The proliferation of digital information in our society has enticed a lot of research into data-embedding techniques that add information to digital content, like images, audio, and video. In this paper, we investigate high-capacity lossless data-embedding methods that allow one to embed large amounts of data into digital images (or video) in such a way that the original image can be reconstructed from the watermarked image. We present two new techniques: one based on least significant bit prediction and Sweldens' lifting scheme and another that is an improvement of Tian's technique of difference expansion. The new techniques are then compared with various existing embedding methods by looking at capacity-distortion behavior and capacity control.

A well-known problem in image analysis is the extraction of thin and elongated features such as edges, fractures, fibres, vessels, etc. In many cases, such features are not linear but curved rather than straight. A well-known morphological tool for the extraction of linear features is the opening by straight line segments of a certain length. In this paper we extend this class of morphological filters with openings with narrow structuring elements which are not necessarily straight line segments, but which can be connected paths defined by an adjacency relation.

Classical linear wavelet representations of images have the drawback that they are not well suited to represent edge information. To overcome this problem, nonlinear multiresolution decompositions are being designed that can take into account the characteristics of the input signal/image. In our previous work [(G. Piella et al., July 2002), (H.J.A.M. Heijmans et al., 2002)] we have introduced an adaptive lifting framework that does not require bookkeeping but has the property that it processes edges and homogeneous regions in an image in a different fashion. The current paper discusses the effects of quantization in such adaptive wavelet decomposition. We provide conditions for recovering the original decisions at the synthesis and for relating the reconstruction error to the quantization error. Such an analysis is essential for the application of these adaptive decompositions in image compression algorithms.

We present three variants of a new quality metric for image fusion. The interest of our metrics, which are based on an image quality index recently introduced by Wang and Bovik in [Z. Wang et al., March 2002], lies in the fact that they do not require a ground-truth or reference image. We perform several simulations which show that our metrics are compliant with subjective evaluations and can therefore be used to compare different image fusion methods or to find the best parameters for a given fusion algorithm.

This paper treats a class of adaptive update lifting schemes which do not require bookkeeping for perfect reconstruction. The choice of the update lifting filter is triggered by a binary threshold criterion based on a generalised gradient which is chosen in such a way that it only smoothes homogeneous regions. More precisely, the criterion can be chosen so that it ‘recognises’ signals that are polynomial up to a given order. The lifting filter updates the signal in these polynomial regions but leaves other parts unaffected.

The paper discusses a method for the construction of nonlinear 2D wavelet decompositions using an adaptive update lifting scheme. A very interesting aspect is that the decomposition does not require any bookkeeping, i.e., it is nonredundant, but that it, nevertheless, allows perfect reconstruction. The major ingredient of the construction is the so-called decision map which triggers the choice of the update filter. Another interesting point is the possibility of better preserving edges, even at low resolutions.

This letter treats a class of adaptive update-lifting schemes that do not require bookkeeping for perfect reconstruction. The choice of the update-lifting filter is triggered by a binary threshold criterion based on a generalized gradient that is chosen in such a way that it only smooths homogeneous regions. This criterion can be chosen so that it ignores portions of a signal that are polynomial up to a given order. The update-lifting filter modifies the signal in these polynomial regions but leaves other portions unaffected.

The lifting scheme provides a general and flexible tool for the construction of wavelet decompositions and perfect reconstruction filter banks. We propose an adaptive version of this scheme which has the intriguing property that it allows perfect reconstruction without any overhead cost. We restrict ourselves to the update lifting step which affects the approximation signal only. The update lifting filter is assumed to depend pointwise on the norm of the associated gradient vector, in such a way that a large gradient induces a weak update filter. Thus, sharp transitions in a signal (eg, edges in an image) will not be smoothed to the same extent as regions which are more homogeneous.

Classical morphology is built up in such a way that all operators occur in pairs, e.g., dilation and erosion, opening and closing, etc. This phenomenon, which is a straightforward consequence of the duality principle, often prohibits the construction of tools that treat the foreground and background of signals in exactly the same way. We discuss an alternative framework for morphological image processing that gives rise to image operators which are intrinsically self-dual.

Mathematical morphology is a geometric approach in image processing and analysis with a strong mathematical flavor. Originally, it was developed as a powerful tool for shape analysis in binary and, later, grey-scale images. But it was soon recognized

Interest in multiresolution techniques for signal processing and analysis is increasing steadily. An important instance of such a technique is the so-called pyramid decomposition scheme. This paper presents a general theory for constructing linear as well as nonlinear pyramid decomposition schemes for signal analysis and synthesis. The proposed theory is based on the following ingredients: 1) the pyramid consists of a (finite or infinite) number of levels such that the information content decreases toward higher levels and 2) each step toward a higher level is implemented by an (information-reducing) analysis operator, whereas each step toward a lower level is implemented by an (information-preserving) synthesis operator. One basic assumption is necessary: synthesis followed by analysis yields the identity operator, meaning that no information is lost by these two consecutive steps. Several examples of pyramid decomposition schemes are shown to be instances of the proposed theory: a particular class of linear pyramids, morphological skeleton decompositions, the morphological Haar pyramid, median pyramids, etc. Furthermore, the paper makes a distinction between single-scale and multiscale decomposition schemes, i.e., schemes without or with sample reduction. Finally, the proposed theory provides the foundation of a general approach to constructing nonlinear wavelet decomposition schemes and filter banks.

For pt.I see ibid., vol.9, no.11, p.1862-76 (2000). In its original form, the wavelet transform is a linear tool. However, it has been increasingly recognized that nonlinear extensions are possible. A major impulse to the development of nonlinear wavelet transforms has been given by the introduction of the lifting scheme by Sweldens (1995, 1996, 1998). The aim of this paper, which is a sequel to a previous paper devoted exclusively to the pyramid transform, is to present an axiomatic framework encompassing most existing linear and nonlinear wavelet decompositions. Furthermore, it introduces some, thus far unknown, wavelets based on mathematical morphology, such as the morphological Haar wavelet, both in one and two dimensions. A general and flexible approach for the construction of nonlinear (morphological) wavelets is provided by the lifting scheme. This paper briefly discusses one example, the max-lifting scheme, which has the intriguing property that preserves local maxima in a signal over a range of scales, depending on how local or global these maxima are.

This paper presents a comprehensive discussion on connected morphological operators for binary images. Introducing a connectivity on the underlying space, every image induces a partition of the space in foreground and background components. A connected operator is an operator that coarsens this partition for every input image. A connected operator is called a grain operator if it has the following “local property”: the value of the output image at a given point x is exclusively determined by the zone of the partition of the input image that contains x . Every grain operator is uniquely specified by two grain criteria, one for the foreground and one for the background components. A well-known criterion is the area criterion demanding that the area of a zone is not below a given threshold. The second part of the paper is devoted to connected filters and grain filters. It is shown that alternating sequential filters resulting from grain openings and closings are strong filters and obey a strong absorption property; two properties that do not hold in the classical nonconnected case.

A binary annular filter removes isolated points in the foreground and the background of an image. Here, the adjective "isolated" refers to an underlying adjacency relation between pixels, which may be different for foreground and background pixels. In this paper, annular filters are represented in terms of switch pairs. A switch pair consists of two operators which govern the removal of points from foreground and background, respectively. In the case of annular filters, switch pairs are completely determined by foreground and background adjacency. It is shown that a specific triangular condition in terms of both adjacencies is required to establish idempotence of the resulting annular filter. In the case of translation-invariant operators, an annular filter takes the form X-->(X [plus sign in circle] A) [intersection] X [union or logical sum] (X [minus sign in circle] B), where A and B are structuring elements satisfying some further conditions: when A [intersection] B [intersection] (A [plus sign in circle] B) not equal Ø, it is an (idempotent) morphological filter; when A [union or logical sum] B [subset] A [plus sign in circle] B, it is a strong filter and in this case it can be obtained by composing in either order the annular opening X-->(X [plus sign in circle] A) [intersection] X and the annular closing X--> [union or logical sum] (X [minus sign in circle] B).

We have been developing general multiresolution signal decomposition schemes that unify traditional (linear) approaches and allow use of nonlinear filtering techniques in the decomposition. This paper summarizes our approach and provides several simple examples. Some of these examples are known in the literature (e.g., Haar pyramid and wavelet) and some of them are new (e.g., morphological Haar pyramid and wavelet).

This paper is devoted to similarity and symmetry measures for convex shapes whose definition is based on Minkowski addition and the Brunn-Minkowski inequality. This means, in particular, that these measures are region-based, in contrast to most of the literature, where one considers contour-based measures. All measures considered in this paper are invariant under translations; furthermore, they can be chosen to be invariant under rotations, multiplications, reflections, or the class of affine transformations. It is shown that the mixed volume of a convex polygon and a rotation of another convex polygon over an angle /spl theta/ is a piecewise concave function of /spl theta/. This and other results of a similar nature form the basis for the development of efficient algorithms for the computation of the given measures. Various results obtained in this paper are illustrated by experimental data. Although the paper deals exclusively with the two-dimensional case, many of the theoretical results carry over almost directly to higher-dimensional spaces.

Connected morphological operators act on the level of the flat zones of an image, i.e., the connected regions where the grey-level is constant. For binary images, the flat zones are the foreground and background grains (connected components) of the image. The flat zones constitute a partition of the underlying space. A connected operator is an operator that makes this partition coarser. A grain operator is a connected operator that is uniquely specified by two grain criteria, one for the foreground and one for the background components. This paper provides a discussion of connected operators and grain operators with an emphasis on connected filters, all in the context of binary images.

A morphological filter is an operator on a complete lattice that is increasing and idempotent. Two well-known classes of morphological filters are openings and closings. Furthermore, an interesting class of filters, the alternating sequential filters, is obtained if one composes openings and closings. This paper explains how to construct morphological filters, and derived notions such as overfilters, underfilters, inf-overfilters, and sup-underfilters by composition, the main ingredients being dilations, erosions, openings, and closings. The class of alternating sequential filters is extended by composing overfilters and underfilters. Finally, it is shown that any composition consisting of an equal number of dilations and erosions from an adjunction is a filter. The abstract approach is illustrated with some experimental results.

This paper introduces a reflection symmetry measure for convex polygons based on Minkowski addition. An interesting property of this measure is that it can be computed in O(k/sup 3/) time, where k is the number of edges of the polygon. The proposed algorithm as well as the symmetry measure computation uses the perimetric measure representation of convex polygons.

This paper presents a unifying approach to the problem of morphologically processing image sequences (or, equivalently, vector-valued images) by means of lattice theory, thus providing a mathematical foundation for vector morphology. Lattice theory is an abstract algebraic tool that has been extensively used as a theoretical framework for scalar morphology (i.e., mathematical morphology applied on single images). Two approaches to vector morphology are discussed. According to the first approach, vector morphology is viewed as a natural extension of the well-known scalar morphology. This approach formalizes and generalizes Wilson's matrix morphology and shows that the latter is a direct consequence of marginal vector ordering. The derivation of the second approach is more delicate and requires careful treatment. This approach is a direct consequence of a vector transformation followed by marginal ordering. When the vector transformation is the identity transformation, the two approaches are equivalent. A number of examples demonstrate the applicability of the proposed theory in a number of image processing and analysis problems.

The median operator is a nonlinear (morphological) image transformation which has become very popular because it can suppress noise while preserving the edges. It treats the foreground and background of an image in an identical way that is, it is a self-dual operator. Unfortunately, the median operator lacks the idempotence property: it is not a morphological filter. This paper gives a complete characterization of morphological operators on discrete binary images which are increasing, translation invariant, and self-dual. Furthermore, it presents a general method for the modification of an increasing operator such that it becomes activity-extensive. Such modifications lead to idempotent operators under iteration. The general procedure is illustrated by giving several modifications of the 3/spl times/3 median operator.>

After a brief discussion of the extension of mathematical morphology to complete lattices, the space of gray-level functions is considered and the concept of a threshold set is introduced. It is shown how one can use binary morphological operators and thresholding techniques to build a large class of gray-level morphological operators. Particular attention is given to the class of so-called flat operators, i.e. operators which commute with thresholding. It is also shown how to define dilations and erosions with nonflat structuring elements if the gray-level set is finite. It is reported that mere truncation yields wrong results.>

The construction of morphological filters, i.e., morphological operators which are increasing and idempo- tent, is an important problem in mathematical morphology. In this paper it is described how one can construct openings and closings and other morphological filters by iteration. It turns out that a sort of order continuity is required to guarantee that iteration leads to idempotence. An important class of operators which yield morphological filters after iteration are the so-called pointwise monotone operators. In the last section the abstract theory is illustrated by a number of concrete examples.