1975 and 1979. respectively

He joined the faculty of Cohnbid University in

1983, where he I \ currently Professor of Electn- cal Engineenng, Associate Director for Multimedia Applications of Columbia University’s Center for

Telecommunications Research (an NSF Engineenng Research Center), and Di- rector of COhimbid University’s Image and Advanced Television Laboratory Prior to joining Columbia University in 1983, he was with the IBM Thomas J Watson Research Center, Yorktown Heights, NY, as a Research Staff Member, working on the development of the internal IBM video-conferencing system Hi5 re\edrCh interests focus on digitdl video telecommunications, with emphasis on mUkimedid applications He has received an IBM Outstanding Innovation Award and an NSF Presidential Young Investigator Award He ha\ been Guest Editor of special issues for vanous joumals and has been involved in the organization of %vera1 workshops and sessions on the subject of digital video and advanced television

Dr Anaqtassiou I \ a member of the Steenng Committee of the yearly Inter- national Workshop on HDTV, an Associate Editor of the IEEE TRANSAC~IONS ON CIRCUITS A N D SYSTEMS FOR VlDbO TECHNOLOGY, and a member of K O ’ \ WGI 1 Moving Picture Experts Group (MPEG) digital video coding stdndard- ization effort, which \pansored the July 1993 MPEG-2 meeting, which was held at Columbia Uniberwy


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