graeco-latin squares design for line detection in the presence of correlated noise

IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 9, NO. 4, APRIL 2000 609

Graeco–Latin Squares Design for Line Detection inthe Presence of Correlated Noise

Gerard J. Genello, Senior Member, IEEE, Julian F. Y. Cheung, Member, IEEE, Steven H. Billis, and Yoshikazu Saito

Abstract—In this paper, the problem of detecting narrow linesembedded in correlated noise is investigated. An approach basedon the contrast theory is adapted to the four-way Graeco–Latinsquares design. The analysis provides a theoretical basis for thecomparison of a line detector to other detectors that ignore noisedependence. A new and compensated algorithm is developed fordetecting lines oriented in any major direction in a two-dimen-sional image plane. The four-way design is also shown to removeany background obscuration, and the actual processor is robust,simple, efficient, and suitable for real-time applications. Extensivecomputer simulations demonstrate the performance of the pro-posed detector relative to some classical mask-based detectors.

Index Terms—Analysis of variance, contrast algorithm, de-pendent Gaussian environments, estimable function, F-statistic,four-way Graeco–Latin squares design, line detection, polynomialapproximation detector, Prewitt detector.

I. INTRODUCTION

I N THE PAST few decades we have seen significant progressin the area of line detection motivated by its importance in

almost all aspects of vision applications, including military, bio-medicine, industrial automation, meteorology, and data com-pression. In particular, Haralick proposed to approximate theneighborhood of a pixel using a functional form that consistsof a cubic polynomial in the variables representing the rows andcolumns [1]. According to the facet model, a line is equivalentto a digital ridge (or valley), which is a simply connected se-quence of pixels with gray-tone intensity values significantlyhigher (or lower) in the sequence than those neighboring the se-quence. Lines are identified by looking for zero crossings of thesmooth surface’s first directional derivative taken in a directionextremizing the second directional derivative.

Most of the vision algorithms model an image based on thegray intensity of individual pixels. Very little work has beendone on exploiting local features. This contradicts the mecha-nism of the human visual system (HVS): when one look at ascene both the gray intensities and features such as lines, edges,contours, etc., are simultaneously processed. If a line is deemedbroken, the HVS will determine whether the cut is natural or not,on the basis of local pixels, and take appropriate actions. Be-

Manuscript received April 14, 1997; revised July 22, 1999. This work wassupported in part by the USAFORS IPA Program and in part by the NYIT Fac-ulty Research Grant. The associate editor coordinating the review of this man-uscript and approving it for publication was Dr. Andrew F. Laine.

G. J. Genello is with the U.S. Air Force Research Laboratory/SNRT, Rome,NY 13441-4514 USA.

J. F. Y. Cheung, S. H. Billis, and Y. Saito are with the Department of ElectricalEngineering, New York Institute of Technology, New York, NY 10023 USA(e-mail: [email protected]).

Publisher Item Identifier S 1057-7149(00)02679-8.

cause of the single-pixel configuration the output consists onlyof the magnitude response, and precise information on the ef-fect of features is lost. The incompatibility of standard visiontechniques to the HVS partly explains that conventional featuredetection, segmentation, recovery, etc., often do not perform sat-isfactorily when applied to a natural scene [2].

Recently, an entirely different philosophical point of viewwas initiated by Haberstroh and Kurz [3], who utilized the anal-ysis of variance (ANOVA) techniques [4] to model an imageusing treatments. The contribution of a treatment to a specifiedgroup of pixels is known as a treatment effect. For any partic-ular treatment a countable collection of its variates is called atreatment group. The modeling of an image (noise-free) usingsome treatment groups with exact one–one mapping is knownas a design. As an example, using column treatments aneighborhood may be represented by , where

is the mean of the neighborhood and is the noise strengthat the th row, th column, . This is the classicalone-way design possessing one column treatment group, withone member along each column and the corresponding treat-ment effect . Pixels of the same column have the same columntreatment, pixels of different columns are not related to eachother. If a line exists between columns 2 and 3, then a signifi-cant difference occurs between the treatment sets and

.As another example, in the classical two-way design a

neighborhood is represented by ,. There are two treatment groups, i.e., row, and column

treatments with the corresponding treatment effects , ,respectively. The additional treatment group makes it possible todetect vertical and horizontal features while taking into accountembedded structures normal to the direction of detection. Also,the estimation space using these orthogonal treatments providesa better estimate of the unknown treatment effects.

In general, a line detector is a two-step device which de-tects the presence of a line from an ANOVA model and thendetermines the line location within a mask frame by usingthe double-edge transition characteristic of a line. Testing forthe pulse distribution of the gray level is a very conservativeway to locate the lines, and causes many Type-II errors. Theuse of the -statistic independently and then the-statisticwhich infers the likelihood of the difference in the trajectorymean and the background mean [5] results in a very largethreshold for a contrast to be detected and located at lowSNR environments.

The extension of the two-way design to include more treat-ment groups while preserving the mask size and hence the res-olution was undertaken by Haberstroh and Kurz [3], who used

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610 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 9, NO. 4, APRIL 2000

a Graeco–Latin squares (GLS) design for image modeling. Byincorporating the single contrast comparison in the-statistic,a more accurate location of a line in a mask is made. However,a shape test in the form of is incorporated, where isa single contrast function (defined in Section II) and is aunit step function. This process conditions the-statistic to im-prove the performance of the line detector. This additional testmay be superfluous in the case of narrow lines.

This motivates the present research, which combines theshape test and the-statistic yielding the contrast algorithmin order to overcome these difficulties. The extension of thenew algorithm to dependent noise environments is developed.The GLS design that we use has the characteristic that hori-zontal treatments cross rows; it is more effective in modelingreal-world textured regions. The new algorithm uses a variablethreshold evaluated in the space of only. A comparison ofthe general contrast algorithm with its counterpart that assumesindependent noise, as well as some classical mask-basedlinear and nonlinear detectors, is verified both analytically andthrough computer simulations.

II. CONTRAST ALGORITHM

In this section, we consider the relationship between the con-trast space and the-statistic. A new detector which comparescontrasts using the -statistic is also derived.

A. Linear Contrast

Suppose is a known coefficient vector andis an unknown parameter vector, then we

define as a parametric function. Aparametric function is said to be estimable if it has anunbiased linear estimate, no matter whatare [4]. Given , if there exists an estimate suchthat

(2.1)

then is an estimable function. is an observationvector and is an coefficient vector. All bold-facedsymbols represent vectors/matrices, and primes denote thetranspose.

In image processing problems, the observation vector asgenerated by row scanning is attributed to some physical fea-tures (treatment groups)1 such as lines, edges, etc., imposed onthe background structures. The gray values contributed by thefeatures (treatment effects) may be represented by a parametervector , subsequently corrupted by a noise vector. A suitableparametric model is : , where ,i.e., is Gaussian distributed with mean and covariancematrix . The effect matrix has entries or ,depending on whether a specific feature (treatment) is presentor not. For example, a image bearing potential line

1Terminologies commonly adopted in ANOVA paralleling those in imageprocessing are given in parentheses.

features imprinted at four major directions on the backgroundis

(2.2)

;is the general mean of the background structures; and, ,, and denote the gray value (treatment effect) of theth

horizontal, the th diagonal, the th vertical, the thdiagonal line feature (treatment), respectively. Undera con-trast is estimable if there exists such that

. That is, is estimable pro-vided .

Since

(2.3)

one prefers a minimum variance unbiased estimate (MVUE). This follows from the Gauss–Markov theorem [4, p.

14], which stipulates that is the MVUE if for any weset

(2.4)

with the least squares estimate of

(2.5)

From (2.1), we obtain , so

(2.6)

An estimable function subject tothe condition is known as a linear contrast. Theconcept of linear contrast imparts important applications in linedetection.

B. Contrast Ellipsoid

Let us choose a set of linearly independentcontrast functions and consider a -Dcontrast space spanned by . We have

and . By virtue of (2.4) and(2.5), the corresponding MVUE is , where

and . Rewriting the set of linearcontrasts as , the most efficient estimateis

(2.7)

GENELLO et al.: GRAECO–LATIN SQUARES DESIGN FOR LINE DETECTION 611

We have , , ,, . It is straightforward to obtain

, ,.

Since and the components of arelinear combinations of Gaussian random variables,is also

-distributed. Therefore,is distributed as , because is the sum of the squares of

independent Gaussian random variables, each with meanand variance . By Corollary 1.2 in Section III the distribu-tion of is , sug-gesting the estimation of by . Equa-tion (a2) in the Appendix confirms that the vector ,which after decorrelation is in the error space, is orthogonal to

, which is in the estimation space. By (2.6) each compo-nent of is a linear combination of . The two quadraticfunctions , are statistically inde-pendent, yielding the ratio that is

distributed. signifies the -distribution with ,degrees of freedom (DF), and its upper -point. Under

the probability is that

(2.8)

A test of the hypothesisversus the alternative : some elements of is equiva-lent to testing against . Makinguse of the contrast ellipsoid defined in (2.8), the test rejects

whenever the confidence ellipsoid does not cover the point. The test statistic becomes,

at the level of significance

if otherwise(2.9)

For the special case of one contrast function , is ascalar, , , and . Thepreceding test reduces to

if otherwise (2.10)

This equation represents the single comparison contrast algo-rithm used in this work to detect line features embedded in cor-ruptive noise.

III. GRAECO–LATIN SQUARESDESIGN

A GLS model is a four-way design which contains four treat-ment groups with levels each. For a completely randomizeddesign, one needs observations, at least one for each cell. AGLS is an incomplete blocks design with only observationswhich results in a significant reduction in processing time andcomplexity. The GLS design is important because of its abilityto detect diagonals and also in modeling the background struc-tures [3].

The treatment layout of a GLS design is illustratedin Fig. 1. In this design, there are four treatment groups: therows, the 45 diagonals, the columns, and the 135diagonals.

Fig. 1. Four-way treatment layout in a5� 5 GLS design.

The treatments are orthogonal in the sense that all treatmentsappear together only once in the entire design. Let, , ,and denote the row, the diagonal, the column, and the

diagonal effects, respectively. Each pixel is represented by, , and is

the set of observations. The parametric model is identical toSection II-A but stated here

(3.1)

where is an observation vector;is

a parameter vector; is an four-way effect matrix; andis a noise vector having the distribution . The

effect matrix using a rectangular mask is shown in (2.2).In general, the effect matrix does not have full rank. For

example, in of (2.2) add to the -column the sum of allother -columns; the like for -, -, and -columns. Sub-tracting each of the resulting -, -, -, and -columns bythe -column, the affected columns become zeros. Evidently,

rank . The rank deficiency implies thatthe estimate of is not unique.

Recall, is a correlation coefficient matrix, it is symmetricand positive-definite. Following [6], there exists a set of-Dorthogonal vectors satisfying for and

. Defining a modal matrix , then.

Premultiplying (3.1) by to get

(3.2)

where , , and . In light of, , the transfor-

mation whitens the noise and, because it is a linear operator,is -distributed. As

, the estimate of is still not unique. Considersubjecting to linear restrictions ,

a matrix of known constants with .Defining the composite matrix , then by [4, Th. 3]


Fig. 2. 5� 5 GLS masks for line features where� (or �) denote background structures (or line features), circled numerical value denotes members of treatmentgroup of background/line, and uncircled value denotes reference row or column in a mask.

Fig. 3. Schematic diagram of the GLS-based line detector.

is unique in the sense that for every possiblein the parametricmodel (3.2) there exists a unique vectorsatisfying

and (3.3)

if and only if 1) and 2) no linear combination ofthe rows of is a linear combination of the rows of exceptthe zero vector.

Suppose conditions 1) and 2) are met. For anylet be theunique solution of (3.3), then . Premulti-plying both sides by , taking the inverse and replacing by

, then . Therefore,an unbiased estimate ofis

(3.4)

where the first equality is due to (3.3). Henceforth,, are usedinterchangeably.


Fig. 4. Lena picture corrupted by Gaussian noise: (a)� = 5; � = � = 0:2 and (b)� = 50; � = � = 0:4.

Fig. 5. Line detection on Fig. 4 (a) using (a) and (b) at� = 0:01.

With the unbiased estimategiven above, an immediate con-cern is: for any linear contrast , is the MVUEof ? This is addressed as follows.

Theorem 1: The set of satisfying (3.4) minimizes thesquares sum .

Proof: Refer to the Appendix.Corollary 1.1: Taking the expectation of (a2), and, in con-

junction with (a1) in the Appendix, yields

, so because is invertible. This, coupled with

Theorem 1, implies that in (3.4) is the least squares estimate of. By the Gauss–Markov Theorem [4, p. 14], for any ,

the corresponding MVUE is .Rewriting (3.4) as , where

, then and . The variance ofthe linear contrast developed in Section II is next evaluated.

Corollary 1.2: As stated in the proof of Theorem 2,are the column vectors of . In view

of , a set of -D orthonormal vectors


Fig. 6. Line detection on Fig. 4(a) using (a) the PA detector at� = 0:01 and (b) the Prewitt detector atthreshold = 34:75.

spanning the estimation space can begenerated, which is extended to form an orthonormalbasis spanning . In par-ticular, . Expressing as ,the coordinates of with respect to the basis vectors

, then . The co-ordinates of are . Itfollows that for ,

because . Rewrite as

,because is the projection of on , andso .

Now for , which implies, because is a diagonal

matrix with all elements equal to . Hence,, so that an unbiased estimate of is

(3.5)

Rewrite as. Upon replacing (3.4) and for

, we have

(3.6)

which by Theorem 1 is the least squares sum under. In brief,an unbiased estimate of is given by (3.5).

The specific form of the linear restrictions isat the discretion of the designer, so long as conditions 1) and2) are met. However, the natural side conditions:

are implementedin this research. In this case, is similar to the effect matrixwith, however, only rows: in the first row all entries equal

and all other entries equal, etc.

1) Some Special Cases:a) Independent Noise:Suppose in model (3.1)

. Then and its transformation are thesame. Equation (3.4) is simplified as

(3.7)

The corresponding least squares sum is evaluated from (3.6) as

(3.8)

b) Noise Dependency Overview:If the noise vector isassumed independent incorrectly, then is the same as in(3.7). For any contrast , the corresponding estimate is

(3.9)

Using (2.3) and Corollary 1.2, the contrast variance estimate is

(3.10)where

.This should be compared to the contrast estimate geared to

dependent noise, which, upon applying (2.4) and (3.4) is

(3.11)

The contrast variance estimate is computed by (2.4) and Corol-lary 1.2

(3.12)


Fig. 7. Line detection on Fig. 4(a) using (a) and (b) atRFO = 0:15.

where.

By the Gauss–Markov Theorem [4, p. 14], is

larger than because, recalling the Proof of Theorem

2, is associated with and the projection of

on the augmented error space, while is the projectionof on . Also, by Theorem 1is smaller than . Decomposing theratio of the contrast variance estimates as

it becomes obvious that in dependent noise environments

is less than .Applying (2.10), at the same-level of significance, the prob-

ability is that

Threshold and

Threshold . Consequently, the threshold for is lowerthan the threshold for .2

IV. SOME CLASSES OFDETECTORS

In this section, the contrast test is extended to four-way linedetection. For comparison purposes, the polynomial approxima-tion (PA) detector and the Prewitt detector are introduced.

1) Contrast Line Detector:The theory of ANOVA is usedin image processing to estimate some qualitative effects andinfer whether they are present in the image. These effects are dueto some sets of treatments which have been applied in some way

2Suppose , were the same for anyyyy then would have a higherprobability of detection as compared to . However, (3.10) and (3.12) revealthat the operating characteristics of , are nonlinear and complicated.

to the samples under consideration. The samples are representedby the pixel gray values, while the treatments are functions ofthe pixel positions within the processing mask. One can regardan image as the outcome of an experiment where we can definedirectional effects in different positions or directions. The detec-tion process entails extracting these effects and evaluating theirrelative positions within the mask. For all the cases of interestin line detection we would like to determine whether a partic-ular contrast is present or not; therefore, and this reducesto the single comparison case. Without loss of generality, onlythe four-major directional line feature detector is studied ana-lytically. Also, a mask size of is adopted.

Following model (3.1), we form the four-major directionalcontrast functions

(4.1)

The superscript denotes the orientation(in degrees) of the line feature passing through the center in theappropriate mask. The masks are shown in Fig. 2.

For diagonal contrasts, a regular Cartesian mask will break upthe off-center lines in various positions of the mask, renderingthe affected pixels irrelevant to the diagonal lines. It is prefer-able to use the inclined masks as shown in Fig. 2(b) and (d) inorder to associate them with physical features. The reason is thatfor each diagonal line a diagonal mask extracts the maximumnumber of pixels in the direction of that line. In diagonal masksthe set denotes the aligned pixels associated with a diag-onal mask, where is the row and the column oriented in thedirection of the mask, and . The set is related

to the physically scanned data via: for a


Fig. 8. Line detection on Fig. 4(a) using (a) the PA detector and (b) the Prewitt detector atRFO = 0:15.

diagonal mask, and for a diagonalmask.3 For or directional contrast the physical line is as-sociated with the row or the column effect, respectively. There-fore, for we have .

Comparing with , the coefficients for the underlyingcontrasts are

and

Upon applying Corollary 2.1 in Section III, the MVUE of thedirectional contrasts are

(4.2)

where , and .The contrast variance estimate is evaluated by means of (3.12).

We now summarize the contrast line detection algorithmshown schematically in Fig. 3. Given an image ,the task is to extract embedded line features. To begin with,both the output plane and are initialized to zero.Following the mapping procedure preceding (4.2), a patch of

3The columns inyyy , yyy are the diagonals inyyy , and so in 45,135 diagonal masks the treatment and effectsf� g, f g are interchangedwith f� g, in order to utilize model (3.1).

pixels is copied from the image plane at position , whichis initialized to , to the -directional mask. Data in thedirectional mask is called . The parameter estimator applies(3.11) in conjunction with (4.2) to evaluate the contrast estimate

, and (3.12) for the contrast variance estimate . If the

ratio is greater than the threshold (tobe defined), declare a line present and paint a-directionalline feature of a specified dimension and at strong intensityalong the -plane centered at position ; otherwise noaction is performed. The processor position indicator isincremented, and the above load-estimate-detect-paint processis repeated until reaches the end of the image plane.This produces the -directional line-feature plane . Theangle is incremented in steps of 45 and is reinitialized,prior to repeating the above procedure, until all line featuresare extracted. Finally, the outputs are superimposed, yieldingthe four-major directional line features of the imagedata , as exemplified by Figs. 5(b) and 4(a), respectively.The multidirectional line detector is known as the contrastalgorithm, . If (3.9) and (3.10) are used in lieu of (3.11)and (3.12), then line features are extracted on the assumptionthat pixel noise is statistically independent. The correspondingcontrast detector is signified by .

We next consider two methods on establishing the thresholdfor the test statistic . If the false alarm

rate (FAR), , is known, then is simply set equal to. Alternatively, suppose the FAR is unknown but

the relative frequency of occurrence (RFO) of-directionalline features is known (RFO is between zero and one). Thescheme in Fig. 3 is modified slightly in order to compute

at every . The outputs are sorted in theascending order; the element corresponding to the indexing

is assigned as . signifiesthe integer part of . With the threshold in hand, the scheme


Fig. 9. Line detection on Fig. 4(a) using (a) and (b) , 4� 4 mask and atRFO = 0:15.

Fig. 10. Line detection on Fig. 4(a) using (a) and (b) , 4� 4 mask and atthreshold = 0:1.

in Fig. 3 is then executed. The RFO approach necessitatesprocessing an image twice.

2) PA Line Detector:The intensity function of ansubarea of the image frame and aligned4 with thelatter may be expressed as a bivariate polynomial of pixel coor-dinates , with respect to the center pixel of the subarea,and corrupted by an additive zero-mean white Gaussian process:

. Here equals for odd in-

4Two 2-D areas are in alignment if their pair of axes are parallel to each other.

dexing , and otherwise; the same for ;and . In vector form this is

(4.3)

where , etc. In particular, the weightingcoefficient matrix for a area is shown in (4.3a) at the bottomof the next page.

Consider a second subarea aligned with the first subarea. Bothsubareas are of the same size but at different locations on the


Fig. 11. Line detection on Fig. 4(b) using (a) and (b) at� = 0:01.

Fig. 12. Line detection on Fig. 4(b) using (a) the PA detector at� = 0:01 and (b) the Prewitt detector atthreshold = 34:75.

image frame. The bivariate polynomial is also applicable withreplaced by . Using the least squares fit, an estimate of the

coefficient vector is . If the gray intensityapproximation of the first subarea, , is not significantly

(4.3a)


Fig. 13. Line detection on Fig. 4(b) using (a) and (b) atRFO = 0:15.

different from that of the second subarea, , then the struc-tures of these two areas are primarily the same, so that there isno vertical line feature bounded by these two areas.5 Therefore,the test statistic for vertical line features is, according to [7]

(4.4)

Comparing (4.3) to (3.1), we see an analogy betweenthe PA detector and the contrast detector. The GLS designrepresents image pixels by a finite set of treatment groupswithout imposing the condition of irradiance continuity on thelocal pixels, so that spatial features (which tend to partition alocal neighborhood into subneighborhoods bearing nonhomo-geneous irradiance) are preserved. This is in contrast to thePA approach, which assumes that local irradiance is smoothand can be approximated by an analytic function, with theconsequence that embedded features are impaired. It is theninteresting to compare the performance of these two detectorsunder various imaging conditions (Section V).

In [1], a bicubic polynomial is considered for edge detection,while a zero-order polynomial is used for detecting edge move-ment in successive frames [8]–[10]. The general framework on

5Suppose both subareas are aligned with the image plane and that the secondsubarea is on the left/right (or top/bottom) of the first subarea, a significant dif-ference in gray intensities between these two subareas would reveal that vertical(or horizontal) line features exist and located equidistantly (most likely) fromthese two subareas. By the same token, if these two aligned subareas are� -di-rectional, line features oriented in the appropriate direction can be detected. Notethat in the literature there is no work on multidirectional PA-based line detec-tion.

the PA detection is developed by Hsuet al. [7], who applied abivariate polynomial for change detection.

3) Prewitt Line Detector:A typical example of the clas-sical linear detectors is the Prewitt detector [11]. Consider aPrewitt mask with the first row equal to , thesecond row equal to zeros, and the last row equal to the negativeof the first row. The mask is convolved with a local neighbor-hood of the image. If the output is greater than a preset threshold,a horizontal line is declared; otherwise no action. The mask isthen rotated by , and the procedure is re-peated. As before, these directional outputs are superimposed.

The Prewitt detector is designed without assuming any noisemodel. For any FAR the corresponding threshold is intractable.This may be resolved by the trial-and-error method. Alterna-tively, one may apply the aforementioned RFO technique tocompute the directional thresholds. The threshold is set equalto the average of the directional thresholds. This is the approachwe use for the Prewitt detector in Section V.

V. COMPUTERSIMULATIONS AND RESULTS

The main purpose is to compare the performance of the pro-posed contrast algorithm, , its counterpart, , that as-sumes uncorrelated noise, and the PA and the Prewitt detectors.Both and apply the detection scheme illustrated inFig. 3. Unless otherwise stated, the PA-based detector uses bi-variate polynomial approximation and a pair of masks(masks’ centers are separated byrows and columns), whileall other detectors use masks. Each detector is endeavoredto extracting four-major directional line features. The FAR,,is set to 0.01. According to Fig. 3, a subarea is copied from thephysically scanned data , at position , to a directionalmask. This is applied to the testing statistic. If is decided, a

directional line feature is painted on the -plane at po-sition . Then advance by 1 column. When reaches the


Fig. 14. Line detection on Fig. 4(b) using (a) the PA detector and (b) the Prewitt detector atRFO = 0:15.

end, is incremented by 1 andis reset. And so forth. Both thePA detector and the Prewitt detector are carried out in the samemanner.

Fig. 4(a) shows the Lena picture contaminated by additivefirst-order Markovian noise of zero mean and variance ,with the row and the column correlation coefficients equal to

, , respectively. The simulation results aregiven in Figs. 5–14 and are self-explanatory. The main points ofinterest are as follows.

1) Figs. 5 and 6 (or Figs. 11 and 12) confirm that bothand are superior to the PA detector and the Prewittdetector. Also, Fig. 6(a) [or Fig. 12(a)] infers that, dueto the complexity of a real image, local regions cannotbe approximated by power-series polynomials. The Pre-witt detector, while rendering extracted line features lessnoisy than that of and , also thickens the linesenormously [see Figs. 6(b) and 12(b)]. Refer to 3) on thefeasibility of the Prewitt detector.

2) Comparing Figs. 5 and 6 with Figs. 7 and 8 (or Figs. 11and 12 with Figs. 13 and 14), it seems that replacingFAR by RFO enhances a detector. The outstanding perfor-mance of , relative to the PA and the Prewitt de-tectors, exhibited in the FAR case previously, carries overfavorably to the RFO case. Moreover, Fig. 7 (or Fig. 13)indicates that the outputs by , are almost thesame. This is not surprising. The RFO approach deter-mines the threshold by ranking the values computed bythe testing statistic. When the same image is reprocessedusing the previously sought threshold, naturally the samenumber of line features results. As the structures ofand are akin to each other, their performances arenearly indistinguishable. The fact that these two detectorsare entirely different from the Prewitt detector and the PAdetector supports the observation line features extracted

by the contrast algorithms are different from the last twodetectors (same RFO level).

3) Inspired by the performance of the RFO approach, thecontrast algorithms are rerun for masks, with theresults shown in Fig. 9. Comparing Fig. 9 to Fig. 7, it isobvious that the contrast algorithms utilizing masksoutperform the same algorithms utilizing masks.This is expected. A GLS design models a maskby 13 independent treatment effects ( # of treatmentgroups # of treatment effects/group # of side-condi-tions), so that the relative complexity efficiency (RCE),which is defined as the ratio of the number of indepen-dent treatment effects to the number of pixels in the mask,is equal to 0.81, while a mask incurs an RCE of

. Correspondingly, the RCE for ,, and masks are equal to ,

, and , respectively, and de-ceases with respect to mask size. In view of the high RCEpossessed by a mask as compared to a mask, asmaller mask is more suitable for real-world images. Theidea is similar to a digital lowpass filter, which ideallysuppresses noise and preserves the spectral content of thedesired signal (no blurring). By the same token, a con-trast algorithm based on masks is better equipped inthe manifestation of changes due to, for example, line fea-tures, in a local neighborhood. We remark that because theRFO (of line features) varies from image to image and isnot knowna priori, detectors based on the RFO approach,including the Prewitt detector, are not very practical.

4) Comparing Fig. 5(b) with Fig. 5(a) [or Fig. 11(b) withFig. 11(a)], it is obvious that in dependent noise envi-ronments is inferior to . The degradation is ex-pounded in Fig. 10 involving masks [high RCE, see3)]. Regrettably, the double-processing characteristic in-


herent in RFO undermines a direct comparison of ,.

5) On comparing Figs. 5–8 with Figs. 11–14, each detector isseemingly susceptible to noise variance. This is consistentwith the detection and estimation theory. In all cases, it isseen that the detectors are in the following order (superiorfirst): , , Prewitt, PA.

The commendable performance of the contrast algorithmssuggests the importance of image representation. In light of thesuperiority of relative to , in the absence of knowl-edge of noise independence one should refrain from assumingan independent detector as the proper one. Using a 500 MHzPentium III Computer, the processing time for out-puts like Figs. 5(b) and 6(a) and 6(b) consume about 57, 25, and14 s, respectively.

VI. CONCLUSIONS

In this paper, a class of line detectors based on the linearcontrast theory is derived. By incorporating the-statistic andthe shape test the new algorithm detects and locates line fea-tures simultaneously without enduring a two-step process, asis commonly the case for ANOVA-based detectors, thereby in-creasing the detection power substantially at the same FAR. Thethreshold of the proposed algorithm is a linear function of the es-timated variance and varies over the neighborhoods [see (2.10)].This alleviates the streaking problem encountered by single- ormultithreshold detectors. Anad-hocmathematical analysis indi-cates that the new algorithm accommodating noise dependenceperforms better than its counterpart ignoring dependent noise.Computer simulations demonstrate that the contrast algorithmsoutperform the PA and the Prewitt detectors.

The inference about one single contrast implied in the newalgorithm and the use of the GLS design as the ANOVA modelpermit the detection and location of lines in any of the maindirections. With slight modification, it may be adapted to thedetection of other types of image features. The new algorithmis robust as related to the correlation coefficient variation and,because of its very general assumptions, it is superior as com-pared to other detectors which are subject to more restrictionsin a corruptive dependent noise environment. Also, the new al-gorithm is fast, efficient, and amenable to real-time processing.

APPENDIX

PROOF OFTHEOREM 1

The proof consists of two parts.1) Let be the column vectors of, . Rewriting

as , the sum of squares error is

Since , are linearly dependent, so thatthere is no unique solution minimizing . We next sub-ject to linear restrictions ,which in matrix form are , ,

. Impose the conditions 1) and 2) stated rightafter (3.3) and, without loss of generality, set . Clearly,

, because by definition every component underneath theouter summand in the second term is zero.

Introduce the variables: for otherwise;and for , otherwise. Then

, ,, , and, because

is associated with the effect matrix , is also associatedwith . Therefore,

is generated from and augmented by the set oflinear restric-tions. By conditions 1) and 2) , the set of vectors

are linearly independent and form the-dimen-sional vector space .

2) Corresponding to the linear restrictions, the set of as-sumptions, , is augmented from (3.1) as (after pre-whitening)

Under we have

(a1)

Rewrite the expectation of the augmented data as. Suppose its estimate

is . Select sothat is minimized, sustainable if andonly if is the projection of on [12]. That is, isthe orthocomplement of in , and is the vector spaceof spanned by .6 Therefore,

, because belongs to

the estimation space , belongs to the error spaceand . Taking into account all the

basis vectors in , then and isequivalent to

(a2)

Since , is invertible. There-fore, the least squares estimate ofis and co-incides with (3.4). From the equality of and ,the theorem follows. Q.E.D.

6The dimensionality ofz isn because its lastt elements are 0. Also,the firstp vectors inV can be regarded as the same setf�� ; � � � ; �� g span-ningV , while the remaining basis vectorsf�� ; � � � ; �� g are generated byGram–Schmidt procedure [13].


ACKNOWLEDGMENT

The authors would like to sincerely thank Associate EditorDr. A. Laine and the referees for their suggestions which sub-stantially improved the content of the paper.

REFERENCES

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[7] Y. Z. Hsu, H. H. Nagel, and G. Rekers, “New likelihood test methodsfor change detection in image sequences,”Comput. Vis., Graph., ImageProcess., vol. 26, pp. 73–106, 1984.

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Gerard J. Genello (SM’91) received the B.S.E.E.degree from the University of Detroit, Detroit, MI, in1973, the M.S.E.E. degree from Syracuse University,Syracuse, NY, in 1979, and the MBA degree fromRensselaer Polytechnic Institute, Troy, NY, in 1989.

In 1973, he began his career at Rome AirDevelopment Center (now Air Force ResearchLaboratory-AFRL) working as a electromagneticcompatibility (EMC) research engineer. His majoractivities included high power microwaves, EMCand weapon system vulnerability test and evaluation.

In 1988, he joined the Surveillance Division as Chief of the Signal ProcessingBranch. In 1997, he became Manager of the Radar Signal Processing Branch,Sensors Directorate, AFRL. His current responsibilities include managing anddirecting the basic research, exploratory, and advanced development programsfor radar signal processing applications. Technologies of interest includespace-time adaptive processing, knowledge-based techniques, spaced-basedradar, subsurface target detection, SAR image detection, and data compressionapplied to airborne and space-based sensor platforms.

Mr. Genello is Chairman of the local Mohawk Valley Aerospace and Elec-tronics Systems Chapter of the IEEE.

Julian F. Y. Cheung (S’84–M’91) was bornin Guangdong, China. He received the B.E.E.,M.S.E.E., and M.S. degrees in applied statistics,and the Ph.D. degree in electrical engineering fromPolytechnic University, Brooklyn, NY, in 1983,1984, 1989, and 1990, respectively.

From 1984 to 1986, he was an Instructor atTechnical Career Institute, New York, NY. FromSeptember 1986 to September 1987, he was anInstructor with the Department of Electrical Tech-nology, New York City Technical College-CUNY,

Brooklyn. He was a Teaching Fellow with the Department of ElectricalEngineering, Polytechnic University, from 1986 to 1989. Since September1990, he has been teaching in the Department of Electrical Engineering, NewYork Institute of Technology, where he is currently an Associate Professor. Heworked as a Consultant at FingerMatrix Corporation in the summer of 1992.He was a Research Associate for the Air Force office of Scientific ResearchSummer Faculty Research Program from 1994 to 1996 (summers). He wason leave-of-absence from October 1996 to September 1997, and again fromJanuary to December 1998, at the U.S. Air Force Research Laboratory in theSensors Directorate, Radar Signal Processing Branch under the Intergovern-mental Personnel Act. His research interests include wireless communication,communication network, adaptive radar system analysis, image processing,and signal processing.

Steven H. Billis received the B.E.E. degree from theCity College of New York in 1966, and the M.S. andPh.D. degrees in electrical engineering from Poly-technic University of New York, Brooklyn, in 1968and 1972, respectively.

He was a National Defense Education Act(NDEA) Graduate Fellow at Polytechnic Universityfrom September 1966 to June 1967, and a ResearchFellow from September 1967 to June 1972. Since1973, he has been a Faculty Member at New YorkInstitute of Technology and currently, a Professor

and Chairman of the Department of Electrical Engineering and the Departmentof Computer Science. He was a Consultant to Rhode Island Institute ofTechnology in 1991 and the United States Army in 1992. He presently servesas a Consultant to the Equitable Life Insurance Company and is a ProjectManager for the Center of Telecommunications and Networks Research at NewYork Institute of Technology, Old Westbury. His major interests are quantumelectronics, image processing, and multimedia network.

Dr. Billis is a member of Tau Beta Pi, Eta Kappa Nu, and Sigma Xi.

Yoshikazu Saito received the B.S. degree fromKeio University, Japan, in 1970, the M.S. degreefrom New York University, New York, NY, in 1973,and the Ph.D. degree from Polytechnic Universityof New York, Brooklyn, in 1977, all in electricalengineering.

From 1975 to 1977, he was a Lecturer at Poly-technic University of New York. He was also aLecturer at Manhattan College, Bronx, NY, from1976 to 1977. He received a teaching fellowshipfrom Polytechnic University from 1974 to 1977.

Since September 1977, he has been a Faculty Member at New York Institute ofTechnology where he is currently an Associate Professor in the Departmentsof Electrical Engineering and Computer Science. He worked as a Consultantfor HZI Research Center from 1983 to 1991, OKI America from 1989 to 1996,PASCO USA from 1993 to 1995, and Japanese Educational Institute from1988 to the present. His main interests are in the areas of VLSI design, imageprocessing, signal processing, telecommunication network, and microcon-troller-based systems.

graeco-latin squares design for line detection in the presence of correlated noise

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