Image enlarging and image segmentation are two fundamental problems in the field of image processing. The main purpose of image enlarging is to let the enlarged image reflect better shape of the original scene, especially the information of the details. An image is a sample of an original scene. The more points sampled, the more details of the original scene will be shown. If we can reconstruct the visible part of the original scene based on the image, and resample the scene, a more clearly enlarged image with more details can be reconstructed. So the problem of image enlarging can be seen as a problem of reconstruct the original scene. Image segmentation is to segment the different part of an image. In image 3D modeling we need curve fitting techniques to join the separate boarders into a continuous boarder. This paper uses image enlarging and image segmentation as the thread to study the curve and surface fitting problem, which is one of the key problems in the field of image processing, computer aided geometric design, computer graphics, computer animation and digital content and so on. This problem is also difficult to solve and has extensive application background in practical work.Focusing on the geometric problems in the field of image enlarging and image segmentation, this paper uses problems based on surface fitting and parameterization of data points as a starting point. We give new theories and methods to four of the problems. Concrete results and innovative research work are as follows:First, we proposed a method for image resizing by piecewise quadratic polynomial with edge constraints. Generally speaking, using methods based on quadratic fitting surfaces, such as Lagrange, spline curves, least squares, etc. to process an image, the edges will usually be blurred. Our experiments show that when using edge information as a constraint to construct polynomial surfaces to resize an image, the enlarged image can have better edge features and higher visual quality. This paper proposes a new method of constructing fitting surfaces to resize an image. Associated with each pixel, a fitting patch is constructed using edge information as a constraint. The fitting patch has the shape suggested by the pixel and its adjacent pixels, and it also has the quadratic polynomial precision. To compute the pixels of the image to be enlarged, we first sample from four fitting patches and obtain four pixels, the enlarged pixel are generated by a weighted combination of the four pixels. The main innovation is proposed that the original scene can be approximated by piecewise polynomial patches. Sampled from different patches, multiple pixels can be obtained, and the enlarged image pixels are computed by the weighted combination of the pixels. Due to the different forms of weighted functions can be used, the generation of enlarged pixels can have higher flexibility. The advantages of the new method are that it is easy to compute and produces the enlarged image with better visual quality and higher PSNR value.Second, we proposed a method of enlarging image by constrained least square approach with shape preserving. From a visual point of view, the visual quality of an image is mainly determined by its edges. Conventional polynomial interpolation of image enlarging methods would produce blurred edges, while edge-directed interpolation based methods would cause distortion in the non-edge areas. This paper proposed a new method for image enlarging. We first construct a surface to interpolate the image data. To remove the zigzagging artifact, for each pixel, a fitting patch is constructed using edge information as constraints. The weighted combination of all the patches forms the interpolation surface which has the shape suggested by the data points. Each point on the interpolation surface can be regarded as a sampling point taken from a unit square domain, which means that when the interpolation surface is used to enlarge the image, each sampling domain of the enlarged pixels is also a unit square, causing the enlarged image to lose some details. To make the enlarged image keep the details as many as possible, the sampling domain of the enlarged pixels should be less than a unit square. So in the second step, for each pixel to be enlarged, nine pixels are computed by the interpolation surface, and new pixels are computed by these nine pixels with constrained optimization methods. The size of the sampling domain of the enlarged pixels is inversely proportional to the size of the enlarged image. The enlarged image by the new method has a quadratic polynomial precision. Its main innovation is basing on idea that the original scene can be approximated by piecewise quadratic polynomials, computing equations for image resizing are presented. And we also gave the solution to solve these equations based on constrained optimization techniques. Comparison results show that the new method produces enlarged image with better quality than other methods.Third, in the field of image processing, to join the discrete edges obtained by image segmentation, into a continuous edge is a common and basic problem. For the problem of constructing fitting curves by discrete data points, we proposed a method of determining knots by optimizing the bending and stretching energies. That is, to choose a knot (parameter value) for each data point. The basic idea of the method is as follows:Associated with each data point, a quadratic polynomial curve passing through three adjacent consecutive data points is constructed. The curve has one degree of freedom which can be used to optimize the shape of the curve. To obtain a better shape of the curve, the freedom degree is determined by optimizing the bending and stretching energies of the curve so that variation of the curve is as small as possible. Between each pair of adjacent data points, two local knot intervals are constructed, and the final knot interval corresponding to these two points is determined by a combination of the two local knot intervals. This method is a local one, and the parametric results have quadratic polynomial precision. The main innovation is the new idea of computing knots by optimizing the bending and stretching energies of the quadratic curve. We also proved that for quadratic curves, optimizing whether the bending or stretching energies will get the same result.Fourth, cubic spline curve becomes one of the most important curve/surface modeling methods due to its minimum norm, best approximation, strong convergence, etc. Cubic spline function curve has cubic polynomial precision, so cubic parameter spline curve should at least has the same precision. From this goal, we discussed how to determine knots with cubic polynomial precision. When determined knots are used to construct parameter spline curves, the constructed curves have cubic polynomial precision. This paper discussed how to construct a cubic polynomial curve with five data points, and present a new method to parameterize the data points. The new method has cubic polynomial precision, which means if the data points are taken from a cubic polynomial function, and the method that constructs the interpolation points has cubic polynomial precision, then the constructed curve that interpolated the data points using knots computed by the new method will reconstruct the cubic polynomial function exactly. Existing methods of data point’s parameterization only have at most quadratic polynomial precision. The main innovation is that we presented the function relation between the points and parameters on the cubic polynomial curve, and proposed a method to compute knots based on cubic polynomial curves. We also provide the concrete numerical calculation method of computing the knots.The shortcomings of the article and the future works:For image enlarging, the main problem is that the edges extraction problem is still difficult to solve so far, especially the judgment of the false edges. How to extract the edges of the image and define the characteristics of the edges will be focused on in the future research work. Moreover, how to apply edge constrains to the object function, to make the constructed surface reflect edge characteristic better will also be a problem needed further research.For the problem of constructing interpolation curves by discrete data points, the paper determined the parameters based on the certain distance and angle between three points, so it is not constant under affine transformation. This is the same as the chord length method, the Foley’s method and the centripetal method. So one of our works in the future is to study how to make the new parameterization method becomes constant under an affine transformation. Though parameterization method with cubic polynomial precision is constant under an affine transformation, it’s too complicated to compute. We’ll make further efforts to simplify the method. |