## 1 Introduction  
Precise building information is increasingly required for urban monitoring [1, 2], urban planning [3, 4], and population estimation [5, 6]. Remote sensing technology is a cost-optimal, efficient, and popular way to acquire large-scale urban information [2, 3, 6, 7, 8, 9, 10]. Over the years, remote sensing data acquisition capacities have greatly improved in both spatial and temporal resolution. High spatial-resolution images provide a basis to study the urban details of an area, such as buildings. However, it also creates new problems such as limited spectral resolution, high heterogeneity, and shadows. In fact, a large number of urban high spatial resolution images, such as those obtained from Google Earth, exist without parameters for solar zenith angles, exhibiting large-area shadows and RGB bands. Thus, building detection is a topic of interest in high resolution imagery research [10]. Many effective algorithms have been proposed for building information extraction. These building detection methods can be roughly classified into two groups: two-dimensional (2D) and three-dimension (3D) building detection methods. The 2D methods generally extract buildings using brightness, shape, texture, and concomitant shadows [11, 12, 13, 14, 15, 16, 17, 18, 19, 20]. [PERSON] and [PERSON] [11] developed a morphological building index (MBI) to extract buildings using brightness, size, and shape. However, the MBI method cannot easily distinguish buildings from bare soil and roads, and it fails to detect dark and heterogeneous roofs. [PERSON] et. al. [21] applied and improved the snake model in initial seeds selection and the external energy function to extract buildings. In 2010, [PERSON] et al. [22] proposed a new active contour model based on level set formulation, avoiding initial curves. Shadow information is frequently used in building detection. [PERSON] et al. [18] introduced a fuzzy landscape generation method to model the relationship between buildings and their shadows and then used GrabCut partitioning to locate regions with buildings. [PERSON] et al. [19] employed morphology and internal gray variance to describe building edges and their accompanying shadows to extract buildings. [PERSON] and [PERSON] [20] used shadows to verify the existence of buildings and confirm building candidates. [PERSON] and [PERSON] [23] added concomitant shadows of buildings to improve detection accuracy. In 3D building detection methods, height information can facilitate building extraction. The earliest 3D methods for building detection focused on a monocular aerial image with detailed tilt angle, swing angle, and sun altitude [24; 25], then calculated the building height according to the shadows and angles. Thus, shadows not only help to extract buildings, but also provide useful information to infer the height of buildings. In recent years, new remote sensing technologies have emerged to monitor cities. light detection and ranging (LiDAR) is a popular way to obtain the height information for ground objects, and buildings are extracted based on height [26; 27; 28; 29]. Stereo imagery and photogrammetry are applied to building detection, but with more precise and detailed information requirements [30]. Generally, in almost all 2D building detection methods, the results are visualized as binary, separating areas into built-up and not-built-up zones. Urban scenes, however, have become more complex, with denser buildings and population. Hence, binary building detection results might not meet the requirements to detect and describe buildings of different appearances and shapes. The 3D building extraction methods yield height information that can provide a more detailed description of buildings. However, more information, data sources, time, and production costs are required, and the coverage of interest area remains limited. Furthermore, the historical data suitable for 3D methods is less extensive than the data available for analysis with 2D methods.