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Journal of Mathematical Imaging and Vision 8, 161–180 (1998) c 1998 Kluwer Academic Publishers. Manufactured in The Netherlands. °

Differentiation-Based Edge Detection Using the Logarithmic Image Processing Model GUANG DENG School of Electronic Engineering, La Trobe University, Bundoora Victoria 3083, Australia [email protected]

JEAN-CHARLES PINOLI Pechiney, Centre de Recherches, BP 27, F38340 Voreppe, France; and Laboratoire Image, Signal et Acoustique, CNRS EP92, Ecole Sup´erieure de Chimie, Physique et Electronique, 31 Place Bellecour, F69288 Lyon Cedex 02, France [email protected]

Abstract. The logarithmic image processing (LIP) model is a mathematical framework which provides a specific set of algebraic and functional operations for the processing and analysis of intensity images valued in a bounded range. The LIP model has been proved to be physically justified by that it is consistent with the multiplicative transmittance and reflectance image formation models, and with some important laws and characteristics of human brightness perception. This article addresses the edge detection problem using the LIP-model based differentiation. First, the LIP model is introduced, in particular, for the gray tones and gray tone functions, which represent intensity values and intensity images, respectively. Then, an extension of these LIP model notions, respectively called gray tone vectors and gray tone vector functions, is studied. Third, the LIP-model based differential operators are presented, focusing on their distinctive properties for image processing. Emphasis is also placed on highlighting the main characteristics of the LIP-model based differentiation. Next, the LIP-Sobel based edge detection technique is studied and applied to edge detection, showing its robustness in locally small changes in scene illumination conditions and its performance in the presence of noise. Its theoretical and practical advantages over several wellknown edge detection techniques, such as the techniques of Sobel, Canny, Johnson and Wallis, are shown through a general discussion and illustrated by simulation results on different real images. Finally, a discussion on the role of the LIP-model based differentiation in the current context of edge detection is presented. Keywords: intensity images, edge detection, logarithmic image processing, differential operators, gray tone vectors

1. 1.1.

Introduction Edge Detection

Edges are areas in an image where rapid changes occur in the intensity function or in spatial derivatives of this intensity function. The goal of edge detection is to recover information about shapes and reflectance or transmittance in an image. It is one of the fundamental

steps in image processing [1], image analysis [2], image pattern recognition [3], computer vision [4], as well as in human vision [5]. Many theories, techniques and algorithms have been proposed for edge detection, in books, e.g., [1–4], and in numerous research articles in different scientific and technical journals, e.g., Computer Vision, Graphics and Image Processing [6–13], IEEE Transactions on Pattern Analysis and Machine Intelligence [14–19],

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IEEE Transactions on Systems, Man and Cybernetics [20–24], Image and Vision Computing [25, 26], International Journal of Computer Vision [27–29], Journal of the Optical Society of America [30], Pattern Recognition [31–35], Proceedings of the IEEE [36], Proceedings of the Royal Society of London [37]. Although a review of edge detection techniques is outside the scope of the present article, it can be inferred from a careful reading of the above literature that it remains a difficult task to find the edges that correspond to a physical origin in the observed scene, such as the change of orientation or reflection or transmission, or objects occluding each other. This task is even more difficult when the image is noisy or/and is obtained under poor or changing illumination conditions. 1.2.

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Edge Detection in the Presence of Locally Small Changes of Scene Illumination

Since in many practical situations the illumination of a scene is changing with space and time, it is therefore necessary to study how to detect edges robustly in uneven or changing illumination conditions. As long as the salient features of a scene do not change, an edge detection technique ideally should continue to generate consistent (or strictly, the same) edges irrespective of the changes in illumination. This is a fundamental requirement for a computer vision system which is used in a situation where the scene illumination is not controllable. Edge detection under the condition of locally small changes in scene illumination has attracted increasing research interest. Johnson [33] has shown that most of the present edge detection techniques, which are based on finding the maxima of the first derivatives or the zero-crossings of the second order derivatives of the intensity function, are not suitable for this problem. Johnson [33] and Robinson [7] have suggested a gradient/mean operator which uses the ratio of the local gradient magnitude to the local mean intensity value. In Johnson’s and Robinson’s edge detection techniques, the gradient operators are the Sobel’s operator [1] and the compass gradient operator [7], respectively. Both techniques have been physically derived and have provided good results. In fact, in developing edge detection techniques or more general image processing techniques, it is of central importance to take into account the nature of the images by using a consistent physical image formation model and a mathematical framework compatible with

the information to be processed, as noted by several authors, e.g., Stockham [38] and Marr [4]. 1.3.

The Logarithmic Image Processing Model

In mid 1980s, an original mathematical framework— the logarithmic image processing (LIP) model—has been introduced by Jourlin and Pinoli [39, 40] for the processing of intensity images valued in a bounded range. The initial aim of developing the LIP model was to define an additive operation closed in the bounded real number interval [0, M), which is both mathematically well defined and physically consistent with some concrete physical image settings. In developing the LIP model, Jourlin and Pinoli argued that, from a mathematical point of view, most of the present image processing techniques are originated from functional analysis theory. These mathematical tools only realize their efficiency when they are put into a well-defined algebraic framework, most of the time of a vectorial nature. They further argued that the usual addition “+” and scalar multiplication “×” operations are not suitable for some important physical settings, such as images formed by transmitted light [41, 42], for human visual perception [5, 43] and in many practical cases of digital images [1, 44]. For the first two nonlinear imaging settings, the major reason is that the addition ‘+” and consequently the scalar multiplication “×” are not consistent with their image formation laws. Regarding digital images, the problem is that a direct addition of two pixel intensity values may result in a value which is out of the range where the image is valued. This is the so-called out-of-range problem. The LIP model is a complete mathematical theory [45, 46] (see [47] for an introductory summary). It consists of an ordered functional framework and provides a set of special operations: addition, subtraction, multiplication, differentiation, integration, convolution, Fourier and wavelet transformations and so on, for the processing of bounded range of intensity images. The LIP model has also been physically welljustified in that it is consistent with the multiplicative transmittance image formation model [40, 48], the multiplicative reflectance image formation model [49, 50], and with several laws and characteristics of human brightness perception [47], namely the brightness scale inversion, the saturation characteristic, Weber’s and Fechner’s laws, and the psychophysical contrast notion [40, 46, 49, 51–55].

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During the last ten years, the LIP model has been successfully applied to a number of image processing problems, e.g., dynamic range and sharpness modification [48, 49, 54–56], image restoration [52, 57], image 3D reconstruction and analysis [58–61], contrast-based edge detection and image segmentation [49, 51, 62, 63], and image filtering [49, 55]. 1.4.

Aim and Outline of This Article

This article aims at introducing, studying, applying and discussing the LIP-model based differentiation in the context of image edge detection, in particular, edge detection in the presence of locally small changes in scene illumination conditions. This article is organised into five main sections. Section 2 introduces the LIP model, summarizing some of its mathematical notions, in particular the gray tones and gray tone functions. Section 3 describes the gray tone vectors and gray tone vector functions. In Section 4, the LIP-model-based differential operators are presented, focusing on their distinctive properties for image processing and analysis. The main characteristics of the LIP-model based differentiation are also highlighted from mathematical, physical and computational viewpoints. In Section 5, the LIP-Sobel based edge detection technique is defined and studied. Its robustness in locally small changing scene illumination condition is shown, and its performance in the presence of noise is reported. Its theoretical and practical advantages over the traditional techniques of Sobel [1], Canny [16], Johnson [33] and Wallis [1] are shown through a general discussion and several simulation results on different digital images. Section 6 presents a general discussion on the role of the LIP-model based differentiation in the current context of edge detection. Issues such as definition of edges, noise and scale are discussed.

2.

The LIP Model

This section presents the symbols, operational rules, mathematical notions and structures of the LIP model, that will be necessary for the following sections. The presentation has been made as concise as possible. First, the relationship between the classical gray level functions and the LIP gray tone functions is given. Then, the vectorial operations on the gray tone function space are introduced. Finally, the gray tone space

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structures and the LIP fundamental isomorphism are presented. 2.1.

Gray Level Functions and Gray Tone Functions

In the LIP model, the intensity of an image is completely represented by its associated gray tone function. Such a function is defined on a non-empty spatial domain D in the Euclidean space R2 , called the spatial support, with values in the bounded real number interval [0, M), where M is strictly positive (in the digital case, M equals 256 for an 8-bit image). A gray tone function is denoted f (x, y). The value of a gray tone function at a spatial location (x, y) is called a gray tone, and the interval [0, M) is thus called the gray tone range. The relationship between a gray tone function f (x, y) and its corresponding classical gray level function [44], denoted f˜(x, y), is given by: f (x, y) = M − f˜(x, y).

(1)

In fact, a gray tone function is nothing else than a classical gray level function, but valued in an inverted intensity scale. Indeed, the limits of the gray tone range [0, M) are anticlassically defined: 0 and M represent the white and black values, respectively, contrary to the usual convention. This scale inversion has been justified on physical [40], psychophysical [47] grounds and mathematical [46] reasons. From a computational point of view, both representations f (x, y) and f˜(x, y) are of the same level of simplicity. 2.2.

The Vectorial Structure on the Gray Tone Function Space

The set of gray tone functions defined on the spatial support D and valued in the real number interval [0, M) is denoted I. The addition of two gray tone functions f (x, y) and g(x, y), and the scalar multiplication of f (x, y) by a positive real number α are defined in terms of the usual real-valued function operations “+” and “×” as: + g(x, y) = f (x, y) + g(x, y) f (x, y) 1

f (x, y)g(x, y) , M

(2)

µ ¶ f (x, y) α × f (x, y) = M − M 1 − . α1 M

(3)

− and

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These two specific operations mathematically describe how images are combined and how the pixel value is amplified. The addition operation has been introduced [40, 45], in order to be close in the real number interval [0, M) and additive, and to physically suit the case of transmitted light images. In fact, the addition expresses the transmittance product law [41, 42]. The scalar multiplication has also been built [40, 45, 46] starting from the addition operation. It has been proved [45, 46] that the set I is a posi+ tive cone [64] for the specific algebraic operations 1 and 1 × , since it is closed for these operations, the addition is both commutative and associative, the scalar multiplication operation obeys the associative and distributive laws, and the null gray tone function is the + . This ensures that neutral element for the addition 1 the addition of two gray tone functions, or the scalar multiplication of a gray tone function by a positive real number, results in a new gray tone function. This is a very desirable property for image processing. In order to enlarge the positive cone I into a vector space, it is necessary to define the opposite of a gray − f (x, y). It is also tone function f (x, y), denoted as 1 necessary to extend the scalar multiplication to any real number. The first task is straightforward: − f (x, y) = −M 1

f (x, y) . M − f (x, y)

(4)

This definition allows the subtraction between two gray tone functions f (x, y) and g(x, y) to be defined as: − g(x, y) = M f (x, y) 1

f (x, y) − g(x, y) . M − g(x, y)

(5)

According to this definition, the gray tone range is extended from [0, M) to (−∞, M), and the positive restriction of the real number in the scalar multiplication operation is removed. This means that a gray tone function can now be multiplied by any real number using the definition shown in Eq. (3). Moreover, the set of gray tone functions, defined on the spatial support D and valued in the real interval (−∞, M), denoted + and 1 × is a real vector space, G, with the operations 1 which thus admits I as its positive cone. Therefore, the elements of the positive cone I appear as positive gray tone functions. The term “positive” is used to emphasise that only positive gray tone functions physically correspond to bounded intensity images addressed by the LIP model, while the extended gray tones belonging to (−∞, M) have only a mathematical meaning.

Note that this extension is not in contradiction with the LIP model. This will be seen in the following sections of this paper. The establishment of a vectorial structure for the set of gray tone functions is the first step in the development of the LIP model. Since the mathematical aim of the LIP model was to develop a functional framework, it was thus necessary to introduce strong structures on the gray tone set. The purpose of the following section is to summarise several of these structures that will be useful in this paper. 2.3.

The Gray Tone Space, the Fundamental Isomorphism and Modulus Notion

In the LIP framework, the elements of the real interval (−∞, M) are called gray tones and are denoted + and 1 f, g, . . . . With the specific operations 1 × , the set of gray tones is a real vector space denoted E. This vector space is algebraically isomorphic to the real number space R by the mapping ϕ defined as: ¶ µ f . (6) ϕ( f ) = −M ln 1 − M In this article, ϕ will be called the fundamental isomorphism, and the isomorphic transform of a gray tone f is denoted by: f¯ = ϕ( f ),

(7)

where f¯ is a real number. The inverse isomorphic transformation is then defined as: µ µ ¯ ¶¶ f −1 ¯ . (8) ϕ ( f ) = M 1 − exp − M The fundamental isomorphism has served as a powerful tool for developing the LIP model. For example, the following operations on gray tones can be defined using the fundamental isomorphism, allowing notions and structures originated from functional analysis [64, 65] to be introduced: (i) Scalar product between gray tones: ( f | g)E = ϕ( f )ϕ(g);

(9)

(ii) Euclidean norm of gray tones: k f kE = |ϕ( f )|R ,

(10)

where |·|R is the real absolute value function;

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(iii) Euclidean distance between two gray tones: dE ( f, g) = |ϕ( f ) − ϕ(g)|R .

(11)

Therefore, the gray tone space E is an Euclidean space, Banach space and metric space. Moreover, it has been shown that the gray tone space E is a Riesz space [66] with the natural order relation, denoted ≥, and the modulus |·|E , defined for a gray tone f by: ½ | f |E =

f, if f ≥ 0 . − f, if f < 0 01

(12)

The modulus notion plays an important role in the LIP model, since for any gray tone f its modulus | f |E is a positive gray tone, i.e., a gray tone belonging to [0, M), and since strong properties occur with respect + and 1 ×: to the vectorial operations 1 + g|E ≤ | f |E 1 + |g|E |f 1

(13)

speaking, the manipulation of gray tones with their LIP operations is equivalent to the manipulation of their corresponding isomorphic transforms with the usual operations. 3.

Gray Tone Vectors and Gray Tone Vector Functions

Deng [49] has extended the gray tone notion to the two dimensional vector form called the gray tone vector. A gray tone vector, denoted by f, is defined as a pair of gray tones f 1 and f 2 : f = ( f 1 , f 2 ).

ϑ(f) = (ϕ( f 1 ), ϕ( f 2 )). (14)

where f and g are two gray tones and α is a real number. The modulus notion appears as the accurate magnitude measure of gray tones [46, 47] and has been used for the introduction of the contrast notion [47, 53]. Its closure in the positive gray tone range [0, M) enables good properties, e.g., for the differentiation notion as it will be seen in Section 4. From an image processing point of view, the LIP model provides a set of specific operations that are valued, intrinsically or by using the modulus notion, in the bounded positive range [0, M). In fact, it has been shown that [46] the gray tone space E is a normed Riesz space for the norm k·kE and the modulus |·|E , in which the norm and the order relation induce the same topology [67]. This mathematical topological equivalence is of fundamental importance in the LIP model theory. It will underlie the well-justified introduction of the differential operators’ magnitude expressions in Section 4. Therefore, the convergent sequences are identical for the modulus and the norm. In summary, it is important to note that the gray tone space E is totally isomorphic to the real number space R, in that it preserves the algebraic, topological and order structures. In other words and roughly

(15)

The mathematical operations and structures defined on the gray tone space E can be easily generalized in the gray tone vector space denoted E2 . In fact, the gray tone vector space is isomorphic to the two dimensional real number space R2 , through the mapping denoted by ϑ:

and × f |E = |α| R 1 × | f |E , |α 1

165

(16)

The gray tone vector representation (15) is called the cartesian representation. It is also useful to define its polar representation. Given a gray tone vector f = ( f 1 , f 2 ), its polar coordinates, denoted ( f, θ ) and called the magnitude and the direction, respectively, are defined as:   q  f¯12 + f¯22  f = M 1 − exp − (17) M and θ = Atan( f¯2 , f¯1 ),

(18)

where f¯1 = ϕ( f 1 ), f¯2 = ϕ( f 2 ) and the function Atan(a, b) is defined in Appendix 1. Note that the magnitude f is a positive gray tone function and the direction θ is an angle belonging to [0, 2π). Inversely, a gray tone vector f can be converted from the polar coordinate representation ( f, θ ) to the cartesian representation ( f 1 , f 2 ) as follows: f 1 = cos θ 1 × f

(19)

f 2 = sin θ 1 × f.

(20)

and

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The gray tone vector notion will be used fruitfully to design the LIP-model-based gradient vector in Section 4. Finally, the gray tone vector functions are defined as being gray tone vector valued functions, that is to say functions defined from the spatial support D into E2 . The mathematical properties of gray tone vector functions can then be established straightforward and will not be developed here. The interested reader is invited to refer to [49] for a complete development. 4.

Differentiation of Gray Tone Functions

The LIP-model based differentiation has been introduced and studied [46, 53], using the general mathematical theory of differentiation of functions valued in Banach spaces. The definition of the gray tone function differentiation is given in Appendix 2. In fact, the useful notion for image processing is the directional derivative. The purpose of this section is to introduce the LIPmodel based first and second order directional differential operators. This section also aims at highlighting the main characteristics of the LIP-model based differentiation from mathematical, physical and computational viewpoints. The discrete expressions of these operations will be given explicitly, whilst the corresponding continuous expressions will be briefly introduced or referenced. In the following, the notations used for designating the notions defined in the LIP model have been chosen as close as possible to the classical notations. Whenever confusion may occur, these notations will be indexed with “1”. Moreover, it will be assumed that the interior set of the spatial support D is non-empty. The points considered below will be taken within this interior set. 4.1.

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The First Order Derivatives

A gray tone function f (x, y) is said to be v-differentiable at a point (x, y) along the direction of a non-zero plane vector v = (vx , v y ) if the following expression exists: ∂1v f (x, y) = lim

t>0 t→0

1 1 × ( f (x + tvx , y + tv y ) tkvkR2

− f (x, y)), 1

(21)

where kvkR2 is the Euclidean norm of vector v. Then, ∂1v f (x, y) is called the directional derivative of the

gray tone function f (x, y) at the point (x, y) along the direction of the plane vector v. This definition is related to the usual directional derivative of the real valued function f¯(x, y), denoted ∂v f¯(x, y), by: ∂1v f (x, y) = ϕ −1 (∂v f¯(x, y)),

(22)

where f¯(x, y) = ϕ( f (x, y)). The two first partial derivatives of a partially differentiable gray tone function f (x, y), denoted ∂1x f (x, y) and ∂1y f (x, y), are then special cases of Eq. (21) (by setting v = (1, 0) and v = (1, 0), respectively), and can also be expressed using Eq. (22) as: ∂1x f (x, y) = ϕ −1 (∂x f¯(x, y)),

(23)

∂1y f (x, y) = ϕ −1 (∂ y f¯(x, y)).

(24)

and

The importance of the directional differentiation of a gray tone function is that it results in another gray tone function. Consequently, using the modulus notion defined by (12), the values of the gray tone function |∂1v f (x, y)|E are positive gray tones. This is a useful property for image processing, especially in edge detection. Indeed, the selection of a threshold is easier in the bounded range [0, M) than in an unbounded range when using a classical differentiation operation. Moreover, such a threshold has a physical meaning: It is a gray tone and thus corresponds to an intensity value. Therefore, the gray tone function modulus corresponds to an intensity image, and can be visualised and processed as such. Using the gray tone vector representation introduced in Section 3, the LIP-model based gradient vector of a differentiable gray tone function f (x, y) at a point (x, y) is then given by: grad1 f (x, y) = (∂1x f (x, y), ∂1y f (x, y)).

(25)

In the discrete case, where the spatial support D is discretized into a lattice of grid points in Z2 , it is possible to obtain a lot of discrete approximations of Eqs. (21) and (25) and their combinations. These socalled discrete differential operators can be built using different “finite difference equations” [68] defined in numerical analysis [69, 70]. For example, similarly to the usual Sobel’s operator, the LIP-Sobel operator is then defined as the mapping which associates the

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is defined, for a second-order differentiable [46] gray tone function f (x, y), as: 2 + ∂ 2 f (x, y) lap1 f (x, y) = ∂1x f (x, y) 1 1y

Figure 1. A (3 × 3) area of a discrete intensity image using the gray tone function representation.

pixels of D with their corresponding LIP-Sobel gray tone vectors. For a (3 × 3) area of a discrete gray tone function (see Fig. 1), the LIP-Sobel gray tone vector, denoted g = (gx , g y ), is given by: + 21 + f7) 1 − ( f3 1 + 21 + f9) gx = ( f 1 1 × f4 1 × f6 1

(26)

and + 21 + f3) 1 − ( f7 1 + 21 + f 9 ). gy = ( f1 1 × f2 1 × f8 1

(27)

Using the Eqs. (17) and (18), the magnitude and direction of the gray tone vector g, denoted by g and θ , respectively, are given by: ¶Á ¶¶ µ µµ q M g = M 1 − exp − g¯ x2 + g¯ 2y

(28)

θ = Atan(g¯ x , g¯ y ),

(29)

where g¯ x = ϕ(gx ) and g¯ y = ϕ(g y ). 4.2.

Its discrete approximation using (3 × 3) neighbourhood (see Fig. 1) is: lap1 f (x, y) =

The Second Order Derivatives

It is also possible to introduce the second order direc2 f (x, y), of a tional derivative (if it exists), denoted ∂1v gray tone function f (x, y) at a point (x, y) along the direction of a non-zero plane vector v = (vx , v y ) as follows: 2 f (x, y) = lim ∂1v

t>0 t→0

1 1 × (∂1v f (x + tvx , y + tv y ) tkvkR2

− ∂1v f (x, y)). 1

(30)

Similarly to the first order case, the second order partial 2 2 f (x, y) and ∂1y f (x, y) are given by setderivatives ∂1x ting the vector v = (1, 0) and v = (0, 1) in Eq. (30), respectively. The LIP-Laplacian operator, denoted lap1 ,

1 + f3 1 + f7 1 + f9) 1 + 2 × (( f 1 1 1 8 + f4 1 + f6 1 + f8) 1 − 12 1 × ( f2 1 × f5) 1 (32)

The importance of the LIP-Laplacian operator is that lap1 f (x, y) is a gray tone function. Consequently, using the modulus notion defined by (12), the gray tone function |lap1 f (x, y)|E is positive and thus corresponds to an intensity image. This is a useful property for image processing, especially in edge detection, as previously discussed in Section 4.1. In fact, the LIP-Laplacian operator corresponds to the classical Laplacian operator. Indeed, replacing in Eq. (32) the LIP model operations with their corresponding usual operations respectively, yields the classical expression known in digital image processing. 4.3.

and

(31)

The Main Characteristics of the LIP-Model Based Differentiation

In this section, the main characteristics of the LIPmodel based differentiation are discussed, from a mathematical, physical and computational viewpoints, successively. Mathematically, the key point of the LIP-model based differentiation is that it is a new differentiation concept which involves the specific vector operations + and 1 − ), instead of the classical (+ and ×)vector (1 operations. Thus, the LIP differential operators, in particular, the LIP-Sobel operator, are defined with those new operations. This is contrary to the traditional Sobel operator or those of other differentiation-based image processing techniques, which are based on the usual operations. In fact, the LIP-model based differentiation is defined in a strong algebraic and functional mathematical framework [46] that provides a lot of compatible notions. Among the available notions, the modulus plays a fundamental role since it allows the definition of positively-valued operators in the LIP model sense (i.e., with values in the range [0, M)). This is the case

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for the LIP-Sobel vector magnitude function studied in Section 4.1. Physically, the LIP-model based differentiation is well adapted to the setting of transmitted and reflected light imaging [40, 48–50] in the sense that the LIPoperations have been shown to be consistent with the corresponding image formation and combination laws. Psychophysically, the LIP model has been connected [47] with several laws and characteristics of human brightness perception. It has been proved that [47, 53] the LIP model allows the definition of a contrast notion that corresponds to the psychophysical notion in the discrete case. The contrast notion is closely linked to the LIP-model based differentiation and the modulus notion. Computationally, the classical differential operators, such as the Sobel operator, are not a satisfactory solution for bounded intensity images. Indeed, the values of a differential magnitude function associated with a given bounded intensity image may be out of the bounded range where it must be in. It has been shown in Section 4.1 that the LIP-Sobel operator together with the modulus notion overcomes this out-of-range problem.

5.

Application to Edge Detection

The aim of this section is to illustrate the application of the above mathematical theory to a practical edge detection problem. First, it is shown that the LIPSobel operator is robust in the presence of locally small changes in scene illumination and this robustness is explained on physical grounds. Then, the performance of the LIP-Sobel edge detection technique is compared with that of other established methods by means of several simulation experiments. Finally, its performance in noisy images is also studied vs. that of Johnson’s technique. 5.1.

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The LIP-Sobel Operator

As claimed in the abstract of the present article, the LIP-Sobel edge detection technique is robust in locally small changing illumination conditions. This assertion will be practically verified in Section 5.2. through several simulation results on different digital test images. Before this practical evaluation, it is possible to explain from a physical viewpoint why the LIP-Sobel operator is invariant to locally small variations in scene illumination.

Indeed, using the classical gray level function representation f˜(x, y) given in (1) and the LIP fundamental isomorphism ϕ defined by (6), the Eqs. (26) and (27) are then expressed as, respectively (see the proof in Appendix 3): Ã ! f˜3 f˜62 f˜9 gx = M − M (33) f˜1 f˜42 f˜7 and

à gy = M − M

! f˜1 f˜22 f˜3 , f˜7 f˜82 f˜9

(34)

where f˜i = M − f i , i = 1 . . . 9. Following the multiplicative reflectance or transmittance image formation models with which the LIP model is known to be compatible [40, 46, 48, 49], the gray level function f˜(x, y) is expressed as : f˜(x, y) = I (x, y)R(x, y) or f˜(x, y) = I (x, y)T (x, y),

(35)

where I (x, y) is the illumination component, R(x, y) and T (x, y) are the reflectance/transmittance components, respectively. Assume that the illumination I (x, y) nearly equals a constant I in a (3 × 3) window of the considered intensity image, then: f˜i = I Ri

or

f˜i = I Ti ,

(36)

for i = 1 . . . 9. Finally, substitution of (36) into (33) and (34) shows that the output of the LIP-Sobel operator is only dependent on the reflectance or transmittance of the scene, and not on locally small changes in illumination. This is because the LIP-Sobel gray tone vector components gx and g y are not dependent on the intensity value, but on the reflectance/transmittance components. In addition, it can be easily inferred from the above demonstration that the robustness is in fact a general property of all the LIP-model-based first order differential operators. Furthermore, the LIP-model based edge detection technique has several other distinctive advantages. For a given gray tone function, the corresponding LIPSobel vector magnitude function defined by Eq. (28) is a positive gray tone function. Indeed, since the positive gray tone function space is structured as a closed set, the magnitude of the output of a LIP-Sobel operator is within the range [0, M). This is a very desirable property for extracting the edges by thresholding. Indeed,

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the selection of a threshold is easier, since the threshold now lies within a given bounded range. Moreover, a threshold is then a positive gray tone, and thus has a physical meaning: it corresponds to an intensity value. Contrary to the LIP-differentiation based edge detection technique, the traditional techniques of Sobel, Canny, Johnson and Wallis failed to have the above properties. Indeed, for these four edge detection techniques, the threshold selection must be done within an unknown range, and their gradient vector magnitude functions cannot be interpreted as intensity images. Although the LIP model and Johnson’s method share the same idea that an image processing system should be consistent with the physical imaging model, there is a remarkable difference between them. Johnson’s method was based on a physical imaging system and was relied on the contrast-based formulation of the edge detector. Although this method has been successfully applied to visible and thermal images, it is limited to edge detection. In contrast, the LIP model was developed as a mathematical theory which is consistent with a number of physical imaging models. As a mathematical tool, the LIP model can be applied to many image processing problems including edge detection. Therefore, in Johnson’s method the above idea was used to solve a single problem, in the LIP model the same idea was used to develop a mathematical theory. 5.2.

Simulation Results and Comparison

To illustrate practical advantages of the proposed edge detection technique, two groups of experiments have been made. The purpose of the first group of experiments is to show that edges extracted by the proposed edge detection technique correspond to the features of the test image, and that these results are comparable to those of other edge detection techniques. The purpose of the second group of experiments is to show that the proposed LIP-Sobel operator is invariant to locally small changes in scene illumination. The above mentioned edge detection techniques can be implemented by two steps: (1) edge magnitude calculation, (2) thresholding. Although there are some adaptive thresholding techniques, they are not used since there are many choices for “adaptively tuning” the threshold. A fixed threshold for a particular technique is used in the following simulation. In addition, using a fixed threshold is of great importance in practical systems where real time computation is needed. In general, there is no definite “rule” to select a threshold for edge detection. To compare the

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performance of different edge detection techniques, one reasonable way is to “tune” these techniques such that they produce edge images which have the same percentage of pixels that are classified as edge pixels. There is also no definite “rule” to determine the percentage for real images. This percentage is actually related to the practical problem and determined by a human operator. The test digital images used in the following simulations are the DMR image (512 × 512 pixels × 8 bits) and the house image (256 × 256 pixels × 8 bits). The original DMR and house images are shown in Figs. 2(a) and 3(a). To simulate the slow changes in scene illumination, the original images are modified by darkening gradually from left to right using the following formula: 0.5 + 5.5 cos g(x, ˜ y) = 6

¡π 2

·

x 512

¢ f˜(x, y),

(37)

where f˜(x, y) and g(x, ˜ y) are the gray level functions of the original and darkened images, respectively. The darkened DMR and house images are shown in Figs. 4(a) and 5(a). In the first group of experiments (see Figs. 2 and 3), various parameters for each edge detection technique are tuned so that the resultant image have 10% of their pixels as edge pixels for the DMR and the house image. In the second group of experiments (see Figs. 4 and 5), each technique is applied to the darkened images, using the same setting of parameters as in the first group of experiments. This is required for evaluating the performance evolution of each technique in presence of change in scene illumination. The resulting edge images are shown in Figs. 2(b)–(f), 3(b)–(f), 4(b)–(f) and 5(b)–(f). Tables 1–2 list the figure numbers and the percentage of edge pixels for each resultant image in the second group of experiments. A comparison of these results shows that: (1) In both groups of experiments, the LIP-Sobel based technique can detect edges that correspond to the important features of an image; (2) The edges generated by the LIP-Sobel based technique are far better than those Table 1. Results of the second group of experiments using the darkened DMR image.

Figure 4 Percentage of edge pixels

LIP-Sobel

Canny

Sobel

Johnson

Wallis

(b)

(c)

(d)

(e)

(f)

10.6%

8.8%

7.0%

9.9%

10.8%

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Figure 2. (a) Original DMR image. Edges of DMR image using the five different techniques, (b) LIP-Sobel-based technique, (c) Canny’s technique, (d) Sobel’s technique, (e) Johnson’s technique, and (f) Wallis’ technique.

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Figure 3. (a) Original house image. Edges of house image using the five different techniques, (b) LIP-Sobel-based technique, (c) Canny’s technique, (d) Sobel’s technique, (e) Johnson’s technique, and (f) Wallis’ technique.

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Figure 4. (a) Darkened DMR image. Edges of darkened DMR image using the five different techniques, (b) LIP-Sobel-based technique, (c) Canny’s technique, (d) Sobel’s technique, (e) Johnson’s technique, and (f) Wallis’ technique.

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Figure 5. (a) Darkened house image. Edges of darkened house image using the five different techniques, (b) LIP-Sobel-based technique, (c) Canny’s technique, (d) Sobel’s technique, (e) Johnson’s technique, and (f) Wallis’ technique.

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Table 2. Results of the second group of experiments using the darkened house image.

Figure 5

LIP-Sobel

Canny

Sobel

Johnson

Wallis

(b)

(c)

(d)

(e)

(f)

10.4%

6.4%

8.4%

10.8%

12.9%

Table 3. The noise variance used to simulate different signal to noise ratio and the FoM for Johnson’s (FoM-J) and LIP-Sobel (FoM-L). σn2

625

125

62.5

31.25

12.5

6.25

1

5

10

20

50

100

FoM-J

16

44

70

95

95

96

FoM-L

17

45

77

95

96

96

SNR Percentage of edge pixels

generated by the Wallis’ technique in the both groups of experiments; (3) The Canny’s and classical Sobel’s techniques fail to detect the edges in the poorly lit areas of the darkened images; (4) In the second group of experiments, the LIP-Sobel based technique, together with the Johnson’s technique, generates almost the same edge binary images as those obtained in the first group of experiments. In order to further evaluate the LIP-Sobel operator, the face portion (256 × 256) of the DMR image is used. Shown in Fig. 6(a) is the image with an area being darkened by a multiplication of 0.3 to each pixel value. This example is a simulation of edge detection in a shaded area of a scene. The above two groups of experiments are conducted by using the same procedure. A similar result to that of the above first and second group of experiments has been obtained. It has been observed that while the LIP-Sobel and Johnson’s techniques produce similar edges, all other methods fail to detect edges in the darkened area. The results of the second group of experiment are illustrated by Figs. 6(b)–(f). 5.3.

LIP-Sobel Technique vs. Johnson’s Technique in the Presence of Noise

Since the performances of the LIP-Sobel and Johnson’s techniques are comparable, and since Johnson’s technique is also based on a physical imaging model, it is interesting to compare their performances in noisy images by using Pratt’s figure of merit (FoM) [36]. The figure of merit is defined as: R=

IA 1 1 X , I N i=1 1 + αd12

(38)

where I N = max(Il , I A ), Il and I A represent the number of edge pixels for the ideal case and the experimental case, respectively, α = 1/9 is a scaling factor and di is the distance between the detected edge pixel and the ideal edge pixel. Following Pratt’s [36] setting of experiment, a (64 × 64) image of a step edge is used.

The brightness value of the left part of the step edge image is 150, and that of the right is 175. The independent, identical distribution Gaussian noise is then added to this image. To simulate different signal to noise cases, different values of noise variance are used. The LIP-Sobel together with Johnson’s technique is applied to these noisy images. In each case the thresholding parameter is tuned so that the maximum value of figure of merit is reached. The results are summarised in Table 3, in which the signal to noise ratio is calculated using Pratt’s definition. It can be seen that the performance of the LIP-Sobel is almost the same as that of Johnson’s technique in all settings of signal to noise ratio. Conceptually, it will be easily seen that when noise is multiplicative, the figure-of-merit of the LIP-Sobel will be improved over the one when noise is addictive. This is because, from a signal processing point of view, the LIP based system is in some sense a multiplicative system (see Eq. (35)) and because, from a mathematical point of view, the LIP model is based on the operations + and 1 × . Thus it is more suitable for smoothing out 1 multiplicative noise. A more general discussion of the noise effect will be given in next section. 6.

Discussion

There are two important issues associated with edge detection: the definition of edges and edge detection in the presence of noise. The purpose of this section is to briefly discuss some fundamental problems and solutions associated with these issues. Another purpose is to clarify that the LIP differentiation method can be incorporated into some well-established techniques and that it is not aimed at solving those fundamental problems. 6.1.

Definition of Edges

Since one of the main purposes of edge detection is to extract features of an image, it is critical that a clear

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Figure 6. (a) Partly darkened DMR image. Edges of Fig. 6(a) using the five different techniques, (b) LIP-Sobel-based technique, (c) Canny’s technique, (d) Sobel’s technique, (e) Johnson’s technique, and (f) Wallis’ technique.

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definition of edges or features be made. Although it is generally accepted that edges are areas where abrupt gray level changes occur, it remains unclear what constitutes an “abrupt gray level change”. In general, the concept-edge is defined by a model. If, at one location, the image data fit the model well, then “an edge” is detected. In most edge detection techniques, gray level change is characterised by differentiation and whether the change is “abrupt” or not is determined by a thresholding process. The present study follows the same strategy. However, the LIP-Sobel based edge detection technique involves another differentiation concept (see comments in Section 4.3), which is set up in a solid and physically well-justified mathematical theory. This new technique has been shown robust in the presence of slow change in scene illumination. It is clear that many well-established thresholding techniques, e.g., [9, 10], can then be used after the LIP-Sobel operation. 6.2.

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The second issue is edge detection in the presence of noise. This issue is closely related to the first one. In fact, there are two problems associated with this issue. One problem is the conflicting requirements for selecting a threshold. A low threshold results in a noisy edge image, while a high one will reject noise as well as important features. Techniques that treat edge detection in noisy images as an optimal linear filter design problem have been extensively studied [15, 16, 23, 27, 71–80]. From a mathematical point of view, the differentiation-based edge detection is an ill-posed problem [15]. A regularisation filtering process is needed and one of the proposed optimal regularisation filter is the Gaussian filter [4, 15]. However, as the Gaussian filter is a low pass filter, it will smooth out noise as well as sharp edges. Several researchers have studied the effects of Gaussian filtering on the detected edges [82–84]. Despite a lot of efforts have been put into studying optimal filters for edge detection in noisy images [85], the following problem is still open. This problem, which is fundamental to edge detection, is how to discriminate noise from features without a priori knowledge. According to Marr’s theory, an image should be described in different scales (resolutions) [4]. This leads to extensive study of scale space filtering [86], where edge detection can be performed on images which are results of the Gaussian low pass filtering with different variances (scale parameters). The

Laplacian-of-Gaussian [4] is a typical approach in this research area. Methods that combine edges from different scales have also been studied. Recent development in multirate signal processing and wavelet transformation provides powerful tools for decomposition of an image into different scales [87, 88]. It has been argued by several researches that one can distinguish noise versus signal only in the scale space [89, 90]. In another word, edge detection can be carried out by (1) decomposing an image into a multiresolution representation and (2) using an appropriate edge detection technique. Depending on the requirement of a problem, the detected edges may be from a particular scale or they can be a combination of edges from several scales. From the above discussion, it is clear that the LIP model based differentiation is a valuable building block (the differentiation step) in the above multiscale edge detection techniques. One example is shown in Fig. 7, where Fig. 7(a) is the original “peppers” image, Figs. 7(b) and (c) are the edges extracted by the same LIP-Sobel operator with and without multiscale noise filtering [90, 91] (wavelet denoising), respectively. In the de-noising step, a simple Daubechies-4 filter is used to perform a three-scale decomposition of an image. A simple hard thresholding (using a fixed threshold) is performed on the first two fine scales. The de-noised image is given by using the inverse wavelet transform. In Fig. 7(b), one can easily see a lot of noises (although small in magnitude) in the dark areas have been detected as edges. In Fig. 7(c), it is clear that the LIP-Sobel edge detection together with the multiscale noise filtering results in a satisfactory edge image. 7.

Conclusion

The LIP model notions and structures have been derived to formulate a new class of differential operators for edge detection. The LIP-Sobel based edge detection technique has been introduced and studied. Its theoretical and practical advantages over the traditional techniques of Sobel, Canny, Johnson and Wallis have been shown: (1) The LIP mathematical framework is powerful and physically consistent with the transmittance and multiplicative reflectance image formation models, and with several laws and characteristics of human brightness perception; (2) The LIP-Sobel vector magnitude function corresponds to an intensity image, and thus can be displayed without rescaling process;

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detection technique is robust in slow changing illumination conditions. In summary, the LIP differentiation provides a new alternative to the usual differentiation for edge detection. In the present study, emphasis has been put on a study of its mathematical properties and its advantages in a practical edge detection problem. When general issues such as scale and noise in edge detection are concerned, it has been shown that the LIP differentiation can be a valuable building block in some well established approaches, such as scale space filtering and multiresolution image processing. Appendices Appendix 1 This appendix gives the definition of the Atan function. Let a and b be two arbitrary real numbers. Then, the Atan function is defined using the classical Arctan function as:  µ ¶ b   Arctan if a > 0 and ≥ 0    a    µ ¶    b   +π if a < 0 Arctan   a    µ ¶ b Atan(a, b) = + 2π if a > 0 and b < 0 Arctan    a     π    if a = 0 and b > 0   2     3π   if a = 0 and b < 0 2 This function is valued in [0, 2π ). Appendix 2

Figure 7. (a) Original peppers image, (b) edges extracted by LIPSobel without multiscale noise filtering, (c) edges extracted by LIPSobel with multiscale noise filtering.

(3) The selection of a threshold for edge detection is easier, since the LIP-Sobel vector magnitudes are within a given range; (4) The edges are detected in both well and poorly lit areas of intensity images in a consistent way, which shows that the LIP-Sobel edge

This appendix gives the definition of the gray tone function differentiation. A gray tone function f (x, y) is differentiable at a spatial location (x, y) belonging to the interior set of the spatial support D, if there exists a linear and continuous mapping, denoted f 10 (x, y), defined on the two dimensional real number space R2 and valued in the gray tone space E, such that: 1 − f (x, y) k f (x + vx , y + v y )1 kvkR2 − f 0 (x, y)(vx , v y )kE 1 1

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tends towards zero when the Euclidean norm kvkR2 of the vector v = (vx , v y ) tends towards zero. The mapping f 10 (x, y) is called the derivative of f (x, y) at spatial location (x, y). Appendix 3 This appendix gives the proof of Eq. (33). The proof of Eq. (34) is then straightforward. Starting from Eq. (26) and using the fundamental isomorphism ϕ defined by (6), yields: ¶ µ ¶ µ M − f6 M − f3 − 2M ln ϕ(gx ) = −M ln M M ¶ µ ¶ µ M − f1 M − f9 + M ln − M ln M M ¶ µ ¶ µ M − f7 M − f4 + M ln . + 2M ln M M Then, using Eq. (1) and the properties of the ln(·) function, yields: ! Ã f˜3 f˜62 f˜9 ϕ(gx ) = −M ln . f˜1 f˜42 f˜7 Next, applying the inverse isomorphic transformation ϕ −1 defined by (8), the previous equation becomes: Ã ! f˜3 f˜62 f˜9 −1 gx = ϕ (ϕ(gx )) = M − M . f˜1 f˜42 f˜7 Acknowledgment The authors of this paper are grateful to anonymous reviewers who provided serval suggestions to improve the presentation of this paper. References 1. W.K. Pratt, Digital Image Processing, 2nd edition, John Wiley: New York, 1991. 2. J. Serra, Image Analysis and Mathematical Morphology, Academic Press, 1982. 3. R.D. Duda and P.E. Hart, Pattern Classification and Scene Analysis, John Wiley & Sons: New York, 1973. 4. D. Marr, Vision: A Computational Investigation into the Human Representation and Processing of Visual Information, W.H. Freeman & Co.: San Francisco, 1982.

5. I.E. Gordon, Theories of Visual Perception, John Wiley & Sons: New York, 1989. 6. L.S. Davis, “A survey of edge detection techniques,” Comput. Graph. Image Proc., Vol. 4, pp. 248–270, 1975. 7. G.S. Robinson, “Edge detection by compass gradient masks,” Comput. Graph. Image Proc., Vol. 6, pp. 492–501, 1977. 8. G.B. Shaw, “Local and regional edge detectors: Some comparisons,” Comput. Graph. Image Proc., Vol. 9, pp. 135–149, 1979. 9. T. Peli and D. Malah, “A study of edge detection algorithms,” Comput. Graph. Image Proc., Vol. 20, pp. 1–21, 1982. 10. L. Hertz and R.W. Schafer, “Multilevel thresholding using edge matching,” Comput. Vis., Graph. Image Proc., Vol. 44, pp. 279– 295, 1988. 11. L.J. VanVliet, I.T. Young, and G.L. Beckers, “A nonlinear Laplacian operator as edge detector in noise images,” Comput. Vis., Graph., Image Proc., Vol. 45, pp. 167–195, 1989. 12. I. Defee and Y. Neuvo, “Median-based zero-crossing edge detectors for closely spaced edges,” CVGIP: Graph. Models Image Proc., Vol. 53, pp. 196–203, 1991. 13. S. Venkatesh and L.J. Kitchen, “Edge evaluation using necessary components,” CVGIP: Graph. Models Image Proc., Vol. 54, pp. 23–30, 1992. 14. R.M. Haralick, “Digital step edges for zero-crossings of second directional derivatives,” IEEE Trans. Patt. Anal. Mach. Intell., Vol. PAMI-6, pp. 58–68, 1984. 15. V. Torre and T.A. Poggio, “On edge detection,” IEEE Trans. Patt. Anal. Mach. Intell., Vol. PAMI-8, pp. 147–163, 1986. 16. J. Canny, “A computational approach to edge detection,” IEEE Trans. Patt. Anal. Mach. Intell., Vol. PAMI-8, pp. 679–698, 1986. 17. J.S. Chen and G. Medioni, “Detection, localization, and estimation of edges,” IEEE Trans. Patt. Anal. Mach. Intell., Vol. PAMI11, pp. 191–198, 1989. 18. H.D. Tagare and R.J.P. DeFigueiredo, “On the localization performance measure and optimal edge detection,” IEEE Trans. Patt. Anal. Mach. Intell., Vol. PAMI-12, pp. 1186–1189, 1989. 19. H. Jeong and C.I. Kim, “Adaptative determination of filter scales for edge detection,” IEEE Trans. Patt. Anal. Mach. Intell., Vol. PAMI-14, pp. 579–585, 1992. 20. M.E. Jernigan and R.W. Wardel, “Does the eye contain optimal edge detection mechanisms?,” IEEE Trans. Syst., Man, Cyber., Vol. SMC-11, pp. 441–444, 1981. 21. L. Kitchen and A. Rosenfeld, “Edge evaluation using local coherence,” IEEE Trans. Systems, Man, Cyber., Vol. SMC-11, pp. 597–605, 1981. 22. W.H.H.J. Lunscher and P. Beddoes, “Optimal edge detector evaluation,” IEEE Trans. Syst., Man, Cyber., Vol. SMC-16, pp. 304– 311, 1986. 23. H.L. Tan, S.B. Gelfand, and E.J. Delp, “A comparative cost function approach to edge detection,” IEEE Trans. Syst., Man, Cyber., Vol. SMC-19, pp. 1337–1349, 1989. 24. P.H. Eichel and E.J. Delp, “Quantitative analysis of a momentbased edge operator,” IEEE Trans. Syst., Man, Cyber., Vol. SMC-20, pp. 59–66, 1990. 25. G.N. Khan and D.F. Gillies, “Extracting contours by perceptual grouping,” Image Vision Comput., Vol. 10, pp. 77–88, 1992. 26. S. Jalali and J.F. Boyce, “Determination of optimal general edge detectors by global minimization of a cost function,” Image Vision Comput., Vol. 13, pp. 683–693, 1995.

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16:50

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27. R. Deriche, “Using Canny’s criteria to derive a recursively implemented optimal edge detector,” Int. J. Comput. Vision, Vol. 1, pp. 167–187, 1987. 28. I.J. Cox, J.M. Rehg, and S. Hingorani, “A bayesian multihypothesis approach to edge grouping and contour segmentation,” Int. J. Comput. Vision, Vol. 11, pp. 5–24, 1993. 29. M.P. Dubuinon and A.K. Jain, “Contour extraction of moving objects in complex outdoor scenes,” Int. J. Comput. Vision, Vol. 14, pp. 83–105, 1995. 30. R.A. Gartenberg, F.O. Huck, and R. Narayanswamy, “Image recovery from edge primitives,” J. Opt. Soc. Am., Pt. A, Vol. 7, pp. 898–991, 1990. 31. P. Meer, S. Wang, and H. Wechsler, “Edge detection by associative mapping,” Pattern Recogn., Vol. 22, pp. 491–503, 1989. 32. J.S.P. Shu, “One-pixel-wide edge detection,” Pattern Recogn., Vol. 22, pp. 665–673, 1989. 33. R.P. Johnson, “Contrast-based edge detection,” Pattern Recogn., Vol. 23, pp. 311–318, 1990. 34. A. Kundu, “Robust edge detection,” Pattern Recogn., Vol. 23, pp. 423–440, 1990. 35. R.M. Haralick and J.S.J. Lee, “Context dependant edge detection and evaluation,” Pattern Recogn., Vol. 23, pp. 1–19, 1990. 36. I.E. Abdou and W.K. Pratt, “Quantitative design and evaluation of enhancement/thresholding edge detectors,” in Proc. IEEE, 1979, Vol. 67, pp. 753–763. 37. D. Marr and E. Hildreth, “Theory of edge detection,” in Proc. Royal Soc. London B, 1980, Vol. 207, pp. 187–217. 38. T.G. Stockham, Jr., “Image processing in the context of a visual model,” in Proc. IEEE, 1972, Vol. 60, pp. 828–842. 39. M. Jourlin and J.C. Pinoli, “Logarithmic image processing,” Acta Stereol., Vol. 6, pp. 651–656, 1987. 40. M. Jourlin and J.C. Pinoli, “A model for logarithmic image processing,” J. Microsc., Vol. 149, pp. 21–35, 1988. 41. J.C. Dainty and R. Shaw, Image Science, Academic Press: London, 1974. 42. M. Born and E. Wolf, Principle of Optics, 2nd edition, Pergamon Press: New York, 1980. 43. L.M. Hurwick and D. Jameson, The Perception of Brightness and Darkness, Allyn & Bacon: Boston, 1966. 44. R.C. Gonzelez and P. Wintz, Digital Image Processing, 2nd edition, Addison-Wesley: Reading, 1987. 45. J.C. Pinoli, Contribution a` la mod´elisation, au traitement et a` l’analyse d’image, D.Sc. thesis, Department of Mathematics, University of Saint-Etienne, France 1987. 46. J.C. Pinoli, “Mod´elisation and traitement des images logarithmiques: Th´eorie and applications fondamentales,” Report No. 6, Department of Mathematics, University of Saint-Etienne, France, 1992. 47. J.C. Pinoli, “The logarithmic image processing model connections with human brightness perception and contrast estimators,” Submitted in a first revised form to JMIV. 48. F. Mayet, M. Jourlin, and J.C. Pinoli, “Justifications physiques du mod`ele LIP pour le traitement des images obtenues en lumi`ere transmise,” to appear in Traitement du Signal. 49. G. Deng, “Image and signal processing using the logarithmic image processing model,” Ph.D. thesis, Department of Electronic Engineering, La Trobe University, Australia, 1993. 50. G. Deng and L.W. Cahill, “Image modelling and processing using the logarithmic image processing model,” in Proc. IEEE

51.

52.

53. 54.

55.

56.

57.

58.

59.

60.

61.

62.

63.

64. 65. 66. 67. 68. 69. 70. 71.

179

Workshop on Visual Signal Process. and Com., Melbourne, Australia, 1993, pp. 61–64. M. Jourlin, J.C. Pinoli, and R. Zeboudj, “Contrast definition and contour detection for logarithmic images,” J. Microsc., Vol. 156, pp. 33–40, 1989. J.C. Brailean, B.J. Sullivan, C.T. Chen, and M.L. Giger, “Evaluating the EM algorithm for image processing using a human visual fidelity criterion,” in Proc. ICASSP, 1991, pp. 2957–2960. J.C. Pinoli, “A contrast definition for logarithmic images in the continuous setting,” Acta Stereol., Vol. 10, pp. 85–96, 1991. M. Jourlin and J.C. Pinoli, “Image dynamic range enhancement and stabilization in the context of the logarithmic image processing model,” Signal Process., Vol. 41, pp. 225–237, 1995. G. Deng, L.W. Cahill, and G.R. Tobin, “A study of the logarithmic image processing model and its application to image enhancement,” IEEE Trans. Image Process., Vol. 4, pp. 506– 5122, 1995. G. Deng and L.W. Cahill, “Multiscale image enhancement using the logarithmic image processing model,” Electron. Lett., Vol. 29, pp. 803–804, 1993. J.C. Brailean, D. Little, M.L. Giger, C.T. Chen, and B.J. Sullivan, “Applications of the EM algorithm to radiographic images,” Med. Phys., Vol. 19, pp. 1175–1182, 1992. C. Bron, P. Gremillet, D. Launey, M. Jourlin, H.P. Gautschi, T. B¨achi, and J. Sch¨upbach, “Three-dimensional electron microscopy of entire cells,” J. Microsc., Vol. 157, pp. 115–126, 1990. P. Gremillet, M. Jourlin, C. Bron, J. Sch¨upbach, H.P. Gautschi, and T. B¨achi, “Dedicated image analysis techniques for threedimensional reconstruction from serial sections in electron microscopy,” Mach. Vision. Appli., Vol. 4, pp. 263–270, 1991. P. Corcuff, P. Gremillet, M. Jourlin, Y. Duvault, F. Leroy, and J.L. Leveque, “3D reconstruction of human air by confocal microscopy,” J. Soc. Cosm. Chem., Vol. 44, pp. 1–12, 1993. P. Gremillet, M. Jourlin, and J.C. Pinoli, “LIP model-based three-dimensional reconstruction and visualisation of HIV infected entire cells,” J. Microsc., Vol. 174, pp. 31–38, 1994. B. Roux and R.M. Faure, “Recognition and quantification of clinker phases by image analysis,” Acta Stereol., Vol. 11, pp. 149–154, 1992. H. Konik, B. Laget, and M. Calonnier, “Segmentation d’images par utilisation de pyramides a` bases locales,” Traitement du Signal, Vol. 10, pp. 283–295, 1993. N. Dunford and J.T. Schwartz, Linear Operators, Part I, General Theory, 2nd edition, John Wiley and Sons: New York, 1988. E. Kreyszig, Introductory Functional Analysis with Applications, 2nd edition, John Wiley & Sons: New York, 1989. W.A.J. Luxemburg and A.C. Zaanen, Riesz Spaces, North Holland: Amsterdam, 1971. G. Choquet, Topology, Academic Press, 1966. A.R. Mitchell and D.F. Griffiths, The Finite Difference Method in Partial Differential Equation, John Wiley: New York, 1980. A. Ralston, A First Course in Numerical Analysis, Mac Graw Hill: New York, 1965. R.W. Hornbeck, Numerical Methods, Quantum Publishers Inc.: New York, 1975. K.S. Shanmugam, F.M. Dickey, and J.A. Green, “An optimal frequency domain filter for edge detection in digital pictures,” IEEE Trans. Patt. Anal. Mach. Intellig., Vol. PAMI-1, pp. 37–49, Jan. 1979.

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72. R. Deriche, “Optimal edge detection using recursive filtering,” in Proc. of 1987 IEEE Int. Conf. on Computer Vision, 1987, pp. 501–505. 73. R.A. Boie, I.J. Cox, and P. Rehak, “On optimum edge detection using matched filters,” in Proc. of 1988 IEEE Int. Conf. on Computer Vision and Pattern Recognition, 1986, pp. 100–108. 74. D. Geman, S. Geman, C. Graffigne, and P. Dong, “Boundary detection by constrained optimisation,” IEEE Trans. Patt. Anal. Mach. Intellig., Vol. PAMI-12, pp. 609–628, July 1990. 75. G.R. Dattatreya and L.N. Kanal, “Detection and smoothing of edge contours in images by one- dimensional Kalman techniques,” IEEE Trans. System Man and Cybernetics, Vol. SMC20, pp. 159–165, Jan./Feb. 1990. 76. S.E. Reichenbach, S.K. Park, and R.A. Gartenberg, “Optimal, small kernels for edge detection,” in Proc. of 1990 IEEE Int. Conf. on Pattern Recognition, 1990, pp. 57–63. 77. S. Sarkar and K.L. Boyer, “Optimal, efficient, recursive edge detection filters,” in Proc. of 1990 IEEE Int. Conf. on Pattern Recognition, 1990, pp. 931–936. 78. H.L. Tan, S.B. Gelfand, and E.J. Delp, “A cost minimisation approach to edge detection using simulated annealing,” IEEE Trans. Patt. Anal. Mach. Intellig., Vol. PAMI-14, pp. 3–18, Jan. 1991. 79. O. Monga, R. Deriche, and J.M. Rocchisani, “3D edge detection using recursive filtering: Application to scanner images,” CVGIP: Graphical Models and Image Processing, Vol. 53, pp. 76–87, Jan. 1991. 80. M. Unser, A. Aldroubi, and M. Eden, “Recursive regularisation filters: design, properties, and applications,” IEEE Trans. Patt. Anal. Mach. Intellig., Vol. PAMI-13, pp. 272–277, March 1991. 81. R. Wilson and A.H. Bhalerao, “Kernel designs for efficient multiresolution edge detection and orientation estimation,” IEEE Trans. Patt. Anal. Mach. Intellig., Vol. PAMI-14, pp. 384–389, March 1992. 82. J.J. Clark, “Authenticating edges produced by zero-crossing algorithms,” IEEE Trans. Patt. Anal. Mach. Intellig., Vol. PAMI11, pp. 43–57, Jan. 1989. 83. Y. Lu and R.C. Jain, “Behavior of edges in scale space,” IEEE Trans. Patt. Anal. Mach. Intellig., Vol. PAMI-11, pp. 337–356, April 1989. 84. D.J. Williams and M. Shah, “Normalized edge detector,” in Proc. of 1990 IEEE Int. Conf. on Pattern Recognition, 1990, pp. 942– 946. 85. M.M. Fleck, “Some defects in finite-difference edge finders,” IEEE Trans. Pattern Anal. Mach. Intellig., Vol. PAMI-14, pp. 337–345, March 1992.

86. A. Witkin, “Scale space filtering,” in Proc. Int. Joint Conf. Artificial Intel., Karlsruhe, 1983, pp. 1019–1021. 87. P.P. Vaidyanathan, Multirate Systems and Filter Banks, Englewood Cliffs, NJ: Prentice-Hall, 1993. 88. I. Daubechies, Ten Lecturers on Wavelets, SIAM, 1992. 89. Y.F. Wong, “Nonlinear scale-space filtering and multiresolution system,” IEEE Trans. Image Processing, Vol. 4, pp. 774–787, 1995. 90. R.R. Coifman and M.V. Wickerhauser, “Adapted waveform ‘deNoising’ for medical signals and images,” IEEE Engineering in Medicine and Biology, pp. 578–586, 1995 91. D.L. Donoho, I.M. Jonhnston, G. Kerkyacharian, and D. Picard, “Wavelet Shrinkage: Asymptopia,” J. Roy. Statist. Soc. B, Vol. 140, pp. 301–369, 1993.

Guang Deng received the Ph.D. degree in Electronic Engineering from La Trobe University at Melbourne, Australia in 1994. He is a lecturer in the School of Electronic Engineering, La Trobe University. His research interests include digital signal processing, communications, multiresolution signal representation, and image processing.

Jean-Charles Pinoli received a M.Sc. degree in Mathematics and a D.Sc. degree in Applied Mathematics, both from the University of Saint-Etienne, France. Since 1990, he has been with the Corporate Research Center of the Pechiney Group, Voreppe, France. He is also an associate research director at the Image, Signal and Acoustics Laboratory, CNRS, Lyon, France. His current research is concerned with 2D and 3D image modeling, processing and analysis.