local 3d shape descriptor

8/3/2019 Local 3D Shape Descriptor

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Scale-Dependent/Invariant Local 3D Shape Descriptors





Scale-space

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Feature detector

– The feature detector is primary part in computer vision

– A specific scale is determined by size of local window

To overcome the limitation

–

Construct an image scale-space – Convolve the image with Gaussian kernels of increasing standard deviation

Scale-space

– Allow us to detect not only the location of a local feature in an image

– But also intrinsic scale

Scale in 3D geometric data

– Construct a representation of the data at different scale

• Multi-resolution representation of a mesh: sensitive to the sampling of the original model• Use smoothing operator similar to 2D scale-space: lead to alternations in the global topology

of the geometric data

– Distance metric for modeling the geometry

• 3D Euclidean distance: lead to creation of erroneous features




Geometric scale-space in range image

– Range image is already a dense and regular projection of a single view of the surfaceof the target 3D shape

Geometric scale-space

– Construct a normal map NNNN

• Triangulate the range image

• Compute a surface normal for each vertex

– Geodesic distance

• Encode accurately the scale-variability of the surface geometry

• Approximate the distance with the sum of Euclidean 3D distances

–

Build the geometric scale-space of a base normal map NNNN• By filtering the normal map with Gaussian kernels of increasing standard deviation

• : local window

Geometric Scale-Space

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: a list of vertex point in the range image

: is a 2D domain in RRRR2

2D normal map




Detect 3D geometric corner

– For a point uuuu in the normal map NNNNσ at scale σ

– The corner response is computed using the Gram matrix

•

NNNNs : horizontal direction, NNNNt : vertical direction• : a free parameter that is empirically set for each particular feature detector

• : weighing factor of the points in the Gram matrix

– Detect corner points at each scale

• Spatial local maxima of the corner detector responses

– Prune corners lying along edge points

• Threshold : the second-order partial derivatives

– Search for local maxima of the corner detector responses across the scales

• The scale of the corners is intrinsic scale of each corner

Scale-Dependent Features

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Limitation of 3D shape descriptor

– Sampling density

– Size of descriptor’s support region

Dense and regular 2D descriptors

– Insensitive to the resolution of the input range images

Support size

– Can be determine by the scale information of corner

Local 3D Shape Descriptors

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Solution in this paper




Determine local neighborhood

– Map the local neighborhood of a corner to a 2D domain

– Use exponential map

Exponential map

–

Map from the tangent space of a surface point to the surface itself (map 2D image to3D surface)

– The exponential map takes a vector wwww on the tangent plane and maps it to the point

on the geodesic curve at a distance of 1 from uuuu, or Exp(wwww)= Γ(1)

– Any point vvvv on the surface in the local neighborhood of uuuu can be mapped to uuuu’s

tangent plane (referred to as Log map) – Geodesic polar coordinates of vvvv

• Geodesic distance

• Polar angle of the tangent to the geodesic at uuuu

Local 3D Shape Descriptors

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Exponential map Geodesic polar angle

uuuu eeee1111

eeee2222

θ

vvvv




Build a scale-dependent local 3D shape descriptor

– Use the geodesic polar coordinates

– : geodesic distance between uuuu and vvvv

– : polar angle of the tangent of the geodesic between uuuu and vvvv defined

relative to a fixed bases {eeee1,eeee2}

–

Radius of the descriptor • Set proportional to the inherent scale σ

Interpolate a geometric entity

– To construct a dense and regular representation of the neighborhood of uuuu at scale σ

– Geometric entity: surface normal

Scale-Dependent Local 3D Shape Descriptor

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Rotation invariant shape descriptor

– Rotate the normal such that the normal at the center point uuuu points in the positive z

direction

– The resulting dense 2D descriptor is invariant up to a single rotation (in-plane rotation

on the tangent plane)

– Align the maximum principal curvature direction at uuuu to the horizontal axis eeee1


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Scale-dependent local 3D shape descriptor

Scale-invariant local 3D shape descriptor

–

Build a set of scale-dependent local 3D shape descriptors – Normalize each descriptor’s size to a constant radius

Assumption for scale-invariant shape descriptor

– The scales of local geometric structures relative to the global scale of a range image

remains constant as the global scale of a range image is altered

– This assumption holds as long as the geometry captured in the range image is rigid

and does not go under any deformation

Scale-Dependent/Invariant Local 3D Shape Descriptor

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for a scale-dependent corner at uuuu and with scale σ

for a scale-dependent corner at uuuu and with scale σ





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Pairwise registration

– Similarity Measure

– A and B are the set of points in the domain of and

Multiview registration

Image registration

– The process of transforming different sets of data into one coordinate system

– Data may be multiple photographs, data from different sensors, from different times,

or from different viewpoints.

– Registration is necessary in order to be able to compare or integrate the dataobtained from these different measurements.

Evaluation of Local 3D Descriptor

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Pairwise registration


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Range images with the same global scale using a

set of scale-dependent local 3D shape descriptor

Range images with inconsistent global scales using

a set of scale-invariant local 3D descriptor






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Initial set of range images Approximate registration

obtained with our

framework

Registration refined with

multi-view ICP [25]

A water tight model using

a surface reconstruction

algorithm [26]





– Multiple objects• 15 views of the Buddha model

• 12 views of the armadillo

• 15 views of the dragon model


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– Random global scale• 15 views of the Buddha and dragon models

• Random scale factor: 1 ~ 4


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– Random global scale• 15 views of the Buddha models

• 12 views of the armadillo models

• 15 views of the dragon models

• Random scale factor: 1 ~ 4


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[1] J. Novatnack and K. Nishino, “Scale-Dependent/Invariant Local 3D Shape

Descriptors for Fully Automatic Registration of Multiple Sets of Range Images,”In Proceedings of the 10th European Conference on Computer Vision

(ECCV2008) , pp. 440-453, 2008.

[2] J. Novatnack and K. Nishino, “Scale-Dependent 3D Geometric Features,” In

IEEE 11th International Conference on Computer Vision (ICCV2007) , 2007.

References

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local 3d shape descriptor

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