Splines are piecewise polynomial functions that are smoothly connected at points called knots. They are widely used in signal and image processing for interpolation and approximation. Unlike traditional sampling theory based on ideal filters and band-limited functions, splines offer a more practical and efficient approach. They are particularly useful for tasks such as edge detection, geometric transformations, and image processing, where smoothness and accuracy are essential. Michael Unser introduced the concept of splines as a viable alternative to Shannon's sampling theory. While Shannon's theory relies on ideal filters and band-limited functions, splines provide a more flexible and computationally efficient method. They are based on B-splines, which are compactly supported and can be used to construct smooth functions. Splines are also closely related to wavelet theory, which has led to significant advancements in signal and image processing. The key advantage of splines is their ability to provide accurate and efficient interpolation and approximation. They are particularly effective in reducing Gibbs oscillations and improving the quality of signal and image representations. Splines can be used for both interpolation and smoothing, making them versatile tools in signal processing. The use of splines in signal and image processing has been greatly enhanced by the development of efficient algorithms for their implementation. These algorithms include digital filtering techniques that allow for fast and accurate computation of spline-based interpolations. Additionally, the connection between splines and wavelet theory has led to the development of multiresolution analysis and wavelet transforms, which are widely used in image and signal processing. In conclusion, splines provide a powerful and efficient alternative to traditional sampling theory in signal and image processing. They offer a more practical and accurate approach to interpolation and approximation, and their connection to wavelet theory has led to significant advancements in the field. The use of splines in signal and image processing is a testament to their versatility and effectiveness in a wide range of applications.Splines are piecewise polynomial functions that are smoothly connected at points called knots. They are widely used in signal and image processing for interpolation and approximation. Unlike traditional sampling theory based on ideal filters and band-limited functions, splines offer a more practical and efficient approach. They are particularly useful for tasks such as edge detection, geometric transformations, and image processing, where smoothness and accuracy are essential. Michael Unser introduced the concept of splines as a viable alternative to Shannon's sampling theory. While Shannon's theory relies on ideal filters and band-limited functions, splines provide a more flexible and computationally efficient method. They are based on B-splines, which are compactly supported and can be used to construct smooth functions. Splines are also closely related to wavelet theory, which has led to significant advancements in signal and image processing. The key advantage of splines is their ability to provide accurate and efficient interpolation and approximation. They are particularly effective in reducing Gibbs oscillations and improving the quality of signal and image representations. Splines can be used for both interpolation and smoothing, making them versatile tools in signal processing. The use of splines in signal and image processing has been greatly enhanced by the development of efficient algorithms for their implementation. These algorithms include digital filtering techniques that allow for fast and accurate computation of spline-based interpolations. Additionally, the connection between splines and wavelet theory has led to the development of multiresolution analysis and wavelet transforms, which are widely used in image and signal processing. In conclusion, splines provide a powerful and efficient alternative to traditional sampling theory in signal and image processing. They offer a more practical and accurate approach to interpolation and approximation, and their connection to wavelet theory has led to significant advancements in the field. The use of splines in signal and image processing is a testament to their versatility and effectiveness in a wide range of applications.