Inspired by Cremers's work, this paper proposes a novel method for detecting open boundaries, such as coastline and skyline in an image. This method is based on B-spline function, curve evolution, and the cartoon model of Mumford-Shah functional (M-S model). Because the object to be detected is an open curve in the image domain, two constraint equations are introduced into the M-S model. Thus, the problem of open boundary detection becomes a minimal partition problem. With the partial differential equations (PDEs) of control points and constraint equations, the curve will stop on the desired boundary. The method can be used to detect automatically a curve that separates an image into two distinct regions and is not necessarily defined by gradient, even if the image is very noisy. In addition, with two open curves, our model can be extended to detect belt-like objects, such as rivers and roads.