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A linear prefilter for image sampling with ringing artifact control

A linear prefilter for image sampling with ringing artifact control
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  ALINEARPREFILTERFORIMAGESAMPLINGWITHRINGINGARTIFACTCONTROL G.Blanchet   1 , 2  , L.Moisan  1 , 2  , B.Roug´ e  2 , 31 MAP5, Universit´e Paris 5, France 2 CMLA, ´Ecole Normale Sup´erieure de Cachan, France 3 Centre National d’´Etudes Spatiales, Toulouse, France {  blanchet, moisan, rouge  } ABSTRACT When sampling a continuous image or subsampling a dis-crete image, aliasing artifacts can be controlled by filteringthe data prior to sampling. Bandlimiting filters completelyavoid aliasing artifacts, but have to find a compromise be-tween blur and ringing artifacts. In this paper, we propose anew joint definition of blur/ringing artifacts, that associatesto a given bandlimited prefilter its so-called  Spread-Ringingcurve . We then build a set of filters yielding the optimalblur/ringing compromise according to the previous defini-tion. We show on experiments that such filters yield sharperimages for a given level of ringing artifact. 1. INTRODUCTION Producing a high quality low-resolution image from a high-resolution one, or building a high quality image from a con-tinuousmodelbothrequireagoodunderstandingofthesam-pling process. The solution to avoid aliasing artifacts, thatmay introduce dramatic texture changes, losses of connect-edness or staircase effects, has been given in 1949 by Shan-non [1]: aliasing can be avoided by bandlimiting the inputimage (that is, setting to 0 its high frequency components),or, equivalently, by filtering the image with a  sinc  filter. Un-fortunately this filter oscillates with slow decay, and intro-duces unwanted oscillations in the vicinity of sharp edges.Since then, other filters were developed such as splines [2],prolate [3] or Ces`aro filter [4] but the necessary trade-off between blur and ringing they introduce was never reallyevaluated.In the literature, the ringing artifact (sometimes referredto –unproperly– as the Gibbs phenomenon [5]) is often de-scribed as noise and is measured with the SNR [6]. Morespecificmeasuresofringing, liketheperceptualringingmet-ric [7] and the visible ringing metric [8], manage to controlthe amount of ringing artifacts put in a signal, but do not tryto define the ringing phenomenon by itself.In this paper, we propose a joint definition of ringingand blur artifacts, yielding what we call a  Spread-Ringingcurve  (Section 2). This curve allows us to compare therelative sharpness of classical bandlimited prefilter for anygiven level of ringing (see Fig. 5). In Section 3 we numer-ically design an optimal filter ( Spread-Ringing  filter (SR))that minimizes both ringing and blur artifacts, and compareit with other filters (Fig. 5). By preserving better edges,this filter reduces the deterioration of image quality usuallyobserved with classical filters, as shown by the image ex-periment reported on Fig. 3 and 4. 2. THESPREAD-RINGINGCURVE In all the following, we shall only consider 1D signals andfilters. Associated 2D filters for images will be built in aseparable way. Let us first define the set of bandlimited sig-nals by B W   =  { g  ∈  L 2 ( R ) ,  supp (ˆ g )  ⊂  [ − W,W  ] } , ˆ g  standing for the Fourier Transform of   g . The ringing arti-fact may occur when a signal  f   is projected onto  B W   (fre-quency cutoff), or more generally when it is  approximated  by a signal  g  ∈  B W  . We propose the following definition. Definition1  When we approximate a signal  f   ∈  L 2 ( R ) by a bandlimited signal  g  ∈  B W   ,  g  may have additionaloscillations the sampled signal  f  ( kπ/W  )  does not have:this is the  continuous ringing phenomenon . Note that the  discrete ringing phenomenon  could be definedas well by considering only oscillations of the sampled sig-nal  g ( kπ/W  ) . Now following Definition 1, we would liketo define a way to  measure  ringing artifacts, in order to beable to constrain them to a non-perceptible level (that is,below the image noise level).Let us consider a filter  ϕ  ∈  B W  , and apply it to theHeaviside function ( H  ), yielding a new signal  g  =  H   ∗  ϕ .Two phenomena may appear for g  : first, the transition from0 to 1 has a certain spread; second, oscillations may appeararound this transition. Defining a measure only for theseoscillations (ringing) is difficult because in general there is  S10 . 2R2R Fig. 1 .  D S,R natural way to define the transition domain (spread) of  g . This is why we choose to simultaneously measure thespread of the transition ( S  ) and the amplitude of the oscilla-tions outside the transition domain ( R ) by constraining thegraph of   g ,  Γ g  =  { ( x,g ( x ));  x  ∈  R }  to be contained in acertain domain (Fig. 1). Definition 2  We denote  D S,R  the subset of   R 2 defined by D S,R  =  x ≤− S 2  and   | y |≤ R ( x,y )  or  | x |≤  S 2  and   − R ≤ y  ≤ 1 +  R or   x ≥  S 2  and   | 1 − y |≤ R  . We observe that a small R prevents too large oscillationsof  g , while a small S   ensures a sharp edge approximation of the discontinuity of   H  . Hence, the couple  ( S,R )  reflectsthe blur/ringing trade-off to be satisfied by  g . This leads to Definition 3  The Spread-Ringing domain associated to a filter   ϕ  ∈  B W   is D ( ϕ ) =  { ( S,R )  ∈  (0 , + ∞ ); Γ ϕ ∗ H   ⊂  D S,R } .  (1) The Spread-Ringing curve associated to  ϕ  is the boundaryof   D ( ϕ ) . If   ( S,R )  ∈  D ( ϕ ) , so does  ( S   +  p,R  +  q  ) . So theboundary of   D ( ϕ )  is obtained by taking the minimal pos-sible value of   R  for any fixed  S  , or the minimal value of   S  for a fixed  R . The Spread-Ringing curve can be describedby the graph of a function r ϕ ( S  ) = min { R  ∈  (0 , + ∞ );( S,R )  ∈  D ( ϕ ) } . Thisdefinitionofringinghasthedrawbackthatnouniquevalue of  S   and R are associated to a given filter since, as wementioned before, it is difficult to distinguish the edge fromthe ringing near the transition. However, this constructionhas the advantage to remain very general, since it does notrely on any arbitrary threshold. Moreover, Definition 3 canbe generalized to a  family  of filters  ϕ α  ( α  being in generala real parameter) and still yields a single  Spread-Ringingcurve , as specified in Definition 4  The Spread-Ringing domain associated to a family of filters  ( ϕ α ) α ∈ A  is D (( ϕ α ) α ∈ A ) =  α ∈ A D ( ϕ α ) .  (2)  Its Spread-Ringing curve is the associated boundary. 3. SR FILTERS We now build a family of bandlimited filters  ( ϕ S  ) , called SR filters , having the best possible  Spread-ringing curve .These filters have minimal ringing  R  for a fixed spread  S  ,or equivalently, minimal spread S   for a fixed ringing R . Forconvexity reasons, this family exists and is unique. For each S  , ϕ S   can be computed by the following iterative algorithm:set  ϕ  =  δ  0  (Dirac)assign a large value to  R repeatset  ϕ S   =  ϕ set  N   = 1 repeatforce  Γ ϕ ∗ H   ⊂  D S,R  by thresholdingforce  ϕ  ∈  B W  , that is  ˆ ϕ ( ξ  ) = 0  for  ξ   ∈  [ − W,W  ] set  N   =  N   + 1 until  convergence test   or  N > N  max reduce  R until  N > N  max The  convergence test   is satisfied when the two forcingsteps have small enough effects on  ϕ , that is, when the firstforcing step changes  ϕ  by less than  ε 1  (according to  L ∞ norm) and the second forcing step changes  ϕ  by less than ε 2  (according to the  L 2 norm). In practice, we checked thatconvergence was undoubtedly attained with  N  max  = 10000 and  ε 1  =  ε 2  = 0 . 001 .Note that the value of  W   is arbitrary since different val-ues of   W   simply mean different scales. Numerically, weused 1024 samples to represent  ϕ  and chose a frequencycutoff corresponding to a reduction by a factor 16.WhenlookingatSRfilters(obtainedwiththisalgorithm),we noticed that their Fourier transform was real symmetric(as expected), but not unimodal (that is, not both increasingon R − and decreasing on R + ), due to large weights on thefrequencies near  W   (and  − W  ). This phenomenon, due tothe fact that the ringing is controlled through a  L ∞ norm,can be undesired in some applications (since for example,natural images generally have unimodal spectra). It canbe avoided by building  unimodal SR filters  (see Fig. 2),obtained by adding a third forcing step in the algorithm(between the  D S,R  forcing step and the  B W   one) whichchanges  ˆ ϕ  into its  L 2 unimodal regression.  Remark:  We chose to control the ringing with the  L ∞ norm (instead of   L 1 or  L 2 norms) because our perceptionof images is more sensitive to large local overshoots thanto small oscillations (hidden by image noise or textures)spread on a large domain. In that sense, the characteriza-tion of ringing we proposed is specific to images and maynot be adequate in a general signal processing framework. W       T      F       (       f       i       l      t     e     r       ) frequencyS=2.5S=2S=1.5S=1 Fig.2 .  Some unimodal SR filters in Fourier domain 4. COMPARISONWITHCLASSICALFILTERS We compare the results of the SR filters with the followingclassical filters: •  sinc  filter:  ∀ ξ   ∈ R ,   sinc( ξ  ) = 1 [ − W,W  ] ( ξ  ) . •  Frequency truncated triangle ( α  ∈  [0 , 1] ):   T  α ( ξ  ) =  1 +  α − 1 W   | ξ  |  · 1 [ − W,W  ] ( ξ  ) •  Cosine filter ( α  ∈ R ):   C  α ( ξ  ) = 1 + cos( αξ  )2  · 1 [ − W,W  ] ( ξ  ) •  Prolate [3] and bandlimited Gaussian filters.As expected, SR filters (Fig. 5) show better results thanothertestedfilters. Theyintroducelessringingforanygivenedge spread. The unimodal constraint reduces the perfor-mance of the filter, as the  Spread-Ringing curve  of the uni-modal filter (a) is on the right hand side of the  Spread- Ringing curve  of the non unimodal filter (b). This diver-gence gets magnified for small spread values.Among other filters, we observe two groups. The firstgroup (c,d,e) contains filters that give relatively good resultsin comparison with the SR filter. For a spread value smallerthan 2 pixels (which is a common level of sharpness), theygive very similar results, especially the prolate (c) and thecosine filter (d). The filters of the second group (f,g) per-form poorly as they produce a high level of ringing. 5. EXPERIMENTS We now apply SR filters to a natural image. We generalizethe 1D filter to 2D by using a separable convolution alongboth coordinates. Fig. 3 shows results obtained on a sub-part  (512  ×  512)  of a natural image. In this experiment, Line 130 Fig.3 .  The reference high-resolution (not bandlimited) town im-age  I   (bottom row) is prefiltered with the prolate filter (yieldingthe top left image:  I  P  ) and with the SR filter (top right image: I  SR ). One can see that the white roof is better preserved with theSR filter (transitions look sharper). we chose the SR filter corresponding to  R  = 0 . 012 , allow-ing  1 . 2%  of overshoot with respect to the transition value,whichapproximatelycorrespondstothelevel of noiseof theimage. The convolution of the srcinal high resolution im-age (bottom of Fig. 3) was made with this SR filter and witha prolate filter (with same value of   R ) as a comparison. Aspredicted by the SR curves, the prolate filter yields a moreblurry image, as shown on Fig. 3 and 4. 0500100015002000250030003500400050 100 150 200 250 300    G  r  e  y   l  e  v  e   l pixelsOriginalProlateSR filter Fig. 4 .  Line 130 from the images of Fig. 3 near the whiteroof. The amplitude of the transitions are better preserved withthe SR filter than with the prolate. The difference between  I   andboth  I  P   and  I  SR  confirms the improvement:  || I   −  I  P  || 1  = 90 , || I   −  I  SR || 1  = 83 ,  || I   −  I  P  || 2  = 150 ,  || I   −  I  SR || 2  = 137 .    R   i  n  g   i  n  g   R   (   M  a  x   i  m  u  m    O  v  e  r  s   h  o  o   t   ) Spread S (pixels)(c)(b)(a)(g)(f)(d)(e)SR unimodal filter (a)SR filter (not unimodal) (b)Prolate (c)cos (Fourier domain) (d)Gaussian (e)truncated triangle (Fourier domain) (f)sinc (g) Fig. 5 .  Spread-Ringing curves  of some bandlimited low-pass filters. These curves display the trade-off between ringing andblur artifacts for each filter. This representation incidentally shows that bandlimited Gaussian filters perform slightly betterthan prolate filters for small spreads ( S <  2 . 5  pixels), which are the most useful values in practice. As expected, SR filtersachieve the best compromise (more than  20%  better than the others for  S   = 1 . 5  pixels). 6. CONCLUSION We introduced a joint measure of blur and ringing artifactsforbandlimitingfilters, andbuiltSRfiltersminimizingtheseartifacts. Compared to classical filters, SR filters yield per-ceptible improvements when bandlimiting an image beforesampling, which could be interesting for applications re-quiring high quality image reduction. This approach alsopushes back the frontier of attainable aliasing/blur/ringingcompromises in image formation processes. In this paper,2D filters are built in a separable way, but generalization tonon-separable filters (radially symmetric or not) could beinvestigated as well. 7. REFERENCES [1] C. E. Shannon, “Communications in the presence of noise”,  Proc.of the IRE  , vol. 37, pp. 10-21, 1949.[2] P. Th´evenaz, T. Blu, M. Unser , “Image Interpola-tion and Resampling”,  Handbook of Medical Imaging,Processing and Analysis , I.N. Bankman Ed., AcademicPress, pp. 393-420, 2000.[3] D. Slepian, “Prolate Spheroidal Wave functions,Fourier Analysis, and Uncertainty- V: The DiscreteCase”,  Bell System Technical Journal , 1978.[4] A. J. Jerri,  The Gibbs Phenomenon in Fourier Analy-sis, Splines and Wavelet Approximations , Kluwer Aca-demic, 1998.[5] E. Hewitt, R. Hewitt, “The Gibbs-Wilbraham phe-nomenon: An episode in Fourier Analysis”,  Hist. Exact Sci. , pp. 129-160, 1979 .[6] A. Mu˜noz, T. Blu, M. Unser “ l  p -Multiresolution Anal-ysis: How to Reduce Ringing and Sparsify the Error”,  IEEE Trans. Image Process. , vol. 11:6, 2002.[7] P. Marziliano, F. Dufaux, S. Winkler, T. Ebrahimi,“Perceptual Blur and Ringing Metrics: Application toJPEG2000”,  SPIC  , vol. 19, pp. 163-172, 2004.[8] S.H. Oguz,  Morphological post-filtering of ringing and lost data concealment in generalized lapped orthogonaltransform based image and video coding , Ph.D disser-tation, Univ of Wisconsin, Madison, 1999.
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