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The goal of urban building mesh simplification is to generate a compact representation of a building from a given mesh. Local smoothness and sharp contours of urban buildings are important features for converting unstructured data into solid models, which should be preserved during the simplification. In this paper, we present a general method to filter and simplify 3D building mesh models, capable of preserving piecewise planar structures and sharp features. Given a building mesh model, a mesh filtering technique is firstly designed to yield piecewise planar regions and extract crease contours. The planar regions are used to constrain the simplification of the mesh. Mesh decimation is achieved through a series of edge collapse operations, which uses regional structural constraints and local geometric error metrics to handle planar and non-planar areas respectively. The proposed method preserves the mesh structure with meaningful levels of detail while reducing the number of faces. The effectiveness of this method is evaluated on various building models generated from different observation scales, and the performance is validated by extensive comparisons to state-of-the-art techniques.

We construct an enhanced version of knot contact homology, and show that we can deduce from it the group ring of the knot group together with the peripheral subgroup. In particular, it completely determines a knot up to smooth isotopy. The enhancement consists of the (fully noncommutative) Legendrian contact homology associated to the union of the conormal torus of the knot and a disjoint cotangent fiber sphere, along with a product on a filtered part of this homology. As a corollary, we obtain a new, holomorphiccurve proof of a result of the third author that the Legendrian isotopy class of the conormal torus is a complete knot invariant.

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As the barcode becomes more widely used, its applications and data capacity demands grow, increasing the need for barcodes with greater data density. Utilizing the quick response (QR) code–one of the many types of barcodes–we developed two algorithms. The first algorithm creates a color QR code that stores more information than a standard QR code and embeds extra data with limited access privilege. The second algorithm denoises a noisy color QR code. These algorithms consist of three techniques: (1) enlarging the data capacity of a compact QR code image by stacking multiple classical QR codes to form a color barcode, (2) embedding information into the color QR code using pseudo quantum signals in an M-band wavelet domain and selecting the discrete 4-band wavelet transforms to compress the QR images, and (3) applying Discrete M-band Wavelet Transform (DMWT) and Patch Group Prior based Denoising (PGPD) methods to denoise noisy QR code images. The peak-signal-to-noise-ratio (PSNR) summary indicates that information in a color QR code can be efficiently stored and retrieved with these methods. Moreover, it shows that our denoising algorithm effectively removes heavy noise from the noisy color QR code. Our algorithms are implemented in a flexible framework, which allows for further modifications to improve both the data capacity of a color QR code and the effectiveness of signal extraction from noisy data to meet future demands.