Cubic splines for image interpolation and digital filtering
01 Dec 1978-IEEE Transactions on Acoustics, Speech, and Signal Processing (IEEE)-Vol. 26, Iss: 6, pp 508-517
TL;DR: Applications to image and signal processing include interpolation, smoothing, filtering, enlargement, and reduction, and experimental results are presented for illustrative purposes in two-dimensional image format.
Abstract: This paper presents the use of B-splines as a tool in various digital signal processing applications. The theory of B-splines is briefly reviewed, followed by discussions on B-spline interpolation and B-spline filtering. Computer implementation using both an efficient software viewpoint and a hardware method are discussed. Finally, experimental results are presented for illustrative purposes in two-dimensional image format. Applications to image and signal processing include interpolation, smoothing, filtering, enlargement, and reduction.
Citations
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TL;DR: A review of recent as well as classic image registration methods to provide a comprehensive reference source for the researchers involved in image registration, regardless of particular application areas.
6,842 citations
Cites background from "Cubic splines for image interpolati..."
...6), quadratic splines [42,191], cubic Bsplines [89], higher-order B-splines [108], Catmull–Rom cardinal splines [100,184], Gaussians [7], and truncated sinc functions [182] belong to the most commonly used interpolants....
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TL;DR: This paper presents a new approach to single-image superresolution, based upon sparse signal representation, which generates high-resolution images that are competitive or even superior in quality to images produced by other similar SR methods.
Abstract: This paper presents a new approach to single-image superresolution, based upon sparse signal representation. Research on image statistics suggests that image patches can be well-represented as a sparse linear combination of elements from an appropriately chosen over-complete dictionary. Inspired by this observation, we seek a sparse representation for each patch of the low-resolution input, and then use the coefficients of this representation to generate the high-resolution output. Theoretical results from compressed sensing suggest that under mild conditions, the sparse representation can be correctly recovered from the downsampled signals. By jointly training two dictionaries for the low- and high-resolution image patches, we can enforce the similarity of sparse representations between the low-resolution and high-resolution image patch pair with respect to their own dictionaries. Therefore, the sparse representation of a low-resolution image patch can be applied with the high-resolution image patch dictionary to generate a high-resolution image patch. The learned dictionary pair is a more compact representation of the patch pairs, compared to previous approaches, which simply sample a large amount of image patch pairs , reducing the computational cost substantially. The effectiveness of such a sparsity prior is demonstrated for both general image super-resolution (SR) and the special case of face hallucination. In both cases, our algorithm generates high-resolution images that are competitive or even superior in quality to images produced by other similar SR methods. In addition, the local sparse modeling of our approach is naturally robust to noise, and therefore the proposed algorithm can handle SR with noisy inputs in a more unified framework.
4,958 citations
Cites methods from "Cubic splines for image interpolati..."
...Ano ther class of SR approach is based on interpolation [29], [6], [27]....
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...We compare our algorithm with bicubic interpolation [29] and back-projection [15]....
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TL;DR: It can be shown that the order of accuracy of the cubic convolution method is between that of linear interpolation and that of cubic splines.
Abstract: Cubic convolution interpolation is a new technique for resampling discrete data. It has a number of desirable features which make it useful for image processing. The technique can be performed efficiently on a digital computer. The cubic convolution interpolation function converges uniformly to the function being interpolated as the sampling increment approaches zero. With the appropriate boundary conditions and constraints on the interpolation kernel, it can be shown that the order of accuracy of the cubic convolution method is between that of linear interpolation and that of cubic splines. A one-dimensional interpolation function is derived in this paper. A separable extension of this algorithm to two dimensions is applied to image data.
3,280 citations
24 Jun 2010
TL;DR: This paper deals with the single image scale-up problem using sparse-representation modeling, and assumes a local Sparse-Land model on image patches, serving as regularization, to recover an original image from its blurred and down-scaled noisy version.
Abstract: This paper deals with the single image scale-up problem using sparse-representation modeling. The goal is to recover an original image from its blurred and down-scaled noisy version. Since this problem is highly ill-posed, a prior is needed in order to regularize it. The literature offers various ways to address this problem, ranging from simple linear space-invariant interpolation schemes (e.g., bicubic interpolation), to spatially-adaptive and non-linear filters of various sorts. We embark from a recently-proposed successful algorithm by Yang et. al. [1,2], and similarly assume a local Sparse-Land model on image patches, serving as regularization. Several important modifications to the above-mentioned solution are introduced, and are shown to lead to improved results. These modifications include a major simplification of the overall process both in terms of the computational complexity and the algorithm architecture, using a different training approach for the dictionary-pair, and introducing the ability to operate without a training-set by boot-strapping the scale-up task from the given low-resolution image. We demonstrate the results on true images, showing both visual and PSNR improvements.
2,667 citations
TL;DR: The article provides arguments in favor of an alternative approach that uses splines, which is equally justifiable on a theoretical basis, and which offers many practical advantages, and brings out the connection with the multiresolution theory of the wavelet transform.
Abstract: The article provides arguments in favor of an alternative approach that uses splines, which is equally justifiable on a theoretical basis, and which offers many practical advantages. To reassure the reader who may be afraid to enter new territory, it is emphasized that one is not losing anything because the traditional theory is retained as a particular case (i.e., a spline of infinite degree). The basic computational tools are also familiar to a signal processing audience (filters and recursive algorithms), even though their use in the present context is less conventional. The article also brings out the connection with the multiresolution theory of the wavelet transform. This article attempts to fulfil three goals. The first is to provide a tutorial on splines that is geared to a signal processing audience. The second is to gather all their important properties and provide an overview of the mathematical and computational tools available; i.e., a road map for the practitioner with references to the appropriate literature. The third goal is to give a review of the primary applications of splines in signal and image processing.
1,732 citations
Cites methods from "Cubic splines for image interpolati..."
...The use of cubic splines in image processing was pioneered by Hou and Andrews [36]....
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References
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01 Jan 1949
TL;DR: A method is developed for representing any communication system geometrically and a number of results in communication theory are deduced concerning expansion and compression of bandwidth and the threshold effect.
Abstract: A method is developed for representing any communication system geometrically Messages and the corresponding signals are points in two "function spaces," and the modulation process is a mapping of one space into the other Using this representation, a number of results in communication theory are deduced concerning expansion and compression of bandwidth and the threshold effect Formulas are found for the maximum rate of transmission of binary digits over a system when the signal is perturbed by various types of noise Some of the properties of "ideal" systems which transmit at this maximum rate are discussed The equivalent number of binary digits per second for certain information sources is calculated
6,712 citations
Book•
01 Jan 1968
TL;DR: In this paper, Spectral Analysis and its Applications, the authors present a set of applications of spectral analysis and its application in the field of spectroscopy, including the following:
Abstract: (1970). Spectral Analysis and its Applications. Technometrics: Vol. 12, No. 1, pp. 174-175.
4,220 citations
"Cubic splines for image interpolati..." refers background in this paper
...The window function Wo(h) is known as a Parzen window [ 27 ] which is nonEegative, has finite second moment, and it can be shown that Wo(o) + 0 as Hence, (19) becomes...
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...Furthermore, as compared with other existing window filters, the Parzen window filter is optimum in the sense that the power spectrum of f($) has the smallest variance [ 27 ] for a random sequence {ck} as shown in (21)....
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01 Jan 1915
TL;DR: In this article, the authors consider a function of a variable x such that its Taylor expansion in any part of the plane of the complex variable x can be derived from its Taylor's expansion in another part by the process of analytic continuation.
Abstract: Let ƒ(x) be a given function of a variable x. We shall suppose that ƒ(x) is a one-valued analytic function, so that its Taylor's expansion in any part of the plane of the complex variable x can be derived from its Taylor's expansion in any other part of the plane by the process of analytic continuation.
753 citations
01 Jun 1973
TL;DR: In this article, the authors examined the relative merits of finite-duration impulse response (FIR) and infinite duration impulse response(IIR) digital filters as interpolation filters and showed that FIR filters are generally to be preferred for interpolation.
Abstract: In many digital signal precessing systems, e.g., vacoders, modulation systems, and digital waveform coding systems, it is necessary to alter the sampling rate of a digital signal Thus it is of considerable interest to examine the problem of interpolation of bandlimited signals from the viewpoint of digital signal processing. A frequency dmnain interpretation of the interpolation process, through which it is clear that interpolation is fundamentally a linear filtering process, is presented, An examination of the relative merits of finite duration impulse response (FIR) and infinite duration impulse response (IIR) digital filters as interpolation filters indicates that FIR filters are generally to be preferred for interpolation. It is shown that linear interpolation and classical polynomial interpolation correspond to the use of the FIR interpolation filter. The use of classical interpolation methods in signal processing applications is illustrated by a discussion of FIR interpolation filters derived from the Lagrange interpolation formula. The limitations of these filters lead us to a consideration of optimum FIR filters for interpolation that can be designed using linear programming techniques. Examples are presented to illustrate the significant improvements that are obtained using the optimum filters.
643 citations