A robust digital method for film contrast correction in

Journal of Periodonml Raearch 1986: 21: 486-495

A robust digital method for film contrastcorrection in subtraction radiography

URS E. RUTTIMANN, RICHARD L, WEBBRR. AND EDGAR SCHMIDT

Diagnostic Systems Branch, National institute of Dental Research, NIH Bethesda, Maryland,and The University of Michigan, School of Dentistry, Ann Arbor. USA.

Digital subtraction radiography requires close matching of the contrast in the films to besubtracted. A digital melhod is presented permitting the retrospective correction of film contrastdifferences. The method is nonparanietric and derives the required gray level transformationdirectly from the histograms associated with the radiographs. This transform is shown to beunique and monotonic. It is based on fewer theoretical assumptions than a previously describedparametric correction method, and it performs significantly better in reducing the contrastmismatch measured by the standard deviation of the gray levels in the subtraction image.

Accepted for publication December 5, 1985

Introduction

Subtraction radiography (Grondahl. Gron-dah! & Webber 1983 a,b) is a potentiallypowerful method for monitoring the resultsof periodontal therapy. Its successful appli-cation in longitudinal studies requires thatthe variations caused by uncontrolledchanges of the x-ray imaging geometry, aswell as by film exposure and processing con-ditions, be small relative to the changes ofdiagnostic interest occurring over the obser-vation interval. While these prerequisitesare difficult to attain, controlled studies(Roshng et ai. 1983. Rethman et al. 1985)have demonstrated that geometrical regis-trations using occlusal templates (Roshnget al, 1975. Duckworth et al. 1983) resuU indetectability of periodontal lesions that issignificantly improved over conventionalradiographic methods. Reproducibility offilm contrast can be maintained by elec-tronic control of exposure time, x-ray tube

current and voltage waveforms, and bycareful film processing with quahty controlof the chemicals. Despite these efforts, con-trast discrepancies do occur in practice. Thisreport presents a new digital method forcorrecting retrospectively film-contrast dif-ferences in radiographs to be subtracted,and compares it to a technique publishedpreviously (Ruttimann et al. 1981).

Material and Methods

The radiographs were obtained from an on-going double-blind clinical investigation atthe University of Michigan, which is testingthe efficacy of metronidazole in the treat-ment of periodontitis. The patients admit-ted to the study had chnicai evidence ofmoderate to advanced periodontitis. Uti-lized for this study were 12 pairs of radio-graphs taken at 2 different occasions in theprotocol, either prior to any treatment, afterthe hygiene phase, or after periodontal sur-

ROBUST CONTRAST CORRECTION 487

gery. Geometrical registration of the radio-graphs was obtained using occlusalregistration/film holder devices (Duckworthet ai. 1983), The film holders were coupledmechanically to the cone of the x-ray ma-chine and the films* were exposed for .3 s(posterior teeth) or .24 s (anterior teeth) at70 kVp and 15 mA. The exposed films weredeveloped in an automated developer! at30' C. Subtraction images of the radio-graphs were obtained by digital subtractionof corresponding 512 x 512 x 8 bit video fra-mes using a computer-interfaced image pro-cessor J as described in (Grondahl et al.1983a).

The digital images were corrected forvariations in contrast prior to subtractionby applying separately both a previouslydescribed parametric method and the newlydeveloped nonparametric method describedin detail below. The parametric contrastcorrection procedure (Ruttimann et al.1981) is based on matching the first 2 mo-ments of the gray level distributions in theradiographic images to be subtracted. Thisis achieved by employing a quadratic trans-form y = ax- + bx -I- c mapping the graylevels X in image 1 into the gray levels y inimage 2 (Fig. 1). The coefficients a, b, andc are determined by 3 equations requiringthat the highest possible gray levels glH,, inboth images map into each other, and thatmean /ii and standard deviation CTI trans-form into the corresponding parameters fiiand (T: of the gray level distribution in thesecond image. Hence, the name parametricmethod was given.

The new contrast correction method pro-posed here is nonparametric and does notrely on matching parameters of correspond-

* Periapical, Ektaspeed, Eastman Kodak Co.,Rochester, NY 14650+ Philips 810. Philips Medical Systems. Inc..Shelton, CT 06484X IP 6400, Gould Inc., DeAnza Imaging &Graphics Division, San Jose, CA 95131

GRAY LEVELS iMAGE 1

Fig. 1. Parametric correction method: gray-ievei trans-formation y is based on the means /^i , /J?, the standarddeviations 01 and a-,, and the highest possibie grayievei giH, in the images.

ing distributions. Its principle is outlinedhere only, a rigorous derivation is presentedin the Appendix. Let us assume that thequantized gray scale in the images rangesfrom 1-N. Then for the image with the his-togram of the desired shape, let the numberof picture elements (pixels) with gray leveli be denoted by pi, forming the p-histogram.Similarly, for the image with the histogramto be modified, q, contains the count ofpixels with gray level j , forming the q-histo-gram. We define for the p-histogram thesum over the counts pi from gray level 1 upto gray level k by Sk, and analogously forthe q-histogram, the sum over the counts q,from 1 up to gray level m by Tm. It is thenrequired that these cumulative sums beclosely matched at every gray level m asderived in the Appendix:

max[Tm-!<Sk<Tn,],k

where m - 1,2 N, and T^-O. This meansthat we find for each gray level m in thehistogram to be modified (q-histogram) thelargest gray level k in the target (p-histo-gram) such that the cumulative sum over

488 R U T T I M A N N ET A L .

the p-histogram up to k is just equal to orsmaller than the cumulative sum over theq-histogram up to m, but strictly larger thanthe value of that sum up to m —1. Thiscorrespondence between k and m defines thegray level transform required. For example,assume that a correspondence has beenfound between m = 3 and k = 5, and similarlybetween m^7 and k —6. Then the graylevels in neighboring bins from m^4 upto and including m = 7 require pooling tomatch the cumulative sum S at k^6; i.e.gray levels 4, 5, 6, and 7 of the q-histogramare mapped into gray level 6 of the p-histogram. This transformation only re-quires pooling; and splitting of gray levelsin the histogram to be modified into two ormore different gray levels in the p-histogramdoes not occur.

It is shown in the Appendix that the gray-level mapping defined above is both uniqueand order-preserving, i.e., the larger of anypair of gray levels before modification re-mains after the transform larger than, or isat least equal to the transformed value ofthe smaller gray level.

Fig. 2 demonstrates the principle of thishistogram modification process for imageswith only 8 gray levels (Row 1), and thesimple case of a uniformly distributed targethistogram; i.e.. for all 8 gray levels P;^16(Row 2). The third row shows the successivevalues of S , the cumulative sum over thevalues of Pi: Si = 16. S^-16+16-32, S j -32+16. etc. The bottom row labelled qdisplays the histogram to be modified, andthe row above it shows the sequence of the

Fig. 2. Principie of histogram modification for imageswith 8 gray levels.

Fig. 3. Exampie from Fig, 2. Bottom: histogram to bemodified. Top: target histogram (open bars), and histo-gram after modification (fiiled bars).

corresponding cumulative sums T^: T^ —1,T2-1 + 7-8 , T3-8 + 21=29, etc. It is seenthat the first 3 gray levels in the q-histogramwith qj=^l, 7. and 21 must be pooled inorder to satisfy the inequality condition forSi: T2-8<S, -16<T3-29 . For the nextgray level in the q-histogram with T4=-63,2 inequality conditions involving S2 and S3hold: 29<32<63, and 29<48<63. Ac-cording to the definition for the mappinggiven above, the larger of the two S 's isselected for the mapping; i.e., gray level 4of the q-histogram is mapped into gray level3. as indicated by the arrow. Further suc-cessive evaluation of the inequality con-ditions for each of the remaining T^ com-pletes the construction of the gray leveltransformation as indicated by the arrowsin Fig. 2.

Fig. 3 shows the resuh of this histogram-rebinning process for the example com-puted above. The open bars display the tar-get histogram on the top, and below thehistogram to be modified. The filled barsrepresent the histogram obtained aftertransformation. While perfect uniformitycannot be achieved in this simple case withonly 8 different gray levels, the resulting


histogram certainly displays a more uni-form distribution than the original shownbeneath it. Furthermore, it is to be notedthat because some bins of the original histo-gram were pooled in the rebinning process,certain gray levels are no longer present inthe image after the modification; i.e., graylevels 2, 4, and 5 are missing. This is due toa general property of the mapping processas defined above, which permits only pool-ing of neighboring gray level bins, but notsplitting of one gray level into severalothers.

Results

Fig. 4 shows a pair of radiographs to besubtracted. The film on the right was under-exposed and. thus, had a higher mean graylevel /(^ 170.58, and a lower contrast asindicated by the standard deviation a^29.93 than the film on the left with / ( -122.51 and cr= 34.67.

The bottom row of Fig. 5 displays the

respective gray level histograms of the 2radiographs. As reflected in the correspond-ing means and standard deviations, the his-togram of the underexposed radiograph isdisplaced to the right, and narrower com-pared to the histogram of the properly ex-posed film. The curves labelled integral inthe 2 graphs represent the respective cumu-lative sums Sk and Tm to be matched by themodification process. The graph on the topright shows the transfer function obtainedby the matching procedure. It maps the graylevels in image 1 associated with the histo-gram beneath into the gray levels in image2, producing the modified histogram on thetop left. This histogram is to be comparedwith its target shown below it. The resultingmean and standard deviation are fi= 122.48and (7 = 34.73, confirming the global matchobtained by a general shift to the left and aslight stretch of the original histogram. Lo-cal effects of the modification proceduremay be seen by comparing the small sidelobes on the left sides of each of the histo-grams. This side lobe is built up in the re-

ig. 4. Pair of radiographs to be subtracted. Left: properly exposed film //m, /z= 170.58, (7 = 29.93.

122.51, <i = 34.67, right: underexposed


Fig. 5. Bottom: tiistograms of properly emodification using gray-level transfer chcumulative sums.

Dsed (left) and underexposed (right) films. Top: histogram after theicteristic shown to the right. Curves labelled integral represent the

suiting histogram and the gap between themain lobe narrowed compared to the orig-inal histogram, in order to obtain a closermatch to the shape of the target. This localshape matching property of the nonparam-etric method constitutes a significant advan-tage over the parametric method, which canonly achieve a global matching of the histo-grams.

Fig. 6 compares the resulting subtractionimages using either of the methods. Subtrac-tion of the unmatched radiographs is shownon the left. Clearly, unacceptably large gray-level difierences exist in the uncorrectedsubtraction image between the areas associ-ated with dentin, bone and soft tissue. Con-trast matching by the parametric methodyields a markedly improved subtraction im-age, shown in the center. However, the root

canal in the bicuspid, and a hint of thetrabecular pattern in the alveolar bone arestill visible. The subtraction after contrastmatching by the nonparametric methodshown on the right results in the most uni-form image with the least amount of visiblestructures present. Note in particular, thatcompared to the subtraction after para-metric matching, the soft tissue region isbetter matched to the adjacent areas repre-senting dentin and alveolar bone. This re-gion occupies gray levels characterized bythe small side lobe on the left side in eachhistogram, as discussed in Fig. 5. Showr.below each of the subtraction images artthe associated histograms, confirming thtresults of the visual inspection. The globamismatch of the histograms in the originaradiographs resulted in a displacement oi


Fig. 6. Top row: Subtraction images, botlom row: corresponding histograms. Left: uncorrected radiographs /j =78.93, tT = 9.78. Middle: cadiographs corrected by parametric method, ;t = ^Z7.3^. (T = 7.35. Right: radiographscorrected by non-parametric method, /j= 127.03, (7 = 6.50.

the mean gray level in the subtraction imagefrom the central value of 127 to /y^ 78.93,and in a standard deviation of 0-^9.78.Both parametric and nonparametric correc-tions resulted in good mean gray level mat-ches with /^^ 127.31 and 127.03, respective-ly, and in standard deviations ordered ac-cording to their potential for localhistogram matching with (T = 7.35 and 6.50respectively.

A further example of the relative per-formance of the 2 methods is demonstratedin Fig. 7. The top row shows a pair oforiginal radiographs with significant con-trast differences. At the bottom left is thesubtraction image after parametric correc-tion, and on the right after nonparametricmatching. The relatively low gray level as-sociated with the regions of soft tissue inthe left image may leave some uncertaintyabout the extent of the crestal bone loss

indicated by the dark band. The local histo-gram matching achieved by the nonpara-metric method resulted in a better correc-tion of the gray levels associated with softtissue (arrow), and, thus, the bone loss isdisplayed with a higher contrast.

The parametric and the nonparametricmatching methods were each applied to 12pairs of radiographs obtained from an on-going clinical study. The standard devi-ations in the gray level histograms of theresulting subtraction images were taken asthe performance measure for comparing themethods. The graph in Fig. 8 summarizesthe results, with the lengths of the bars de-picting the observed standard deviations.Except for one case, the standard deviationsusing the nonparametric method were low-er, yielding statistical significance (p<.01)of the difference by the paired Wileoxontest.

492 RUTTIMANN ET AL.

Fig. 7. Top row Pair of radiographs to be subtracted. Bottom row: subtraction image after parametric correction(left) cf=15 15 and after nonparametric correction (right), (T= 13.86. Arrow points out improved detail contrastindicating bone loss

DiscussionThe parametric method has been shown towork well in experimental {Ruttimann eta!. 1981, Rethman et al. 1985) and clinical(Rosling et al. 1983) subtraction radiogra-phy. For theoretical reasons, the parametricmethod is hmited to, and thus, expected towork best for. "smooth" histograms thatare relatively well concentrated around theirmean gray levels. When the histograms arisefrom Gaussian distributions this method is,of course, exact. However, the presence ofmultiple strong peaks and/or large skew inone of the histograms violates the math-ematical assumptions underlying the para-metric method, and, thus, suboptimal re-sults or even occasional gross errors mustbe expected under these circumstances. In

SUBTRACTION IMAGES

Fig. 8. Standard deviations in 12 subtractionafter parametric (open bars) and nonparame'rection (filled bars).

ROBUST CONTRAST OORRECTtON 493

order to alleviate possible problems, a new,nonparametric method was proposed thatdoes not rely on matching parameters ofsmooth distributions approximating the ob-served histograms. Instead, the proposedmethod operates directly on the histogramsthemselves. It permits modifying the shapeof a histogram associated with a given im-age into any other desired shape, withoutposing any restricting assumptions regard-ing the particular shapes of the respectivehistograms involved. In particular, the his-tograms may be multimodal, and separatematching of the individual modes can beachieved. The latter property was referredto above as local histogram matching capa-bility.

The gray-level mapping procedure isbased on shifting and/or pooling bins as-sociated with the histogram to be modifiedsuch that the cumulative sum over thecounts in its bins extending from one upto each gray level is matched as closely aspossible to the corresponding sum in thetarget histogram, without splitting existingbins into different gray levels. This modifi-cation principle is rigorous mathematicallyand has been proven to generate a gray-level mapping which is both unique andmonotonic. Uniqueness means that for eachgray level in the original histogram there isonly one possible gray level in the targetinto which it can be mapped. Monotonicityimplies that the transform cannot introducegray-level reversals into a sequence of, say,monotonicalty increasing gray levels takenfrom a string of neighboring pixels in theimage to be modified. Without this assur-ance the transform could generate false con-lours in the modified image. Monotonicityis demonstrated in the upper right graph ofFig. 5. showing that for any increase of thegray level in image 1 the curve representing'he transfer function either increases, or atieast stays at the same level.

While both contrast correction proce-

dures achieve global matching of the histo-grams concerned, i.e., the respective meansand standard deviations are sought to beequalized, only the nonparametric methodhas a local matching capability. The import-ance of this property is demonstrated inFigs. 6 and 7 by the superior matching ofthe gray levels associated with soft tissueareas in the radiographs. These gray levelsare generally located in the left tails of thehistograms, and may at times display strongmodes. Proper matching of side lobes inthe histograms is important for achieving abackground as uniform as possible in thesubtraction image, in order to better discernevidence of true tissue changes.

The standard deviations of the gray levelsin the subtraction images are a function ofboth the global and local histogram match-ing attained. Although it is theoreticallypossible that one matching could be tradedfor the other while maintaining a constantstandard deviation., it is highly unlikely thatthis will happen with the proposed non-parametric method for images of practicalconcern. Hence, the reduction in standarddeviation attained by the nonparametricmethod may be considered Sargely the resultof the local histogram matching achieved.Also, the standard deviations may not fullyreflect the often substantial visual improve-ments achieved by local histogram match-ing. With due consideration of these limi-tations of the standard deviation as a meansfor comparing the methods, the resultsshown in Fig. 8 clearly demonstrate superi-ority of the nonparametric procedure.Hence, smaller standard deviations can beexpected to be attained in the majority ofsubtractions of clinically obtained radio-graphs. These smaller standard deviafionssignify the better gray level matchingsachieved for the respective areas of enamel,bone, dentin, and soft tissue, enhancing thechance for the detection of more subtle ana-tomical changes.


In conclusion, the nonparametric methodis based on fewer theoretical assumptionsand, thus, is more robust in practical appli-cations than the parametric approach; andit yields statistically significantly better re-sults in terms of standard deviations in thesubtraction images, indicating a closermatching of the respective gray levels as-sociated with different anatomical areas inthe radiographs.

Appendix

The digital images are quantized into Ngray levels. Let p, denote the number ofpixels with gray level i in the target histo-gram (p-histogram), and similarly let qj de-note the number of pixels with gray level j inthe histogram to be modified (q-histogram).Thus,

From the right side of (5) it follows that

Sk-T«<0. (6)

Subtracting T from the left side of (4), andusing (6) yields

^„.,-^„<s,-T„<o. (7)

If m>n, then m— 1 >n, and because Tj ismonotonic it follows that T^.i—Tn>0,which is a contradiction to (7). Similarly, ifm<n it can be shown that T^.i—Tm>0leads to a contradiction with (4) and (5).Hence, (4) and (5) cannot hold simul-taneously, unless m^n, (Q.E.D.).

Using the same reasoning as in the proofabove, it can be shown that the gray leveltransform defined by (3) preserves the order-ing of the gray levels in the 2 histograms,i.e..

p,>0,qj>0;iJ-L2 N. (I)

ifk'<k" thenm'<m". (8)

The cumulative sums over the respectivehistograms are given by

S.-i;p,-,T™-X qMk,mE[l.N], (2)

From (1) and (2) it follows immediately thatSj and T^ are monotonic increasing func-tions of the integers k and m, respectively;i.e., for all gray levels if k'<k" then Sfc.<St..(or if m'<m" then T . <T^..). We define amapping of gray level k from the p-histo-gram into gray level m of the q-histogramsubject to the constraints

Assertion: For each k e [1, N] there existsonly one m E [I, N] which satisfies (3).

Proof Assume (3) is satisfied by m and byn /m. Henee,

T^_i<Sj<T^ (4)

T„-,<S,<T„. (5)

Note, so far we have shown that for each kthere is only one m satisfying (3). There may,however, exist several gray levels k' < k" <...,for which (3) is satisfied by the same valueof m as indicated in (8), when equality existsbetween m' and m". Hence, in order to makethe mapping between k in the p-histogramand m in the q-histogram unique, we selectthe largest possible gray level from amongthe admissible gray levels k' < k" <... associ-ated with the same m. Consequently, themapping satisfying the constraints

m=l,2 N, To-0, (9)

ensures uniqueness of the relation betweenthe gray levels k and m.

References

Duckworth, J. E., Judy, P. F,, Goodson. J. M. &Socransky, S. S. 1983, A method Tor the geo-metric and densitometric standardization of in-traorai radiographs. Journal of Periodontoiogy54: 435-440.


Grondahl. H.-G., Grondahl, K. & Webber, R.L. 1983a. A digital subtraction technique fordental radiography. Oral Surgery. Oral Medi-cine, Oral Pathology 55: 96-102.

Grondahl, H.-G., Grondahl, K.. & Webber, R.L. 1983b. Digital subtraction radiography fordiagnosis of periodontal bone lesions wilhsimulated fast speed systems. Oral Surgerv.Oral Medicine. Oral Palhologv 55: 313-318. '

Rethman, M. P., Ruttimann, U. E.. O'Neil, R.B.. Webber. R. L., Davis, A. A. & Woodyard.S. G. 1985. Diagnosis of bone lesions by sub-traction radiography. Journal of Periodontolo-gy 56: 324^329.

Rosling. B. G., Slots, J., Webber, R. L..Christersson, L. A. & Genco. R. J. 1983. Mi-crobiological and clinical effects of topicalsubgingival antimicrobial treatment on humanperiodontal disease. Journai of Clinical Peri-odontology 10: 487-514.

Rosling, B. G., Hollender. L.. Nyman. S. &

Olsson, G. 1975. A radiographic method forassessing changes in alveolar bone height fol-lowing periodontal therapy. Journai of ClinicalPeriodontotogy 2: 211-217.

Ruttimann, U. E., Okano, T., Grondahl, H.-G..Grondahl, K. & Webber, R. L. 1981. Exposuregeometry and film contrast differences as basesfor incomplete cancellation of irrelevant struc-tures in dental subtraction radiography. Pro-ceedings of the International Society for OpticalEngineering fSPIE) 314: 372-377. '

Address:

Dr. Urs E. RuttimannSenior Staff FellowDiagnostic Systems Branch, NIDRBuilding 10. Room 5N-2689000 Rockville PikeBethesda. MD 20892USA

Documents

A robust digital method for film contrast correction in