A.D. Calway, The Multiresolution Fourier Transform: A General Purpose Tool for Image Analysis, Ph.D. Thesis, Warwick Univ., 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... an implementation of the popular Canny edge detector [16] 2 A Two stage, Region Boundary Model for Segmentation The image model is derived from one quad tree image model introduced in [27] forms of which have been applied in the areas of image estimation [28] coding [29] and curve extraction [30]. The image can be visualised as being built up by successive approximation through scale, beginning with a coarse description at a low resolution and progressively refining it through scale. To introduce multiple regions, however, a more complex scheme is necessary: images are realised by a ....

A. Calway. The Multiresolution Fourier Transform: A general Purpose Tool for Image Analysis. PhD thesis, Department of Computer Science, The University of Warwick, UK, September 1989. 26

....of all appropriate discrete coordinates. A linear feature in the image can then be found by looking for peaks in these counter values. 5 1.2.2 Frequency spectrum method A different way in which parameterised features can be extracted from an image is by analysing its frequency spectrum. Calway [5] and Davies [7] have used a method of extracting linear features directly from the Fourier spectrum of an image. Given that an image contains a single linear feature, certain properties of the spectrum are known: 1) the magnitude spectrum will have a preferred orientation, perpendicular to the ....

....domain. One of the reasons for this is that in the video coding process the feature extraction needs to be combined with other aspects like the affine motion estimation and texture analysis. All these aspects are preferably dealt with within one unified framework and it has been established [5] that the Multiresolution Fourier Transform (MFT) is a suitable frequency domain image representation to act as a basis for this framework. This section will discuss some much used frequency domain image representations, and compare them to the MFT. The standard frequency domain representation is ....

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Andrew Calway. The Multiresolution Fourier Transform: A general Purpose Tool for Image Analysis. PhD thesis, Department of Computer Science, The University of Warwick, UK, September 1989.

....linear and translational components of T 12 . Bearing in mind the need for a suitable scale for the texture patches, a multiresolution windowed FT, the MFT, is used as the representation. Recall that a given level l of the MFT may be described as a set of estimated local spectra and is defined as [7] [11] s( i (l) j (l) l) X k w l ( i (l) Gamma k )s( k ) exp[ Gamma j (l) k ] 12) where 2 = Gamma1 and, for notational simplicity, the spatial and scale coordinates will be suppressed except where required in the sequel. An MFT block at a given ....

....l representing a local Fourier spectrum is first selected as the prototype at that scale. The affine transform of the prototype to each 11 of the remaining MFT blocks is identified. On the basis of the identified transform, the original texture image can be resynthesised. As in Calway s work [7], the dominance of low frequencies is countered by a pre emphasis step. In the present work, 4 lowpass coefficients per block are used to represent low frequencies; these are removed from the image before analysis and added after synthesis. The implementation consists of three main components: ....

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A. D. Calway. The Multiresolution Fourier Transform: A general Purpose Tool for Image Analysis. PhD thesis, Department of Computer Science, The University of Warwick, UK, September 1989.

.... likely to contain more than one motion [134, 16] The above problem is often referred to as the generalised aperture problem [16, 127] and represents a manifestation of the uncertainty principle in image analysis which has been investigated by a number of workers for different aspects of the field [147, 149, 121, 25, 14, 6, 107]. In terms of motion analysis the uncertainty principle can be thought of as asserting that joint localisation in motion space and in region space cannot be achieved, ie it is not possible to have simultaneously high spatial resolution and accurate motion estimation. It can be argued that all the ....

....be explained using a single motion hypothesis. Similar observations have been made by a number of others [127, 16] The uncertainty principle is fundamental to image analysis in general, and multiresolution techniques have been proposed to address its effect in a range of image analysis tasks [6, 14, 147, 25], including motion analysis [40, 17] It seems intuitively obvious that multiresolution techniques can be beneficial to motion analysis image sequences are characterised by motion events over a range of spatial resolutions, ie multiple objects of different sizes moving with different motions, so ....

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A. D. Calway. The multiresolution Fourier transform: a general purpose tool for image analysis. PhD thesis, University of Warwick, UK, September 1989. BIBLIOGRAPHY 155

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A.D. Calway, The Multiresolution Fourier Transform: A General Purpose Tool for Image Analysis, Ph.D. Thesis, Warwick Univ., 1989.

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Calway, A.D. 1989, The Multiresolution Fourier Transform: A General Purpose Tool for Image Analysis, Ph.D. Thesis, Warwick University.

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A.D. Calway,The Multiresolution Fourier Transform: A General Purpose Tool for Image Analysis, Ph.D. Thesis, Warwick Univ., 1989.

....to elementary geomatric transformation, such as rotation, scaling and translation, with the capability of capturing both local and global texture information simultaneously. Towards this end, a new model for texture feature representation based on the multiresolution Fourier transform (MFT) [5] is described. The MFT is a linear transform providing local frequency estimation over multiple spatial resolutions and is thus ideally suited to the analysis of local frequency properties of textural elements at different scales. Local spectra within the MFT corresponding to image regions at ....

A. D. Calway. The Multiresolution Fourier Transform: A general Purpose Tool for Image Analysis. PhD thesis, Department of Computer Science, The University of Warwick, UK, September 1989.

....reflect the orientation properties of local regions. This has led to several investigations into such representations, particularly within the traditional form of the WT [15] 16] In a similar manner, given that the previous implementation of a 2 d MFT was based on a cartesian separable scheme [11], it is pertinent to inquire whether orientation selectivity might also be incorporated within an MFT framework, while still retaining the advantages of generalisation mentioned above. The purpose of this report is to describe one such class of MFT. Orientation selectivity is incorporated by ....

.... and the number of spatial samples are constrained to be a power of 2, each of the bandpass filtering operations required to calculate the transform coefficients can be implemented using a separable radix 2 FFT algorithm in a similar manner to that adopted for the cartesian separable case [11]. 3.3 An Example The purpose of this section is to present an example of an orientation selective MFT based on the class of transforms described above. Suitable values for the various parameters involved are discussed and examples given of the associated 2 d window functions. Examples of the ....

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A.D. Calway, The Multiresolution Fourier Transform: A General Purpose Tool for Image Analysis, Ph.D. Thesis, Warwick Univ., 1989.

....centered around i and k in frames m and m 1 respectively, at scale n (cf. equation 23) To implement the above algorithm we need local spectral estimates over different sized regions. An ideal tool for this is the Multiresolution Fourier Transform (MFT) extensively described by Calway [24] and Wilson et al. 25] The MFT of a discrete 2 D image x( i ) at scale oe n , frequency j and spatial position i is defined as x( i ; j ; oe n ) X k w n ( k Gamma i )x( k )e Gamma k Delta j (20) Estimation Motion Estimation Motion Motion ....

A. D. Calway, The Multiresolution Fourier Transform: A General Purpose Tool for Image Analysis. PhD thesis, University of Warwick, UK, September 1989.

....of a least squares minimisation problem over all spatial frequencies and is shown to be equivalent to calculating spatial correlations via the frequency domain. Moreover, by defining the scheme within the framework of a generalised wavelet transform, the multiresolution Fourier transform (MFT) [7, 8], the analysis can be based upon local spatial regions over a range of sizes and so enable a fast coarse to fine matching strategy to be adopted. This approach avoids the problem of tying disparity to scale and removes the dependence on a known centre frequency associated with phase differencing. ....

.... local correlations to be dependent upon a scale parameter oe, ie ae( 1 ; 2 ; oe; d) 1 4 2 Z 1 Gamma1 xL ( 1 ; oe)x R ( 2 ; oe) exp[ d] d (8) where the local spectra are now scale dependent and correspond to multiresolution Fourier transforms (MFT) [8, 7] x( oe) oe Z 1 Gamma1 w(oe( Gamma ) x( exp[ Gamma : d (9) ie a stack of windowed FTs in which the locality of the spectral estimates is varied as a function of oe. Using (8) it is therefore possible to derive local correlation fields, and thus disparity ....

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A.D. Calway, The Multiresolution Fourier Transform: A General Purpose Tool for Image Analysis, Ph.D. Thesis, Warwick University, 1989.

....assumes the image to consist of a set of line and edge features defined at different spatial resolutions. The estimator is implemented by making use of the multiresolution Fourier transform (MFT) 3] an invertible transform that has been shown to be useful for a variety of image analysis problems [4, 5]. A curve representation is defined within the image model and a hierarchical extraction process is used to identify the curves. Results are presented for a natural image. Presented at the IEEE Int. Conf. on Acoustics, Speech and Signal Processing, Albuquerque, NM, 1990 2 A Multiresolution ....

....estimation of three parameters for each feature: its scale index n, its orientation nxy and its offset j nxy . These are obtained from the original image by using a combined estimation decision procedure. A brief outline of the scheme is provided here, a more detailed description can be found in [4, 5]. 3.1 Local Feature Modelling and Estimation The estimation at a given scale index n is based upon a frequency domain model of oriented, localised features such as lines and edges. These features are characterised by a concentration of energy in the Fourier transform along a line passing through ....

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A.Calway, The Multiresolution Fourier Transform: A General Purpose Tool for Image Analysis, University of Warwick Ph.D Thesis, 1989.

....by a node. image data. This paper presents a summary of the basic properties of the MFT and the associated modelling techniques, which will illustrate its effectiveness in capturing the image information which is important to viewers. Fuller descriptions of the work can be found in the references [7, 6, 10]. 2 IMAGE MODELLING BASED ON THE MFT In order to model image features such as boundary contours and textures, two aspects of structure have to be captured: local structure relating the luminance at one pixel to that of its neighbours and global structure, which typically reflects the large scale ....

....defining the basis vectors for the patch and v i is a vector of white noise samples. The matrix B i will typically be parameterised to such properties as the orientation and location of the features in the patch. 3. The scale selection model is a binary Markov process defined on the quadtree ([7, 8, 9]) which is defined by the conditional probabilities P (i; j; kji=2; j=2; k Gamma 1) which are the probabilities that a given node (i; j; k) in the tree is a leaf node, given the state of its father at scale k Gamma 1, the node covering four siblings at level k. In general, the image ....

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A.D. Calway, The Multiresolution Fourier Transform: A General Purpose Tool for Image Analysis, Ph.D. Thesis, Univ. of Warwick, Ph.D. Thesis, 1989.

....these limitations. With this in mind, the Image and Signal Processing Group at the University of Warwick has developed a novel form of WT in which the link between scale and frequency is removed, and which provides a degree of shift invariance. Known as the Multiresolution Fourier Transform (MFT) [4][5] this resembles a stack of windowed Fourier transforms (WFT) 6] in which the window size is varied systematically to give a multiresolution representation of the spacefrequency plane. As such, it constitutes a superset of the WT and WFT, providing a complete representation of the frequency ....

....MFT is invertible if the window is chosen carefully, although unlike the former there are Department of Computer Science, University of Warwick. several ways in which the original signal can be recovered, either from the coefficients at a given scale or via some form of multiple scale approach [4][5] When dealing with discrete signals, the three coordinate axes , and oe are sampled at appropriate intervals, the latter usually at powers of 2 in much the same way as the standard WT. The sampling in position and frequency follows that employed in a discrete WFT [6] with the intervals in ....

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A.D. Calway, The Multiresolution Fourier Transform: A General Purpose Tool for Image Analysis, Ph.D. Thesis, Warwick Univ., 1989.

....from the disparity field D( oe m ) defined at scale oe m . Practical implementation of the scheme is based upon a discrete version of the multiresolution Fourier transform (MFT) a linear transform providing Gabor like representations such as that defined in eqn (3) over multiple scales (Calway 1989, Wilson et al. 1992) Each level of the discrete transform consists of sampled bandpass versions of the original image at uniformly distributed frequencies. To optimize computation, the scale parameter is varied by a factor of 2 between levels and the sampling intervals in each domain are ....

....are assigned frequency estimates from each bandpass image corresponding to the underlying spatial region (Fig. 1) The window functions used are based on cartesian products of bandlimited prolate spheroidal sequences and each level is computed using standard fast Fourier transform (FFT) techniques (Calway 1989). Discrete MFTs of each binocular image are generated and disparity estimates obtained using a discrete version of the focusing algorithm defined within the quadtree framework. Estimates based on corresponding nodes at some high level are used to select which nodes are compared at the next level ....

Calway, A.D. 1989, The Multiresolution Fourier Transform: A General Purpose Tool for Image Analysis, Ph.D. Thesis, Warwick University.

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Andrew Calway. The Multiresolution Fourier Transform: A General Purpose Tool for Image Analysis. PhD thesis, Department of Computer Science, The University of Warwick, September 1989.

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