The article discusses the use of splines in signal and image processing, highlighting their advantages over traditional methods based on Shannon's sampling theory. Splines are piecewise polynomials that offer a more flexible and efficient approach to signal representation and processing. The article covers several key aspects of spline theory, including: 1. **Spline Interpolation**: Splines are used to interpolate data points, with cubic splines being particularly popular due to their minimum curvature property. The article provides a tutorial on spline interpolation, including the use of B-splines and digital filtering techniques. 2. **Cardinal Splines**: Cardinal splines are introduced as the spline analogs of the sinc function, providing a conceptual understanding of the spline interpolation process. They are useful for understanding the relationship between spline interpolation and band-limited functions. 3. **Spline Sampling Theory**: The article explains how spline sampling can be achieved using appropriate prefiltering and sampling, similar to Shannon's sampling theory. It discusses the error bounds and the role of the prefilter in controlling the approximation error. 4. **Multiresolution Spline Processing**: Splines are shown to have multiresolution properties, making them suitable for constructing wavelets and pyramids. The article details the two-scale relation for splines and how it can be used to create efficient algorithms for signal processing. 5. **Spline Wavelets**: Splines are discussed as the only wavelets with a closed-form formula, providing a more efficient and non-redundant representation of signals compared to other wavelet bases. 6. **Optimality Properties**: The article highlights the extremal properties of splines, such as the minimum curvature property and the best approximation properties among wavelets. It also discusses smoothing splines and their relation to wavelet denoising techniques. 7. **Applications**: The article provides an overview of various signal and image processing applications that benefit from the use of splines, including image interpolation, geometric transformations, and wavelet analysis. Overall, the article emphasizes the practical advantages of using splines in signal and image processing, particularly in terms of computational efficiency, flexibility, and theoretical robustness.The article discusses the use of splines in signal and image processing, highlighting their advantages over traditional methods based on Shannon's sampling theory. Splines are piecewise polynomials that offer a more flexible and efficient approach to signal representation and processing. The article covers several key aspects of spline theory, including: 1. **Spline Interpolation**: Splines are used to interpolate data points, with cubic splines being particularly popular due to their minimum curvature property. The article provides a tutorial on spline interpolation, including the use of B-splines and digital filtering techniques. 2. **Cardinal Splines**: Cardinal splines are introduced as the spline analogs of the sinc function, providing a conceptual understanding of the spline interpolation process. They are useful for understanding the relationship between spline interpolation and band-limited functions. 3. **Spline Sampling Theory**: The article explains how spline sampling can be achieved using appropriate prefiltering and sampling, similar to Shannon's sampling theory. It discusses the error bounds and the role of the prefilter in controlling the approximation error. 4. **Multiresolution Spline Processing**: Splines are shown to have multiresolution properties, making them suitable for constructing wavelets and pyramids. The article details the two-scale relation for splines and how it can be used to create efficient algorithms for signal processing. 5. **Spline Wavelets**: Splines are discussed as the only wavelets with a closed-form formula, providing a more efficient and non-redundant representation of signals compared to other wavelet bases. 6. **Optimality Properties**: The article highlights the extremal properties of splines, such as the minimum curvature property and the best approximation properties among wavelets. It also discusses smoothing splines and their relation to wavelet denoising techniques. 7. **Applications**: The article provides an overview of various signal and image processing applications that benefit from the use of splines, including image interpolation, geometric transformations, and wavelet analysis. Overall, the article emphasizes the practical advantages of using splines in signal and image processing, particularly in terms of computational efficiency, flexibility, and theoretical robustness.