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.

/pdf/splines-a-perfect-fit-for-signal-and-image-processing-50x3vrh3wj.pdf

Splines: a perfect fit for signal and image processing

By interpreting the Green-function reproduction property of exponential splines in signal processing terms, we uncover a fundamental relation that connects the impulse responses of allpole analog filters to their discrete counterparts. The link is that the latter are the B-spline coefficients of the former (which happen to be exponential splines). Motivated by this observation, we introduce an extended family of cardinal splines-the generalized E-splines-to generalize the concept for all convolution operators with rational transfer functions. We construct the corresponding compactly supported B-spline basis functions, which are characterized by their poles and zeros, thereby establishing an interesting connection with analog filter design techniques. We investigate the properties of these new B-splines and present the corresponding signal processing calculus, which allows us to perform continuous-time operations, such as convolution, differential operators, and modulation, by simple application of the discrete version of these operators in the B-spline domain. In particular, we show how the formalism can be used to obtain exact, discrete implementations of analog filters. Finally, we apply our results to the design of hybrid signal processing systems that rely on digital filtering to compensate for the nonideal characteristics of real-world analog-to-digital (A-to-D) and D-to-A conversion systems.

/pdf/cardinal-exponential-splines-part-ii-think-analog-act-4d6bgwjvvr.pdf

Cardinal exponential splines: part II - think analog, act digital

A data interpolating system by which the operating quantity can be decreased and no truncation error is produced. A data processor comprises a discrete value extracting section (10), a sampling function operating section (20), and a convolution operating section (30) so as to perform interpolation between discrete values. The discrete value extracting section (10) extracts just previous four values from successively inputted discrete data, and the sampling function operating section (20) calculates, when the data interpolating position is specified, the value of the interpolating position based on the distances between the data interpolating position and the discrete values by using a sampling function of finite support which can be differentiated only once over the whole region. The convolution operating section (30) multiplies the values of the four sampling functions by the discrete values, and adds up the products, thus performing convolution operation and outputting the interpolation value.

Data interpolating system

A digital-to-analog converter for producing less distorted output waveforms without the need for an increase in the operating speed of components. A D/A converter (1) comprises a memory (10), an address counter (12), a B spline function generator circuit (14), four sampling function generator circuits (16), three delay circuits (18), four amplifiers (20), and three summing circuits (22, 24 and 26). Four items of digital data supplied one after another in the predetermined time interval T are stored in the memory (10). The gain of each of the four amplifiers (20) is set according to the corresponding digital data. The four sampling function generator circuits (16) generate signal waveforms of the sampling function, which appear individually in points a time T away from one another. The signal waveforms are amplified in the amplifier (20) and summed to produce an analog signal associated with the interpolation value.

Digital to analog converter

A sampling function generator capable of providing continuous output corresponding to a sampling function. The sampling function generator (1) comprises a B-spline function generator circuit (10), delay circuit (12 and 14), inverting amplifiers (16 and 18), and summing circuits (20 and 22). The B-spline function generator circuit (10) continuously produces signal waveforms according to a third-order B-spline function. After delayed by predetermined time or attenuated to 1/4 amplitude and inverted, the signal waveforms are combined in the summing circuits (20 and 22) to form a series of signal waveforms of a sampling function that is differentiable once over the range and has finite values.

Sampling function generator

A simple system is proposed which generates a quadratic spline function interpolating a given discrete-time signal. It works as a D-A converter which is free from phase distortions and whose output is smooth. The output is constructed as two-fold integration of a staircase signal. A kind of sampled data feedback is employed to stabilize the system. Its circuit is mainly composed of a pair of D-A and A-D converter chips and two analogue integrators. This hardware complexity is about a quarter of current systems.

Quadratic spline interpolator

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Quadratic spline interpolator