Algorithms for Extracting Correct Critical Points and Constructing Topological Graphs from Discrete Geographical Elevation Data

Takahashi, Shigeo; Ikeda, Tetsuya; Shinagawa, Yoshihisa; Kunii, Tosiyasu L.; Ueda, Minoru

doi:10.1111/j.1467-8659.1995.cgf143_0181.x

Cited by 154 publications

(81 citation statements)

References 9 publications

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“…Given this assumption, the formula that a critical point must satisfy is given (Takahashi et al. ),

\frac{\partial f}{\partial x} = \frac{\partial f}{\partial y} = 0

with the corresponding three types of critical points being:

f = {\begin{matrix} - x^{2} - y^{2} ​, peak (index = 2) \\ x^{2} + y^{2} ​, pit (index = 0) \\ - x^{2} + y^{2} ​, pass (index = 1) \end{matrix}

where the index indicates the number of negative eigenvalues of the Hessian matrix (denoted by Hf , Rana ):

H f = (\begin{matrix} f_{x x} & f_{x y} \\ f_{y x} & f_{y y} \end{matrix}) = (\begin{matrix} \frac{\partial^{2} f}{\partial x^{2}} & \frac{\partial^{2} f}{\partial x \partial y} \\ \frac{\partial^{2} f}{\partial y \partial x} & \frac{\partial^{2} f}{\partial y^{2}} \end{matrix})

where f xx , f xy , f yx and f yy are the partial derivatives of the function f . The basic three types of critical points are shown in Figure .…”

Section: Methodsmentioning

confidence: 99%

“…However, Takahashi et al. () claim that a raster elevation model lacks smoothness, affecting surface feature extraction. Therefore, we convert the raster to a triangulated irregular network (TIN) surface model to address this problem.…”

Section: Methodsmentioning

confidence: 99%

“…A critical line is defined as follows (Takahashi et al. ). A curve on the topological surface is denoted by C ( t ), it can be represented in the following form:

C (t) = (C_{x} (t), C_{y} (t))

where t ( t ∈ ℝ + ) is a parameter.…”

Section: Methodsmentioning

confidence: 99%

“…This relates to the Euler‐Poincare formula that specifies the number of vertices, edges and faces in a manifold (Okabe and Masuda ; Takahashi et al. , Sadahiro ). According to this formula, the value should equal one for a smooth and continuous surface.…”

Section: Examplementioning

confidence: 99%

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Detecting and Analyzing Mobility Hotspots using Surface Networks

Miller

2014

Transactions in GIS

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Capabilities for collecting and storing data on mobile objects have increased dramatically over the past few decades. A persistent difficulty is summarizing large collections of mobile objects. This article develops methods for extracting and analyzing hotspots or locations with relatively high levels of mobility activity. We use kernel density estimation (KDE) to convert a large collection of mobile objects into a smooth, continuous surface. We then develop a topological algorithm to extract critical geometric features of the surface; these include critical points (peaks, pits and passes) and critical lines (ridgelines and course-lines). We connect the peaks and corresponding ridgelines to produce a surface network that summarizes the topological structure of the surface. We apply graph theoretic indices to analytically characterize the surface and its changes over time. To illustrate our approach, we apply the techniques to taxi cab data collected in Shanghai, China. We find increases in the complexity of the hotspot spatial distribution during normal activity hours in the late morning, afternoon and evening and a spike in the connectivity of the hotspot spatial distribution in the morning as taxis concentrate on servicing travel to work. These results match with scientific and anecdotal knowledge about human activity patterns in the study area.

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“…Given this assumption, the formula that a critical point must satisfy is given (Takahashi et al. ),

\frac{\partial f}{\partial x} = \frac{\partial f}{\partial y} = 0

with the corresponding three types of critical points being:

f = {\begin{matrix} - x^{2} - y^{2} ​, peak (index = 2) \\ x^{2} + y^{2} ​, pit (index = 0) \\ - x^{2} + y^{2} ​, pass (index = 1) \end{matrix}

where the index indicates the number of negative eigenvalues of the Hessian matrix (denoted by Hf , Rana ):

H f = (\begin{matrix} f_{x x} & f_{x y} \\ f_{y x} & f_{y y} \end{matrix}) = (\begin{matrix} \frac{\partial^{2} f}{\partial x^{2}} & \frac{\partial^{2} f}{\partial x \partial y} \\ \frac{\partial^{2} f}{\partial y \partial x} & \frac{\partial^{2} f}{\partial y^{2}} \end{matrix})

where f xx , f xy , f yx and f yy are the partial derivatives of the function f . The basic three types of critical points are shown in Figure .…”

Section: Methodsmentioning

confidence: 99%

Section: Methodsmentioning

confidence: 99%

“…A critical line is defined as follows (Takahashi et al. ). A curve on the topological surface is denoted by C ( t ), it can be represented in the following form:

C (t) = (C_{x} (t), C_{y} (t))

where t ( t ∈ ℝ + ) is a parameter.…”

Section: Methodsmentioning

confidence: 99%

Section: Examplementioning

confidence: 99%

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Detecting and Analyzing Mobility Hotspots using Surface Networks

Miller

2014

Transactions in GIS

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“…Now we are ready to consider how to partition the overall terrain surface into feature areas , which serve as landmarks for guide‐map editing. To this end, our system uses the previously proposed algorithm 15 that extracts critical points ( peaks, passes , and pits ) and their associated feature lines ( ridge and ravine lines ) from the given terrain surface. Figure 6 shows the features extracted from the discrete samples shown in Figure 5, where a ridge line goes from a pass to a peak on the terrain surface while a ravine line from a pass to a pit.…”

Section: Algorithm For Partitioning a Terrain Surfacementioning

confidence: 99%

Modeling Surperspective Projection of Landscapes for Geographical Guide-Map Generation

Takahashi

Ohta

Nakamura

et al. 2002

Computer Graphics Forum

Self Cite

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It is still challenging to generate hand‐drawn pictures because they differ from ordinary photographs in that they are often drawn as seen from multiple viewpoints. This paper presents a new approach for modeling such surperspective projection based on shape deformation techniques. Specifically, surperspective landscape images for guide‐maps are generated from 3D geographical elevation data. Our method first partitions a target geographical surface into feature areas to provide designers with landmarks suitable for editing. The system takes as input 2D visual effects, which are converted to 3D geometric constraints for geographical surface deformation. Using ordinary perspective projection, the deformed shape is then transformed into a target guide‐map image where each landmark enjoys its own vista points. An algorithm for calculating such 2D visual effects semi‐automatically from the geographical shape features is also considered.

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