Another rather obvious approach is to apply a desired edge detection method arately to each color component and construct a cumulative edge map.. The Laplacian of Gaussian approach can a
Trang 119.5 Approaches for Color and Multispectral Images 519
FIGURE 19.10
Canny edge detector of Eq (19.22) applied after Gaussian smoothing over a range of :
(a) ⫽ 0.5; (b) ⫽ 1; (c) ⫽ 2; and (d) ⫽ 4 The thresholds are fixed in each case at TU⫽ 10
and TL⫽ 4
only computational cost beyond that for grayscale images is incurred in obtaining the
luminance component image, if necessary In many color spaces, such as YIQ, HSL,
CIELUV , and CIELAB, the luminance image is simply one of the components in that
representation For others, such as RGB, computing the luminance image is usually easy
and efficient The main drawback to luminance-only processing is that important edges
are often not confined to the luminance component Therefore, a gray level difference in
the luminance component is often not the most appropriate criterion for edge detection
in color images
Trang 2(a) (b)
FIGURE 19.11
Canny edge detector ofEq (19.22)applied after Gaussian smoothing with ⫽ 2: (a) TU ⫽ 10,
TL ⫽ 1; (b) TU ⫽ TL ⫽ 10; (c) TU ⫽ 20, TL ⫽ 1; (d) TU ⫽ TL ⫽ 20 As TLis changed, notice theeffect on the results of hysteresis thresholding
Another rather obvious approach is to apply a desired edge detection method arately to each color component and construct a cumulative edge map One possibility
sep-for overall gradient magnitude, shown here sep-for the RGB color space, combines the
component gradient magnitudes[24]:
ⵜf c (x,y) ⫽ ⵜf R (x,y)⫹ⵜf G (x,y)⫹ⵜf B (x,y).The results, however, are biased according to the properties of the particular color spaceused It is often important to employ a color space that is appropriate for the target
Trang 319.5 Approaches for Color and Multispectral Images 521
application For example, edge detection that is intended to approximate the human
visual system’s behavior should utilize a color space having a perceptual basis, such as
CIELUV or perhaps HSL Another complication is the fact that the components’ gradient
vectors may not always be similarly oriented, making the search for local maxima of
|ⵜf c| along the gradient direction more difficult If a total gradient image were to be
computed by summing the color component gradient vectors, not just their magnitudes,
then inconsistent orientations of the component gradients could destructively interfere
and nullify some edges
Vector approaches to color edge detection, while generally less computationally
effi-cient, tend to have better theoretical justification Euclidean distance in color space
between the color vectors of a given pixel and its neighbors can be a good basis for
an edge detector [24] For the RGB case, the magnitude of the vector gradient is as
follows:
ⵜf c (x,y) ⫽ ⵜfR (x,y)2⫹ⵜf G (x,y)2⫹ⵜf B (x,y)2
Trahanias and Venetsanopoulos [29]described the use of vector order statistics as the
basis for color edge detection A later paper byScharcanski and Venetsanopoulos [26]
furthered the concept While not strictly founded on the gradient or Laplacian, their
techniques are effective and worth mention here because of their vector bases The basic
idea is to look for changes in local vector statistics, particularly vector dispersion, to
indicate the presence of edges
Multispectral images can have many components, complicating the edge detection
problem even further.Cebrián et al [6]describe several methods that are useful for
mul-tispectral images having any number of components Their description uses the second
directional derivative in the gradient direction as the basis for the edge detector, but other
types of detectors can be used instead The components-average method forms a
gray-scale image by averaging all components, which have first been Gaussian-smoothed, and
then finds the edges in that image The method generally works well because
multispec-tral images tend to have high correlation between components However, it is possible
for edge information to diminish or vanish if the components destructively interfere
Cumani [8]explored operators for computing the vector gradient and created an
edge detection approach based on combining the component gradients A multispectral
contrast function is defined, and the image is searched for pixels having maximal
direc-tional contrast Cumani’s method does not always detect edges present in the component
bands, but it better avoids the problem of destructive interference between bands
The maximal gradient method constructs a single gradient image from the component
images[6] The overall gradient image’s magnitude and direction values at a given pixel
are those of the component having the greatest gradient magnitude at that pixel Some
edges can be missed by the maximal gradient technique because they may be swamped
by differently oriented, stronger edges present in another band
The method of combining component edge maps is the least efficient because an edge
map must first be computed for every band On the positive side, this method is capable
of detecting any edge that is detectable in at least one component image Combination
Trang 4of component edge maps into a single result is made more difficult by the edge locationerrors induced by Gaussian smoothing done in advance The superimposed edges canbecome smeared in width because of the accumulated uncertainty in edge localization.
A thinning step applied during the combination procedure can greatly reduce this edgeblurring problem
19.6 SUMMARY
Gray level edge detection is most commonly performed by convolving an image, f , with
a filter that is somehow based on the idea of the derivative Conceptually, edges can be
revealed by locating either the local extrema of the first derivative of f or the zero-crossings
of its second derivative The gradient and the Laplacian are the primary derivative-basedfunctions used to construct such edge-detection filters The gradient,ⵜ, is a 2D extension
of the first derivative while the Laplacian,ⵜ2, acts as a 2D second derivative A variety
of edge detection algorithms and techniques have been developed that are based on thegradient or Laplacian in some way Like any type of derivative-based filter, ones based onthese two functions tend to be very sensitive to noise Edge location errors, false edges,and broken or missing edge segments are often problems with edge detection applied tonoisy images For gradient techniques, thresholding is a common way to suppress noiseand can be done adaptively for better results Gaussian smoothing is also very helpfulfor noise suppression, especially when second-derivative methods such as the Laplacianare used The Laplacian of Gaussian approach can also provide edge information over arange of scales, helping to further improve detection accuracy and noise suppression aswell as providing clues that may be useful during subsequent processing
Recent comparisons of various edge detectors have been made byHeath et al [13]
andBowyer et al [4] They have concluded that the subjective quality of the results ofvarious edge detectors applied to real images is quite dependent on the images themselves.Thus, there is no single edge detector that produces a consistently best overall result.Furthermore, they found it difficult to predict the best choice of edge detector for a givensituation
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[11] Q Ji and R M Haralick Efficient facet edge detection and quantitative performance evaluation.
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prefilters IEEE Trans Image Process., 4:1572–1577, 1995.
[13] M Heath, S Sarkar, T Sanocki, and K Bowyer Comparison of edge detectors, a methodology and
initial study Comput Vis Image Underst., 69(1):38–54, 1998.
[14] A Huertas and G Medioni Detection of intensity changes with subpixel accuracy using
Laplacian-Gaussian masks IEEE Trans Pattern Anal Mach Intell., PAMI-8(5):651–664, 1986.
[15] S R Gunn On the discrete representation of the Laplacian of Gaussian Pattern Recognit., 32:
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[16] A K Jain Fundamentals of Digital Image Processing Prentice-Hall, Englewood Cliffs, NJ, 1989.
[17] J S Lim Two-Dimensional Signal and Image Processing Prentice-Hall, Englewood Cliffs, NJ, 1990.
[18] D Marr Vision W H Freeman, New York, 1982.
[19] D Marr and E Hildreth Theory of edge detection Proc R Soc Lond B, 270:187–217, 1980.
[20] B Mathieu, P Melchior, A Oustaloup, and Ch Ceyral Fractional differentiation for edge detection.
Signal Processing, 83:2421–2432, 2003.
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Vis Pattern Recognit., 652–659 IEEE, New York, 1996.
[22] V S Nalwa and T O Binford On detecting edges IEEE Trans Pattern Anal Mach Intell.,
PAMI-8(6):699–714, 1986.
[23] W K Pratt Digital Image Processing, 2nd ed Wiley, New York, 1991.
[24] S J Sangwine and R E N Horne, editors The Colour Image Processing Handbook Chapman and
Hall, London, 1998.
[25] S Sarkar and K L Boyer Optimal infinite impulse response zero crossing based edge detectors.
Comput Vis Graph Image Process Image Underst., 54(2):224–243, 1991.
[26] J Scharcanski and A N Venetsanopoulos Edge detection of color images using directional
operators IEEE Trans Circuits Syst Video Technol., 7(2):397–401, 1997.
[27] P Siohan, D Pele, and V Ouvrard Two design techniques for 2-D FIR LoG filters In M Kunt,
editor, Proc SPIE, Visual Communications and Image Processing, Vol 1360, 970–981, 1990.
[28] V Torre and T A Poggio On edge detection IEEE Trans Pattern Anal Mach Intell.,
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[30] A P Witkin Scale-space filtering In Proc Int Joint Conf Artif Intell., 1019–1022 William
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Trang 720
Diffusion Partial Differential
Equations for Edge
Detection
Scott T Acton
University of Virginia
20.1 INTRODUCTION AND MOTIVATION
20.1.1 Partial Differential Equations in Image and Video Processing
The collision of imaging and differential equations makes sense Without motion or
change of scene or changes within the scene, imaging is worthless First, consider a static
environment—we would not need vision in this environment, as the components of the
scene are unchanging In a dynamic environment, however, vision becomes the most
valuable sense Second, consider a constant-valued image with no internal changes or
edges Such an image is devoid of value in the information-theoretic sense
The need for imaging is based on the presence of change The mechanism for change
in both time and space is described and governed by differential equations.
The partial differential equations (PDEs) of interest in this chapter enact diffusion In
chemistry or heat transfer, diffusion is a process that equilibrates concentration differences
without creating or destroying mass In image and video processing, we can consider the
mass to be the pixel intensities or the gradient magnitudes, for example.
These important differential equations are PDEs, since they contain partial derivatives
with respect to spatial coordinates and time These equations, especially in the case
of anisotropic diffusion, are nonlinear PDEs since the diffusion coefficient is typically
nonlinear
20.1.2 Edges and Anisotropic Diffusion
Sudden, sustained changes in image intensity are called edges We know that the human
visual system makes extensive uses of edges to perform visual tasks such as object
recog-nition[1] Humans can recognize complex 3D objects using only line drawings or image
edge information Similarly, the extraction of edges from digital imagery allows a
valu-able abstraction of information and a reduction in processing and storage costs Most 525
Trang 8definitions of image edges involve some concept of feature scale Edges are said to exist
at certain scales—edges from detail existing at fine scales and edges from the boundaries
of large objects existing at large scales Furthermore, large-scale edges exist at fine scales,leading to a notion of edge causality
In order to locate edges of various scales within an image, it is desirable to have animage operator that computes a scaled version of a particular image or frame in a videosequence This operator should preserve the position of such edges and facilitate the
extraction of the edge map through the scale space The tool of isotropic diffusion, a
linear lowpass filtering process, is not able to preserve the position of important edgesthrough the scale space Anisotropic diffusion, however, meets this criterion and hasbeen used effectively in conjunction with edge detection
The main benefit of anisotropic diffusion is edge preservation through the imagesmoothing process Anisotropic diffusion yields intra-region smoothing, not inter-regionsmoothing, by impeding diffusion at the image edges The anisotropic diffusion processcan be used to retain image features of a specified scale Furthermore, the localizedcomputation of anisotropic diffusion allows efficient implementation on a locally-
interconnected computer architecture Caselles et al furnish additional motivation for
using diffusion in image and video processing[2] The diffusion methods use localizedmodels where discrete filters become PDEs as the sample spacing goes to zero ThePDE framework allows various properties to be proved or disproved including stability,locality, causality, and the existence and uniqueness of solutions Through the establi-shed tools of numerical analysis, high degrees of accuracy and stability are possible
In this chapter, we introduce diffusion for image and video processing We cally concentrate on the implementation of anisotropic diffusion, providing severalalternatives for the diffusion coefficient and the diffusion PDE Energy-based variationaldiffusion techniques are also reviewed Recent advances in anisotropic diffusion proce-sses, including multiresolution techniques, multispectral techniques, and techniques forultrasound and radar imagery, are discussed Finally, the extraction of image edges afteranisotropic diffusion is addressed, and vector diffusion processes for attracting activecontours to boundaries are examined
specifi-20.2 BACKGROUND ON DIFFUSION
20.2.1 Scale Space and Isotropic Diffusion
In order to introduce the diffusion-based processing methods and the associated processes
of edge detection, let us define some notation Let I represent an image with real-valued
intensity I (x) image at position x in the domain ⍀ When defining the PDEs for diffusion,
let It be the image at time t with intensities I t (x) Corresponding with image I is the edge map e—the image of “edge pixels” e (x) with Boolean range (0 = no edge, 1 = edge), or
real-valued range e (x) ∈ [0,1] The set of edge positions in an image is denoted by ⌿.
The concept of scale space is at the heart of diffusion-based image and video
processing A scale space is a collection of images that begins with the original, fine
Trang 920.2 Background on Diffusion 527
scale image and progresses toward more coarse scale representations Using a scale space,
important image processing tasks such as hierarchical searches, image coding, and image
segmentation may be efficiently realized Implicit in the creation of a scale space is the
scale generating filter Traditionally, linear filters have been used to scale an image In
fact, the scale space ofWitkin [3]can be derived using a Gaussian filter:
The Marr-Hildreth paradigm uses a Gaussian scale space to define multiscale edge
detection Using the Gaussian-convolved (or diffused) images, one may detect edges
by applying the Laplacian operator and then finding zero-crossings [5] This popular
method of edge detection, called the Laplacian-of-a-Gaussian (LoG), is strongly
moti-vated by the biological vision system However, the edges detected from isotropic diffusion
(Gaussian scale space) suffer from artifacts such as corner rounding and from edge
localization error (deviation in detected edge position from the “true” edge position) The
localization errors increase with increased scale, precluding straightforward multiscale
image/video analysis As a result, many researchers have pursued anisotropic diffusion as
a viable alternative for generating images suitable for edge detection This chapter focuses
on such methods
20.2.1.1 Anisotropic Diffusion
The main idea behind anisotropic diffusion is the introduction of a function that inhibits
smoothing at the image edges This function, called the diffusion coefficient c (x),
encour-ages intra-region smoothing over inter-region smoothing For example, if c (x) is constant
at all locations, then smoothing progresses in an isotropic manner If c (x) is allowed
to vary according to the local image gradient, we have anisotropic diffusion A basic
anisotropic diffusion PDE is
⭸I t (x)
with I0⫽ I[6]
Trang 10The discrete formulation proposed in[6]will be used as a general framework forimplementation of anisotropic diffusion in this chapter Here the image intensities areupdated according to
direc-is given by t ⌬T is the time step—for stability, ⌬T ⱕ1
2 in the 1D case, and⌬T ⱕ1
4 inthe 2D case using four diffusion directions For 1D discrete-domain signals, the simpledifferencesⵜI d (x) with respect to the “western” and “eastern” neighbors, respectively
(neighbors to the left and right), are defined by
and
The parameters h1 and h2 define the sample spacing used to estimate the directionalderivatives For the 2D case, the diffusion directions include the “northern” and “south-ern” directions (up and down), as well as the “western” and “eastern” directions (left andright) Given the motivation and basic definition of diffusion-based processing, we willnow define several implementations of anisotropic diffusion that can be applied for edgeextraction
20.3 ANISOTROPIC DIFFUSION TECHNIQUES
20.3.1 The Diffusion Coefficient
The link between edge detection and anisotropic diffusion is found in the edge-preservingnature of anisotropic diffusion The function that impedes smoothing at the edges isthe diffusion coefficient Therefore, the selection of the diffusion coefficient is the mostcritical step in performing diffusion-based edge detection We will review several possiblevariants of the diffusion coefficient and discuss the associated positive and negativeattributes
To simplify the notation, we will denote the diffusion coefficient at location x by
c (x) in the continuous case For the discrete-domain case, c d (x) represents the diffusion coefficient for direction d at location x Although the diffusion coefficients here are
defined using c (x) for the continuous case, the functions are equivalent in the
discrete-domain case of c d (x) Typically c(x) is a nonincreasing function of |ⵜI(x)|, the gradient magnitude at position x As such, we often refer to the diffusion coefficient as c (|ⵜI(x)|).
For small values of|ⵜI(x)|, c(x) tends to unity As |ⵜI(x)| increases, c(x) decreases to
zero.Teboul et al [7]establish three conditions for edge-preserving diffusion coefficients.These conditions are (1) lim
|ⵜI(x)|→0 c (x) ⫽ M where 0 < M < ⬁, (2) lim
|ⵜI(x)|→⬁ c (x) ⫽ 0,
Trang 1120.3 Anisotropic Diffusion Techniques 529
and (3) c(x) is a strictly decreasing function of |ⵜI(x)| Property 1 ensures isotropic
smoothing in regions of similar intensity, while property 2 preserves edges The third
property is given in order to avoid numerical instability While most of the coefficients
discussed here obey the first two properties, not all formulations obey the third property
In[6], Perona and Malik propose
as diffusion coefficients Diffusion operations using(20.9)and(20.10)have the ability
to sharpen edges (backward diffusion), and are inexpensive to compute However, these
diffusion coefficients are unable to remove heavy-tailed noise and create “staircase”
arti-facts[8, 9] See the example of smoothing using(20.9)on the noisy image inFig 20.1(a),
producing the result inFig 20.1(b) In this case, the anisotropic diffusion operation leaves
several outliers in the resultant image A similar problem is observed inFig 20.2(b), using
the corrupted image inFig 20.2(a)as input You et al have also shown that(20.9)and
(20.10) lead to an ill-posed diffusion—a small perturbation in the data may cause a
significant change in the final result[10]
The inability of anisotropic diffusion to denoise an image has been addressed by
Catte et al [11] andAlvarez et al [12] Their regularized diffusion operation uses a
modification of the gradient image used to compute the diffusion coefficients In this
case, a Gaussian-convolved version of the image is employed in computing diffusion
coefficients Using the same basic form as(20.9), we have
This method can be used to rapidly eliminate noise in the image as shown inFig 20.1(c)
In this case, the diffusion is well posed and converges to a unique result, under certain
conditions [11] Drawbacks of this diffusion coefficient implementation include the
additional computational burden of filtering at each step and the introduction of a linear
filter into the edge-preserving anisotropic diffusion approach The loss of sharpness due
to the linear filter is evident inFig 20.2(c) Although the noise is eradicated, the edges
are softened and blotching artifacts appear in the background of this example result
Another modified gradient implementation, called morphological anisotropic
diff-usion, can be formed by substituting
Trang 12into (20.11), where B is a structuring element of size m ⫻ m, I ◦ B is the
morpho-logical opening of I by B, and I • B is the morphological closing of I by B In [13],the open-close and close-open filters were used in an alternating manner between itera-tions, thus reducing grayscale bias of the open-close and close-open filters As the result
inFig 20.1(d)demonstrates, the morphological anisotropic diffusion method can beused to eliminate noise and insignificant features while preserving edges Morphological
Trang 1320.3 Anisotropic Diffusion Techniques 531
anisotropic diffusion has the advantage of selecting feature scale (by specifying the
structuring element B) and selecting the gradient magnitude threshold, whereas
pre-vious anisotropic diffusions, such as (20.9)and(20.10), only allowed selection of the
gradient magnitude threshold
You et al introduce the following diffusion coefficient in[10]:
Trang 14(d) (e)
FIGURE 20.2
(a) Corrupted “cameraman” image (Laplacian noise, SNR⫽ 13dB) used as input for results
inFigs 20.2(b)–(e); (b) after 8 iterations of anisotropic diffusion with(20.9), k⫽ 25; (c) after
8 iterations of anisotropic diffusion with(20.11) and (20.12), k⫽ 25; (d) after 75 iterations
of anisotropic diffusion with(20.14), T ⫽ 6, e ⫽ 1, p ⫽ 0.5; (e) after 15 iterations of multigrid
anisotropic diffusion with(20.11)and(20.12), k⫽ 6[35]
where the parameters are constrained by > 0 and 0 < p < 1 T is a threshold on the gradient magnitude, similar to k in(20.9) This approach has the benefits of avoidingstaircase artifacts and removing impulse noise The main drawback is computationalexpense As seen inFig 20.2(d), anisotropic diffusion with this diffusion coefficientsucceeds in removing noise and retaining important features from Fig 20.2(a), butrequires a significant number of updates
The diffusion coefficient
standard anisotropic diffusion coefficient as in(20.9)continues to smooth over edges
Trang 1520.3 Anisotropic Diffusion Techniques 533
while iterating, the robust formulation(20.16)preserves edges of a prescribed scale
and effectively stops diffusion
Here seven important versions of the diffusion coefficient were given that involve
tradeoffs between solution quality, solution expense, and convergence behavior Other
research in the diffusion area focuses on the diffusion PDE itself The next section
reveals significant modifications to the anisotropic diffusion PDE that affect fidelity to
the input image, edge quality, and convergence properties
20.3.2 The Diffusion PDE
In addition to the basic anisotropic diffusion PDE given inSection 20.1.2, other diffusion
mechanisms may be employed to adaptively filter an image for edge detection.Nordstrom
[18]used an additional term to maintain fidelity to the input image, to avoid the selection
of a stopping time, and to avoid termination of the diffusion at a trivial solution, such as
a constant image This PDE is given by
⭸I t (x)
⭸t ⫺ div {c(x)ⵜIt (x)} ⫽ I0(x) ⫺ I t (x). (20.17)Obviously, the right-hand side I0(x) ⫺ I t (x) enforces an additional constraint that pena-
lizes deviation from the input image
Just asCanny [19]modified the LoG edge detection technique by detecting
zero-crossings of the Laplacian only in the direction of the gradient, a similar edge-sensitive
approach can be taken with anisotropic diffusion Here, the boundary-preserving
diffu-sion is executed only in the direction orthogonal to the gradient direction, whereas the
standard anisotropic diffusion schemes impede diffusion across the edge If the rate of
change of intensity is set proportional to the second partial derivative in the direction
orthogonal to the gradient (called), we have
This anisotropic diffusion model is called mean curvature motion, because it induces a
diffusion in which the connected components of the image level sets of the solution image
move in proportion to the boundary mean curvature Several effective edge-preserving
diffusion methods have arisen from this framework including[20] and[21].Alvarez
et al [12]have used the mean curvature method in tandem with the regularized diffusion
coefficient of(20.11)and(20.12) The result is a processing method that preserves the
causality of edges through scale space For edge-based hierarchical searches and multiscale
analyses, the edge causality property is extremely important
The mean curvature method has also been given a graph theoretic interpretation
[22, 23].Yezzi [23]treats the image as a graph inn—a typical 2D grayscale image would
be a surface in3 where the image intensity is the third parameter, and each pixel is a
graph node Hence a color image could be considered a surface in 5 The curvature
motion of the graphs can be used as a model for smoothing and edge detection For
example, let a 3D graph s be defined by s(x) = s(x, y ) = [x, y, I(x, y)] for the 2D image I