Multiscale Edge Detection and Fiber Enhancement Using Differences ofOriented Means

Meirav Galun Ronen Basri Achi BrandtThe Weizmann Institute of Science

Dept. of Computer Science and Applied MathematicsRehovot 76100, Israel

Abstract

We present an algorithm for edge detection suitable forboth natural as well as noisy images. Our method is basedon efficient multiscale utilization of elongated filters mea-suring the difference of oriented means of various lengthsand orientations, along with a theoretical estimation of theeffect of noise on the response of such filters. We use ascale adaptive threshold along with a recursive decisionprocess to reveal the significant edges of all lengths andorientations and to localize them accurately even in low-contrast and very noisy images. We further use this algo-rithm for fiber detection and enhancement by utilizing sto-chastic completion-like process from both sides of a fiber.Our algorithm relies on an efficient multiscale algorithmfor computing all “significantly different” oriented meansin an image inO(N log ρ), whereN is the number of pix-els, andρ is the length of the longest structure of interest.Experimental results on both natural and noisy images arepresented.

1. Introduction

One of the most intensively studied problems in com-puter vision concerns with how to detect edges in images.Edges are important since they mark the locations of discon-tinuities in either depth, surface orientation, or reflectance.Edges and fibers constitute features that can support seg-mentation, denoising, recognition, matching and motionanalysis tasks.

Accurate detection of edges can be challenging, particu-larly when low contrast edges appear in the midst of noise.

Research was supported in part by the European Commission Project IST-2002-506766 Aim Shape, by the Institute for Futuristic Security Researchand by the APNRR grant. Research was conducted at the Moross Labo-ratory for Vision and Motor Control at the Weizmann Institute of Science.We thank Ziv Reich and Eyal Shimoni for the electron microscopy im-ages. We thank Michael Fainzilber and Ida Rishal for the light microscopyimages.

Figure 1.Electron microscope images: demonstrating the chal-lenge of detecting edges (at different scales) embedded in noise.

Figure 2.Two adjacent noisy intensity profiles (right) parallel to along edge (left) in an image acquired by an electron microscope.

The images in Figure1, acquired by an electron micro-scope, exemplify this kind of challenge. Despite the noise(see profile plots in Figure2), such low contrast edges areevident to the human eye due to their consistent appear-ance over lengthy curves. Accurate handling of low contrastedges is useful also in natural images, where boundaries be-tween objects may be weak due to similar reflectance prop-erties on both sides of an edge, shading effects, etc.

This paper presents an algorithm for edge detection andfiber enhancement designed to work on both natural as wellas very noisy images. Our method is based on efficient uti-lization of filters measuring difference of oriented meansof various lengths and orientations, along with a theoreticalestimation of the effect of noise on the response of such fil-ters. In particular, we derive a scale adaptive threshold anduse it to distinguish between significant real responses andresponses due to noise at any length. We complement thiswith a recursive decision process to distinguish between re-

1

sponses due to consistent, long edges and responses due toseveral scattered sub-edges. These enable us to consider-ably reduce the effect of noise without relying on a priorstep of isotropic smoothing, and consequently reveal andaccurately localize very noisy, low-contrast edges. More-over, short gaps can be overcome, and despite the use oflong filters the ”correct” ends of an edge can be delineateddue to the scale multiplicity. We further present an applica-tion of this algorithm to fiber detection and enhancement byemploying a stochastic completion-like process from bothsides of a fiber. This process is quite robust to varying fiberwidth and fiber branching.

Our algorithm relies on a fast, bottom-up procedure, fol-lowing [1], to construct all “significantly different” orientedmeans at all locations, lengths, and orientations. By this,our method utilizes the same number of filters at everylength, so that as we increase the length of the filter we re-duce its spatial resolution and proportionally increase its di-rectional resolution. This, along with our recursive decisionprocess, yield efficient runtime complexity ofO(N log ρ),whereN is the number of pixels andρ is the length of thelongest structure of interest, typicallyρ ≤ O(

√N). Exper-

imental results on various images are presented.

2. Previous work

Common algorithms for edge detection (e.g., [3]) over-come noise by applying a preprocessing step of smoothing,typically using a gaussian kernel of a user specified width.Scale-space representations extend this approach, allowingfor a simultaneous delineation of edges from all scales bycombining spatially varying gaussian smoothing with auto-matic scale selection [10, 18]. Such isotropic smoothing,however, often reduces the contrast of weak edges, mayblend adjacent edges, and may result in poor localizationof edges.

To avoid the pitfalls of isotropic smoothing, anisotropicdiffusion schemes [15, 19, 9] were proposed as a means foredge-preserving smoothing and image enhancement. Thesemethods utilize a diffusion tensor designed to avoid smooth-ing in the direction of the intensity gradients, while allow-ing smoothing in coherence directions. These approachesimprove edge localization considerably, and edges at differ-ent scales are revealed at different iterations of the diffusionprocess. These edges remain visible in their original loca-tion for many iterations before they finally fade out. A scaleselection mechanism however is required to extract edgesof different scales. As the reliance on local gradients intraditional diffusion processes may lead to accumulation ofnoise, recent methods [19, 9] modify the diffusion tensorthrough isotropic spatial averaging or resetting of its eigen-values. Such spatial smoothing and eigenvalue modifica-tions are, however, adapted to a single scale. Moreover, byusing an averaged diffusion tensor, these methods accumu-

late squaredlocal intensity differences, and this may leadto smoothing across noisy, low contrast edges. The relianceon local gradients is a common problem also in both singleand multiscale variational methods [14, 2].

Another stream of work utilizes filters of various lengths,widths, and orientation, yieldingcurvelet and contourletimage decomposition [17, 4, 7]. The focus of these meth-ods, however, is mainly on achieving sparse representationsof images and not on edge extraction or fiber enhancement.

Finally, recent methods for edge detection in natural im-ages [16, 11] compare histograms of intensity, color (andalso texture in [11]) in two half disks on either sides of anedge, while in [16] the size of the disc is related to the lengthof the edges. While this approach avoids smoothing acrossa measured edge, the use of large discs may lead to smooth-ing across nearby edges. Furthermore, the histograms arecalculated at all locations and orientations leading to inef-ficient schemes. Our decision process can be related to thelogical-linear application proposed in [6].

3. Multiscale edge detection algorithm

Our goal is tosimultaneouslyextract edges of all lengths,in both natural and noisy images. LetI(x, y) denote a con-tinuous function representing image intensities given in atwo-dimensional domain. We denote byoriented meansthefamily of averagesof I(x, y) along rectangular patches ofa given center locationx = (x, y), lengthL, width w, andorientationθ. Typically in our implementationw is set toa small constant, yieldingelongatedoriented means. Thefamily of oriented means can be obtained via integrals ofthe form

F (x, L, w, θ) =1

wL

∫ w/2

−w/2

∫ L/2

−L/2

(1)

I(x + γ cos θ − δ sin θ, y + γ sin θ + δ cos θ)dγdδ.

Our approach utilizesresponses of differences of orientedmeans (briefly denotedresponses). This family of re-sponses is defined as

D(x, L, w, s, θ) =12F (x +

s

2(− sin θ, cos θ), L,

w

2, θ)

−12F (x− s

2(− sin θ, cos θ), L,

w

2, θ), (2)

with s ≥ w/2 to avoid overlap.D defines an average differ-ence of neighboring oriented means of thesame orientation.

Given a collection of responses, our first task (Sec-tion 3.1) is to determine which response is significant. Thisis achieved by applying a scale adaptive threshold deter-mined by measuring the magnitude of the noise in the imageand by considering the length and width of the measured re-sponse. Next (Sec.3.2) we examine the set of significant re-sponses at each length. While some of these responses may

indeed indicate the presence of an edge of the appropriatelength, others may be due to scattered collections of shorteredges. We therefore apply a recursive decision process todistinguish between these two cases . Finally, our algorithmrelies on a framework and a data structure for efficient com-putation of oriented means [1] (Sec.3.3).

3.1. Multiscale noise estimation

Our first task is to identify differences of oriented meansthat elicit significant responses. In this section we derivea scale adaptive threshold that takes into account the prop-erties of the noise as well as the length and width of ourfilter. Below we assume that the noise does not vary signifi-cantly across the image and that the pixel noise is normallydistributed asN(0, σ2).

Suppose we apply differences of oriented meansD(x, L, w, s, θ) to an image composed of white noise. De-note byDL,w the discrete form ofD(x, L, w, s, θ), soDL,w

is the average difference of two averages ofwL/2 normaldistributions. Consequently,DL,w itself distributes nor-mally, DL,w ∼ N(0, σ2

L), whereσ2L = σ2/(wL).

In order to differentiate between responses due to real-edges and those that are due to noise, we wish to estimatethe chance that given a real numbert À σL, a valued drawnfrom the distributionDL,w ∼ N(0, σ2

L) satisfies|d| < t,

i.e., p(|d| < t) = 1√2πσL

∫ t

−texp

(− x2

2σ2L

)dx. Therefore,

we first estimate the following integral fort À σL

∫ ∞

t

exp(− x2

2σ2L

)dx =∫ ∞

0

exp(−−(t + x)2

2σ2L

)dx

= exp(−t2

2σ2L

) ∫ ∞

0

exp(−2tx + x2

2σ2L

)dx (3)

≈ exp(−t2

2σ2L

) ∫ ∞

0

exp(− tx

σ2L

)dx =

σ2L

texp

(−t2

2σ2L

).

With this estimate, we obtain

p(|d| < t) ≈ 1−√

2π

σL

texp(− t2

2σ2L

) .= 1− ε. (4)

Suppose we produceO(N) responses of lengthL, whereN is the number of pixels in the image. To differentiate be-tween true edge responses and noise responses, we wouldlike to determine an adaptive thresholdt(w,L, N) such thatwith high probability the values ofall noise responsesd ob-tained for a noise image will fall in the range[−t, t]. Ac-cording to (4), this means(1− ε)N = O(1), which implies1 − εN = O(1), in other words to assure high-probabilitywe demand1 − εN ≥ 1/2 or ε <≈ 1

2N . Plugging the def-inition of ε (4) into the latter relation and taking the naturallogarithm leads to the following approximate relation

t2(w, L,N)2σ2

L

= ln N + ln(σL

t) ≈ ln N, (5)

Figure 3.Scale-adaptive noise estimation: length-L vs. the ratiobetween the scale-adaptive threshold andσ.

where the approximation is due to the relationln(σL/t) ¿t2/(2σ2

L), under the assumptionσL ¿ t.This approximation yields the relation

t(L,w, N) = σ

√2 ln N

wL, (6)

which means that a response which exceedst(L,w,N) canpotentially indicate the existence of a real edge. This is thescale-adaptive thresholdthat we have used throughout ourexperiments. Clearly, it can be very useful only if we areable estimateσ, the standard deviation of the pixel noise.We suggest to estimateσ in the following simple way: foreach pixel we calculate the minimal standard deviation ob-tained from the collection of3× 3 windows containing thepixel. We then construct a histogram summarizing the em-pirical distribution of this minimal standard deviation ob-tained for all the pixels.σ is determined as the90th per-centile value of this histogram.

We have further confirmed this adaptive threshold empir-ically. We generated synthetic white noise images (200 ×200 pixels) with different values of standard deviationσ,and measured for a widthw = 4 the maximal response overall length-L responses, forL = 2, 4, 8, 16, 32, 64. Figure3shows that the ratio obtained between the maximal responseandσ is comparable to the theoretical estimate of the ratio,

i.e.,√

2 ln NwL .

3.2. Recursive decision process

A responseD(x, L, w, s, θ) of length L (and fixedw)is considered significant if it exceeds the length adaptivethreshold, i.e.,D ≥ TL

.= t(L,w, N). However, while thethreshold is designed to eliminate responses due to noise,long responses may still exceed the threshold due to scat-tered, short edges of high contrast. In this section we de-scribe a recursive procedure whose objective is to distin-guish between responses due to long edges from responsesdue to scattered sub-edges.

A natural way to determine the composition of a re-sponse is to consider sub-portions of the edge and testwhether they too give rise to a significant response. For a re-sponse of lengthL consider an arbitrary sub-edge of lengthcL (0 < c < 1). Based on our analysis, such a response is

considered significant if it exceeds the thresholdTcL. How-ever, sinceTcL > TL, requiring every portion of an edge toexceed its respective threshold would amount to requiringthe complete length-L edge to exceed the higher thresholdTcL, making the scale-adaptive threshold useless. A morerelaxed approach is to require each sub-edge of lengthcL topass the threshold for the complete edgeTL. This approachtoo is unsatisfactory since, due to noise, short portions of along edge may fall below the threshold.

We address this problem by requiring that most of thesub-edges will pass some lower threshold and that the re-maining gaps will be short relative to the length of the edge.We therefore define two parameters,α andβ (0 ≤ α, β ≤1). We setαTL to be a low threshold for length-L responses.Namely, responses of lengthL that exceedαTL and do notexceedTL are considerednearly significant, i.e., they arenot significant enough to form an edge on their own, butthey can form sub-portions of longer edges. In addition, wesetβL to be a threshold on the total gap size for length-Ledges, i.e., a length-L edge cannot be considered significantif it contains sub-portions of total length> βL that do notexceed their respective low threshold. With such a long gapthe response cannot be marked as an edge even if it exceedsits corresponding thresholdTL.

We implement these principles in a recursive procedureas follows. We setw and s to constants and begin withthe set of responsesD(x, L, w, s, θ) obtained for filters ofsome predetermined short length, e.g.,L = 1. We classifyeach response to one of the following three classes: (1)sig-nificant (D > TL) (2) nearly-significant(αTL < D ≤ TL)and (3)non-significant(D ≤ αTL). For the first two classesthe total gap size is set to “0”, while for the latter the gapsize is set to “L”.

We proceed by sequentially considering lengths of pow-ers of 2. The classification of every length-L responseat locationx, D(x, L, w, s, θ), is affected by the classi-fication of its associated length-L/2 responses,D(x ±(L/4)(cos θ, sin θ), L/2, w, s, θ) as described below. In ad-dition, the total gap size is set to be the sum of gap sizesassociated with these responses.

• A length-L responseD is classifiedsignificantif it ex-ceeds the respective threshold,D > TL, and its to-tal gap size is belowβL. In this case the response ismarked as apotential edgeand its gap size is reset tozero.

• A length-L responseD is classifiednearly-significantif αTL < D ≤ TL and its total gap size is belowβL.In this case, its gap size is not updated.

• A length-L responseD is classifiednon-significantifD ≤ αTL or if its total gap size exceedsβL. In thiscase its total gap size is reset toL.

Marked potential edges are further tested for statistical sig-nificance. We consider the intensity profiles along the twosides of a potential edge and denote byσlocal the maximumof the two empirical standard deviations on either side. We

then remove edges for whichD < c√

2 ln NwL σlocal, typi-

cally c = 1. This condition generalizes (6) by replacingthe global noiseσ with a local estimate of the noise,σlocal,yielding a scale-dependent local threshold. This test allowsus to be less strict with the selection ofα andβ.

Finally, we apply a process ofangular non-maximal sup-pressionfollowed by spatial non-maximal suppressionateach lengthL. The non-maximal suppression is a necessarystep to ensure well-localized edges, expressed by a singleresponse. Moreover, we perform an inter-scale suppression,to ensure that each edge is identified with its maximal per-ceptual length.

This inter- and intra-scale decision process along withthe scale-adaptive threshold allow us to overcome noisewithout prior smoothing, and therefore to reveal and accu-rately localize very noisy, low-contrast edges.

3.3. Hierarchical construction of oriented means

We construct our differences of oriented means byadding and subtracting width-1 oriented means of the ap-propriate length. The family of width-1 oriented means,F (x, L, w, θ) is defined over an infinite set of locations,lengths, and orientations, but under minimal smoothness as-sumptions ofI(x, y) it is possible to obtain any orientedmean in this family from only a finite collection of orientedmeans. Oriented means at interim locations, lengths, andorientations can then be computed by interpolation with anaccuracy determined by the spatial and angular resolutions.

Following [1], the minimal spatial and angular resolu-tions depend on the lengthL as follows.

• The minimal spatial resolution in the direction of in-tegration is inversely proportional to the integrationlength. In particular, when doubling the integrationlength, the number of evaluation points may be halved.

• The spatial resolutionperpendicularto the direction ofintegration is independent of the integration length.

• The minimal angular resolution is proportional to theintegration length. In particular, if the integrationlength is doubled, the number of angles computed persite must also be doubled.

A direct consequence of these observations is that thetotalnumber of integrals at any lengthL is independent ofL.

Figure4(left) depicts the commondirect approachto ap-proximating discrete oriented means of orientation|θ| ≤ π

4 .Using the trapezoid rule, the oriented mean for, e.g., theline segment depicted in Figure4(left), is given by the

Figure 4.Left panel: Direct calculation of oriented mean (w =1, L = 4): an interpolation from the given data points to the redpoints is followed by the trapezoid rule. Right panel: Base levelinitialization: stencil of four length-1 integrals for each pixel (left)cover the whole grid (right).

sum over the three interior points (obtained by interpolationfrom nearby data points) plus the average of the endpoints,divided by four. Similar calculation applies for orientedmeans withπ

4 ≤ θ ≤ 3π4 . However, this direct approach

is quite expensive to approximate many oriented means.In the remainder of this section we follow [1] and de-

scribe a fast,hierarchical recursive calculationof “all sig-nificantly different” oriented means in a discrete image. Inthe context of a discrete image, we define thelength of anoriented meanon a uniform grid as the length of its maxi-mal projection onto the x- and y-axis, where the length unitsare pixels. The hierarchical construction is as follows. Atthe base level, for each pixel four length-1 oriented meansare calculated (see Figure4(right)). The total number oforiented means at this level is therefore4N . Recursively,given4N oriented means of lengthL we proceed to com-puting new4N oriented means of length2L. Following theprinciples outlined above, the angular resolution should bedoubled. Consequently, the new oriented means can be di-vided into two equally sized sets. Half of the new orientedmeans follow the same directions of those of the previouslevel, while the other half of the oriented means follow in-termediate directions. The first set of oriented means canbe computed simply by taking the average of two, length-Loriented means with one coinciding endpoint (see Figure5,left panel). The second set of oriented means can be ob-tained by interpolation of four length-L oriented means ofnearby directions. These four oriented means form a tightparallelogram around the desired length-2L integral. This isillustrated in Figure5 (left panel), where the average of thefour length-1 oriented means is used to construct a length-2oriented mean. This can be viewed as first linearly inter-polating the two nearest directions to approximate the newdirection at length-1, then creating a length-2 oriented meanby averaging two adjacent interpolated oriented means.

It should be emphasized that the numerical error intro-duced by this fast calculation, relative to the slow, directapproach, is smaller than the error induced by discretiz-ing I(x, y). The algorithm is very efficient, it calculates“all significantly different” oriented means inO(N log ρ),whereN is the number of pixels in the image andρ is thelength of the longest oriented mean, typicallyρ ≤ O(

√N).

Figure 5.Left panel: Building integrals of length-2 from integralsof length-1, for existing directions (left), by taking the average oflength-1 adjacent integrals (dashed lines) in that direction, and fora new direction (right), by averaging four nearest integrals (dashedlines). Right panel: The red lines denote length-4 vertically ori-ented means, which are calculated at each2nd row (pink rows).The blue lines denote length-4 horizontally oriented means, whichare calculated at each2nd column (blueish columns).

3.4. Implementation

For our implementation we maintain ahierarchical datastructureof the angular and spatial resolutions as follows.At the base level, for each pixel length-1 oriented meansat four directions are calculated. The angular resolution oflength-2L oriented means is twice the angular resolution oflength-L oriented means. The spatial resolution is halvedas the length is doubled as follows. Length-L verticallyori-ented means (π4 ≤ θ ≤ 3π

4 ) are calculated atevery(L/2)th-row, at each pixel in this row. Length-L horizontallyori-ented means (|θ ≤ π

4 |) are calculated atevery (L/2)th-column, at each pixel in this column. In this manner, eachlength-L oriented mean has anoverlapof lengthL/2 withanother length-L oriented mean of the same orientation, inorder to improve accuracy. As a result, at each scale (length)the number of oriented means is8N , except for the short-est level (length-1) for which4N oriented means are calcu-lated. Note that at each length only4N of the8N orientedmeans are used to construct the subsequent level. The gridsetting forL = 4 is illustrated in Figure5(right).

Our recursive decision process is evoked during the con-struction of oriented means after the construction of everylength, maintaining the low complexity of the algorithm.For this decision process we associate with each length-Lresponse the two (if the longer response follows an orien-tation of the previous level) or four (if the longer responseis of a “new” orientation) length-L/2 responses from whichit was generated. Finally, the local standard deviation testcan be applied by accessing the raw pixels, assuming onlya sparse set of responses are considered significant. Alter-natively,σlocal can be recursively accumulated by comput-ing in addition oriented means for the squared intensities,I2(x, y), doubling the overall complexity.

4. Experiments

Our algorithm employs differences of oriented meansD(x, L, w, s, θ) at lengthsL = 1, 2, 4, 8, 16, 32, 64, withs = 2, w = 2 ands = 5, w = 4 for the edge detection in

natural images and noisy images, respectively.

4.1. Noisy images

We have implemented our edge detection algorithm andtested it on very noisy images acquired by electron micro-scope. For this implementation we useα =

√0.5 and

β = 3/8. Figure 6 contains images of a plant undercertain photosynthetic conditions. Depending on lighting,the membranes, which are sub-cell organs (organelles), aresometimes very close to each other. As demonstrated in thefigure, our algorithm performs very well in detecting theedges of the membranes, independently of the scale (length)and the membranes’ density (Figure6, middle row). Cannyedge detector (with optimized parameters) cannot revealthese dense edges, due to the pre-smoothing (Figure6, bot-tom row). We compare our algorithm with our own imple-mentation (following [8]) of the Beltrami flow anisotropic-diffusion. The enhanced images and their respective inversegradients are shown in Figure7. It can be seen that differentstructures are enhanced in different iterations, yet the denseorganelles structure is not revealed at any iteration, and theoverall result is severely affected by the amount of noisepresent in this image. Figure8 contains images of bacte-ria environment. Our algorithm extracts edges at all scales,both elongated and circular.

4.2. Natural images

We have further applied our algorithm to the 100graylevel test images of the Berkeley segmentation dataset andhuman annotation benchmark [12]. For this implementa-tion we useα =

√0.8 andβ = 0. Although this dataset

is not ideal for evaluating our method as (1) human anno-tation is affected to a large extent by semantic considera-tions, (2) subjects only marked subsets of the boundaries,and (3) our method does not deal with texture; it never-theless demonstrates that our method achieves competitiveresults on natural images. To determine boundary classifi-cation we associate a response with every edge point in amanner similar to that used in [11]. Specifically, we com-pute two intensity histograms for the two half-discs (usingdiscs of radius 7 pixels) centered in each edge center andoriented according to the edge direction. We then use aχ2 histogram difference measure to compare the two his-

togramsχ2(g, h) = 12

∑ (gi−hi)2

gi+hi. We finally normalize

these values (for each image) between 0 and 1. Figure9shows some example images, demonstrating the exact lo-calization of edges that can be achieved with our method.

We further used the standard F-measure test, which pro-vides a trade-off between precision(P ) and recall(R) de-fined asF = PR/(γR + (1 − γ)P ), with γ set to0.5.These are computed using a distance tolerance of2 pixels toallow for small localization errors in both the machine and

Figure 6.Edge detection. Top row: original images. Middle row:edges obtained by algorithm overlaid on the original images. Bot-tom row: edges obtained by Canny edge detector.

Figure 7.Image enhancement and inverse gradient by Beltramiflow. Top row: image enhancement after 8, 11 and 20 iterations,respectively (from left to right). Bottom row: respective inversegradients.

human boundary maps. Table1 shows the F-measure val-ues achieved by our algorithm compared to other edge de-tection algorithms. Note that somewhat better performance

Figure 8.Edge detection. Left column: original images. Rightcolumn: edges obtained by our algorithm overlaid on the originalimages.

is achieved by methods that use color and texture featuressuch as [11]. Code and results of other algorithms are ob-tained from Berkeley athttp://www.eecs.berkeley.edu/Research/Projects/

CS/vision/grouping/segbench/ .

Algorithm F-measure

our multiscale edge detection algorithm 0.61

Brightness Gradient 0.60

Multiscale Gradient Magnitude 0.58

Second Moment Matrix 0.57

Gradient Magnitude 0.56

Random 0.41Table 1.F-measure comparison over edge detection algorithms.

Our non-optimized (combined Matlab and C++) imple-mentation requires 3, 11, and 18 seconds for171 × 246,481×321, and512×512 images respectively on a Pentium4, 2.13GHz PC.

5. Fiber detection and enhancement

Our method is further extended to detect and enhanceelongated fibers. Fibers appear in many types of images,particularly in biomedical imagery and airborne data. De-tection of fibers can be difficult due to both imaging con-ditions (noise, illumination gradients) and shape variations(changes in widths, branchings). We propose a method thatuses edges, along with their sign (transition from dark tolight or vice versa) to highlight the fibers. Specifically, webegin by detecting all edges in the image. We next markthose edges as either “red” or “blue”, by classifying theirgradient orientation relative to a pre-specified canonical ori-entation. We then construct two diffusion maps, one for thered and the other for the blue edge maps, by convolving thebinary edge map with a gaussian filter. Finally, we multi-ply the two diffusion maps to obtain the enhanced fibers.

This process is analogous to stochastic completion of con-tours [13] with each edge point emitting particles to its sur-roundings. The product of the diffusion maps reflects theprobability of beginning in one edge and ending in a neigh-boring edge of opposite gradient sign. This process is quiterobust to varying fiber width and fiber branching, and al-though many red and blue responses may appear in the im-age, only those that can be paired are enhanced.

We applied our fiber enhancement process to fluorescentimages of nerve cells. Comparison of these cells under dif-ferent conditions requires quantitative measurements of cel-lular morphology, and a commonly used measure is the totalaxonal length. Typically, a single experiment produces hun-dreds of images, many of which may suffer from low signalto noise ratio. Therefore, there is an obvious need for fastand accurate automatic processing of such images. Usingour multiscale fiber enhancement method we identify thebranching axonal structures and measure their total length.Figure10shows examples of images of nerve cells and theirenhanced axonal structures. The total axonal length wasfurther computed automatically from the enhanced imagesand compared to a manual length estimation applied to theoriginal images. The two measures match with slight devia-tion, see Table2. For this implementation we useα =

√0.5

andβ = 0.0.

Manual estimation 11940 4467 3407 7347

Automatic estimation 11180 4106 3620 9092

Relative error (percents) -6.37 -8.08 6.25 23.75Table 2.Measuring total axonal length: manual length estimationvs. automatic length estimation (in pixel units), for the four left-most images in Figure10.

6. Conclusion

We have presented an algorithm for edge detection suit-able for both natural as well as noisy images. Our methodutilizes efficient multiscale hierarchy of responses measur-ing the difference of oriented means of various lengths andorientations. We use a scale-adaptive threshold along with arecursive decision process to reveal the significant edges atall scales and orientations and to localize them accurately.We have further presented an application to fiber detec-tion and enhancement by utilizing stochastic completion-like process from both sides of a fiber. Our current methodidentifies curved edges as concatenation of short straight re-sponses. We plan to construct a direct framework to detectcurved edges. We further plan to extend our algorithm tohandle edges of varying widths and to estimate and incor-porate local noise.

Figure 9.Edge detection results. Top: original images. Middle: our results. Bottom: human-marked boundaries.

Figure 10.Neuron enhancement results. Top: original images acquired by light microscope. Bottom: enhanced neurons.

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