# Shape Anchor Guided Holistic Indoor Scene Understanding

Mingyue Dong1, Linxi Huan1, Hanjiang Xiong1, Shuhan Shen2, Xianwei Zheng1
1The State Key Lab. LIESMARS, Wuhan University
2Institute of Automation, Chinese Academy of Sciences
{dmy25148, whu_hlx, xionghanjiang<EMAIL_ADDRESS>
<EMAIL_ADDRESS>Corresponding author.

###### Abstract

This paper proposes a shape anchor guided learning strategy (AncLearn) for
robust holistic indoor scene understanding. We observe that the search space
constructed by current methods for proposal feature grouping and instance
point sampling often introduces massive noise to instance detection and mesh
reconstruction. Accordingly, we develop AncLearn to generate anchors that
dynamically fit instance surfaces to (i) unmix noise and target-related
features for offering reliable proposals at the detection stage, and (ii)
reduce outliers in object point sampling for directly providing well-
structured geometry priors without segmentation during reconstruction. We
embed AncLearn into a reconstruction-from-detection learning system (AncRec)
to generate high-quality semantic scene models in a purely instance-oriented
manner. Experiments conducted on the challenging ScanNetv2 dataset demonstrate
that our shape anchor-based method consistently achieves state-of-the-art
performance in terms of 3D object detection, layout estimation, and shape
reconstruction. The code will be available at https://github.com/Geo-
Tell/AncRec.

##  1 Introduction

Holistic indoor scene understanding from partial observations (_e.g_., single-
view images or 3D scans) is a comprehensive task that provides 3D semantic
scene models for indoor applications. Early works studied this task with a
reconstruction-from-detection framework that recovers the geometries of room
structures and objects from the corresponding detection in a separate way.
Later, end-to-end learning methods were proposed to simultaneously perform
layout estimation, object detection, and shape prediction in one forward pass
for semantic scene reconstruction [27, 37, 23]. With the recent success of
point-based 3D detection and instance reconstruction, surging interest has
been witnessed in detecting and modeling objects directly from sparse point
clouds [28, 33].

Figure 1: Comparison between different feature grouping strategies. (a) the
original scene model. (b)-(e) The different feature grouping operations all
suffer from the issue of confusing non-target noise with useful features. (f)
The proposed shape anchor guided grouper directly generates anchors at the
instance surface to merge instance-related features, which largely alleviates
noise interference.

Benefiting from the use of rich geometry information, current scan-based deep
methods have improved the performance of semantic scene reconstruction.
However, they still left two issues that bottleneck high-quality semantic
reconstruction: (1) the noisy instance feature learning at the detection
phase, and (2) the difficulty in retrieving instances from sparse point clouds
for reconstruction.

At the detection phase, it is required to group features for instance
representation learning. The ball query [30, 31] and 3D convolution[39, 20]
are two basic operations for point feature grouping, but they often mix
massive noise with informative features due to the fixed grouping range, as
shown in Fig. 1 (b) and (c). To compensate for the deficiency caused by the
fixed range, VoteNet [14] and its variants [34, 29, 38, 35] adopt a voting
strategy to cluster object features by moving surface points towards object
centers. Albeit more flexible than the basic operations, the voting-based
strategy often generates an unconstrained grouping area that brings quantities
of outliers as illustrated in Fig. 1 (d). BRNet [10] hence restricts the
grouping space by sampling around representative points given by coarse box
proposals. Nevertheless, limited by box-like grouping area, the sampling
points can still fall far beyond targets when the objects are irregularly
shaped, as depicted in Fig. 1 (e).

At the reconstruction stage, the retrieval of outlier-free object points is a
prerequisite for object recovery. Due to the noise introduced by feature
grouping in the previous detection phase, the points grouped for localizing
objects can hardly serve as ideal reconstruction priors as indicated by Fig.
1. Consequently, the current methods are forced to employ an additional
foreground classifier [28] or replace the detector with a complex instance
segmentation backbone to sample object points from the raw scans [33].
However, the existence of numerous non-target outliers in the search space
challenges instance segmentation, resulting in increased risks of gluing
different instances and misclassifying background points.

Based on the discussion above, the two noise interference issues during
detection and reconstruction are actually highly coupled and can be resolved
together as long as outliers are excluded during feature grouping. To this
end, we are motivated to propose a shape anchor-guided learning strategy
(_AncLearn_) that generates surface anchors to fit the feature grouping areas
to object shape distributions, as displayed in Fig. 1 (f). With the geometry
constraint provided by surface anchors, it is able to merge local target-
focused features for predicting reliable object proposals and construct a
shape-aware search space for robustly sampling instance points without
segmentation during reconstruction. The proposed anchor-guided learning
strategy can be easily embedded into an end-to-end learning system to
accomplish object detection, layout estimation, and instance reconstruction
for holistic scene understanding. The main contributions are summarized as
follows:

  1. -

We present a shape anchor guided learning strategy to simultaneously address
the issues of noisy feature learning in detection and instance point retrieval
during reconstruction.

  2. -

We embed the proposed anchor-guided learning strategy into an end-to-end
learning system to accomplish object detection, layout estimation, and
instance reconstruction for holistic scene understanding in a purely instance-
oriented way.

  3. -

Extensive experiments demonstrate that our AncRec achieves high-quality
semantic scene reconstruction with state-of-the-art performance in instance
detection and mesh prediction on the challenging ScanNetv2 dataset [12] (some
ground truths provided by Scan2CAD [1] and SceneCAD [3]).

Figure 2: Overview of the AncRec framework. At the detection stage, seed point
features for walls and objects are first learned with a modified dual-branch
PointNet++. Following each branch, a voting module and an anchor-guided
feature grouper respectively generate proposal features
$\boldsymbol{f}_{\text{obj(wall)}}^{\text{vote}}$ and
$\boldsymbol{f}_{\text{obj(wall)}}^{\text{anchor}}$, which are fed into the
decoders to predict object bounding boxes and wall quads. The room layout is
then constructed by processing the wall quads into connected corners. At the
reconstruction stage, objects with high objectness scores are reconstructed
under the guidance of $\boldsymbol{f}_{\text{obj}}^{\text{vote}}$,
$\boldsymbol{f}_{\text{obj}}^{\text{anchor}}$, and the geometry priors sampled
by shape anchors. Finally, object models are arranged in the scene with the
predicted layout according to the spatial alignment provided by predicted
bounding boxes.

##  2 Related Work

Semantic scene understanding has been extensively studied over the past years.
Many methods focused on acquiring the semantics of 3D scenes [22, 15, 8],
while others recovered scene geometries by scene completion [18, 13, 36].
Recently, increasing interest has emerged in semantic scene reconstruction,
which recovers both the semantics and geometric shapes of objects. By treating
semantic reconstruction as a problem of holistic scene understanding,
promising progress has been made based on a reconstruction-from-detection
principle [28]. In the following, we review the research from the two core
aspects of the reconstruction-from-detection pipeline, _i.e_., 3D object
detection and scene-aligned instance reconstruction.

###  2.1 3D Object Detection

Over the past years, deep detectors have gained great success in 2D object
detection [40, 25]. The progress of 2D deep detection has inspired the
development of deep learning techniques for recognizing objects from scene
point clouds. Compared to image data, point clouds provide rich surface
geometry clues for locating objects in real scenes. Nevertheless, the sparse,
irregular, and orderless characteristics of point clouds make them hard to be
handled by the grid-based convolution model.

Works in the early stage leveraged the 2D proposals as 3D detection
constraints or projected point clouds into regular 2D/3D grids. Although these
3D detectors are applicable, their performance is still limited by the 2D
detectors and the missing geometric details brought by projection. To directly
learn the rich geometry features from the raw points, PointNet [30] and
PointNet++ [31] used a ball query operation for grouping point features.
Later, PointRCNN [32] adopted the point-wise features extracted by PointNet++
for producing instance proposals with point clouds. Hindered by the fixed
feature grouping range of grid-based convolutions or ball query operation,
these methods often mixed massive noise with informative features, which
impairs the reliability of proposals. Considering that the observed object
surface points usually lie far away from the object centers, VoteNet [14]
introduced a voting mechanism for proposal generation. Based on the voting
mechanism, numerous variants further refined proposal features with context
relationships [34], hierarchical clues [35], and hybrid geometric primitives
[38]. However, the issue of outliers in votes still blocks the learning of
representative box features. BRNet [10] derived proposal features based on the
generated virtual points given by coarse box proposals, whereas the virtual
points can fall into non-target areas and bring noise. In this paper, we
directly merge features anchored on target surfaces into robust proposal
representations for indoor instance learning.

###  2.2 Scene-aligned Instance Reconstruction

Scene-aligned instance reconstruction requires not only modeling 3D object
shapes but also correctly arranging the shapes in the 3D scene space. The
semantic model of an indoor scene was early constructed with retrieval
techniques, which search for CAD shapes or geometric primitives from an
offline database and align the approximate object models to input scene data
[1, 2, 17, 24]. Albeit able to present delicate scene models, the retrieval-
based methods lack generalization ability to various scenes due to the
inefficient inference and the limited database scale [37].

The promising advances in deep learnable shape representations motivated
scene-aligned instance reconstruction without a finite model pooling [16, 11].
Many prior arts in literature generate 3D objects with explicit or implicit
representations with learned features derived from 2D recognition results [19,
27, 37, 23]. Unlike previous single-view modeling methods, RevealNet [21] and
the following works [5, 4] predicted the occupancy grids of semantic instances
with 3D features extracted from voxelized RGB-D scans. For high scene
resolution that is hard to acquire by volumetric representation, RfD-Net [28]
and DIMR [33] extracted instance points from scans with segmentation
techniques for subsequent single object reconstruction with deep implicit
functions [26, 9]. Because of the existence of non-target points, these two
state-of-the-art methods have to use an extra foreground classifier or a
complex instance segmentation backbone to sample object points for
reconstruction. In this paper, we utilized the previously generated surface
anchors to localize instance points in scans. Hereby, we introduced an easy
but effective anchor-guided sampling strategy to offer geometry priors for the
following instance reconstruction without an extra segmentation stage.

##  3 Method

We illustrate the proposed AncRec in Fig. 2. AncRec achieves semantic scene
reconstruction with the proposed anchor-guided learning strategy in an
instance-oriented way. It first localizes objects and walls via a dual-branch
detector that is equipped with shape anchors learned for grouping point
features at the detection stage. The object shape anchors are subsequently
leveraged to sample instance geometry priors for predicting object meshes.
Finally, the complete semantic scene model is reconstructed by arranging
object meshes in the post-processed room layout with alignment to the parsed
bounding boxes and poses. In the following, we elaborate on how the shape
anchors are learned and leveraged to address the noise interference during
instance detection and mesh prediction for high-quality semantic scene
reconstruction.

###  3.1 Shape Anchor-guided Instance Detection

We modify the VoteNet into a dual-branch instance detector to simultaneously
localize objects and walls in one forward pass (Fig. 2). Although the voting
mechanism of VoteNet is workable for generating proposals, the reliability of
voting-based predictions is often impaired by the outliers in votes. We hence
design an anchor-guided feature grouper to learn target-focused features for
robust instance detection.

####  3.1.1 Anchor-guided Feature Grouper

We illustrate the mechanism of the anchor-guided feature grouper in Fig. 3. In
the following, by taking the object detection branch as an example, we
describe how it works with the voting module in three steps to predict the
oriented 3D bounding boxes.

Figure 3: The anchor-guided feature grouper. The anchor-guided proposal
feature $\boldsymbol{f}_{i}^{\text{anchor}}$ is the average of features at
shape anchors. The shape anchors are deformed from initial anchors
$\left\\{\boldsymbol{a}_{j}\right\\}$ with offsets derived from the
concatenation of the voting-based proposal feature
$\boldsymbol{f}_{i}^{\text{vote}}$ and $\left\\{\boldsymbol{a}_{j}\right\\}$.
The anchors are then translated to the cluster center $\boldsymbol{c}_{i}$.

Step 1: Anchor Generation. Given the $i^{th}$ voting-based proposal feature
$\boldsymbol{f}^{\text{vote}}_{i}\in\mathbb{R}^{128}$, the anchor-guided
grouper first generates shape anchors that depict the geometry of the $i^{th}$
object candidate via template deformation. With $N$ initial anchors
$A=\left\\{\boldsymbol{a}_{j}\in\mathbb{R}^{3}|j=1,...,N\right\\}$ uniformly
selected from a unit ball surface, a deformation layer is applied to translate
the initial anchors to the $i^{th}$ candidate object surface by

$\hat{\boldsymbol{a}}_{j}=\boldsymbol{a}_{j}+\underbrace{\text{Tanh}(\text{MLP}([\boldsymbol{a}_{j},\boldsymbol{f}^{\text{vote}}_{i}]))}_{\text{deformation
offset}\ \Delta\boldsymbol{a}_{j}}+\boldsymbol{c}_{i}.$ (1)

As indicated by Eq. 1, the deformation layer moves $\boldsymbol{a}_{j}$ with
offsets $\Delta\boldsymbol{a}_{j}$ inferred by a multi-layer perceptron (MLP)
and a Tanh activation function and then places the deformed anchors at the
candidate center $\boldsymbol{c}_{i}$ clustered by the voting module. The
training of the deformation layer is supervised by the Chamfer distance loss
defined as

$\displaystyle\mathcal{L}_{\mathrm{anchor}}=$
$\displaystyle\frac{1}{|P_{\text{gt}}|}\sum_{\boldsymbol{p}\in
P_{\text{gt}}}\min_{\boldsymbol{\hat{a}}\in\hat{A}}\|\boldsymbol{p}-\boldsymbol{\hat{a}}\|_{2}^{2}$
(2)
$\displaystyle+\frac{1}{|\hat{A}|}\sum_{\boldsymbol{\hat{a}}\in\hat{A}}\min_{\boldsymbol{p}\in
P_{\text{gt}}}\|\boldsymbol{\hat{a}}-\boldsymbol{p}\|_{2}^{2},$

where $\hat{A}$ and $P_{\text{gt}}$ denote the sets of deformed anchors and
surface points sampled from the corresponding scene-aligned 3D meshes.
Thereby, the anchors are learned to fit the $i^{th}$ object surface, and the
clustering center $\boldsymbol{c}_{i}$ is also refined with shape constraint
provided by Eq. 2. Moreover, due to the supervision with complete surface
points, the shape anchors can assist in exploiting context for recovering
unobserved structures, _e.g_., the missing object bottom that is supported by
the observed floor areas.

Step 2: Feature Grouping. The anchor-guided grouper propagates the seed point
features (extracted by the dual-branch PointNet++ backbone) to each anchor via
interpolation followed by an MLP. As the deformed anchors are mainly located
on the object surface, the seed point features for propagation can be reliably
selected from target-related areas. The $i^{th}$ noise-reduced proposal
representation $\boldsymbol{f}^{\text{anchor}}_{i}\in\mathbb{R}^{128}$ is
obtained by averaging the anchor features.

Step 3: Prediction Fusion. We employ two decoders to predict the object
category and oriented box parameters from $\boldsymbol{f}^{\text{vote}}_{i}$
and $\boldsymbol{f}^{\text{anchor}}_{i}$ respectively for the $i^{th}$
candidate. This is based on the consideration that the voting-based proposal
feature $\boldsymbol{f}^{\text{vote}}_{i}$ contains more contextual
information while the anchor-guided proposal feature
$\boldsymbol{f}^{\text{anchor}}_{i}$ is more target-focused. The estimated
parameters, denoted as $\boldsymbol{\Theta}^{\text{vote}}_{i}$ and
$\boldsymbol{\Theta}^{\text{anchor}}_{i}$, are averaged with learnable weights
to obtain the final object parameters:

$\boldsymbol{\Theta_{i}}=\boldsymbol{w_{1}}\cdot\boldsymbol{\Theta}^{\text{vote}}_{i}+\boldsymbol{w_{2}}\cdot\boldsymbol{\Theta}^{\text{anchor}}_{i}.$
(3)

From the three steps above, the attributes of objects are parsed with
robustness to noise. The anchor-guided detection of wall instances works in a
similar way. Considering that walls are generally connected to each other, we
additionally deploy the attention operation used in [35] to enhance the
anchor-guided wall proposal features with the strong relationship between
walls for precise layout estimation.

####  3.1.2 The Detection Training Loss

The total loss function of the anchor-guided instance detector is defined as

$\mathcal{L}=\mathcal{L}_{\text{obj}}+\mathcal{L}_{\text{wall}}+\mathcal{L}_{\text{obj}}^{\text{anchor}}+\mathcal{L}_{\text{wall}}^{\text{anchor}}.$
(4)

In Eq. 4, $\mathcal{L}_{\text{obj}}$ is the object detection loss given by
[14] while $\mathcal{L}_{\text{wall}}$ is the wall quad loss used in [7].
$\mathcal{L}_{\text{obj}}^{\text{anchor}}$ and
$\mathcal{L}_{\text{wall}}^{\text{anchor}}$ are the chamfer distance losses
that supervise the anchor deformation learning in terms of positive object and
wall candidates.

###  3.2 Anchor-guided Object Reconstruction

Instance points from the input scan are desirable geometry priors for
reconstruction. Previous works utilized segmentation operations for instance
point sampling. Despite being applicable, accurate instance segmentation is
difficult due to the existence of massive non-target noise in the search
space. In contrast, we take advantage of the shape anchors to construct a
shape-aware search space, in which object points can be directly localized
with little noise interference. Fig. 4 illustrates the workflow of our anchor-
guided instance point sampling.

Figure 4: The anchor-guided instance point sampling. Object points are
iteratively sampled from the input scan under the guidance of shape anchors.
The sampled object points serve as the geometry prior for shape
reconstruction.

We first add the shape anchors of positive proposals to the original scene
scan to enhance indoor objects’ structural information. From the enhanced
scan, we select points that lie within a given radius of each anchor and
update the anchor set with these selected points to expand the searching
space. With the searching space compactly fitting the object shape, we can
further sample more instance points with high coverage. In this way, our
anchor-guided sampling efficiently generates object geometry prior without
reliance on an extra segmentation module. In the experiments, the anchor-
guided sampling process iterates twice with the sampling radius set to the
minimum distance between the anchors.

The geometry priors are next concatenated with the corresponding proposal
features $\boldsymbol{f}^{\text{vote}}_{\text{obj}}$ and
$\boldsymbol{f}^{\text{anchor}}_{\text{obj}}$ and encoded into shape
embeddings $\boldsymbol{f}^{\text{shape}}_{\text{obj}}$ by a ResPointNet [28,
30]. Based on $\boldsymbol{f}^{\text{shape}}_{\text{obj}}$, the decoder of
BSP-Net [9] is adopted to predict the signed distances of query points at the
canonical coordinates. Following the BSP-Net, we apply the Constructive Solid
Geometry (CSG) technique to extract the shape surfaces.

###  3.3 Semantic Scene Reconstruction

We now build the semantic scene model with the anchor-guided results of the
instance detector and shape predictor. The room structure is first constructed
by transforming the detected wall quads into orderly connected corners with
the merging technique [7]. Next, the models of objects with high objectness
are chosen to be arranged in the scene with alignment to the predicted 3D
bounding boxes [28]. In the end, the semantic scene model is built as a
combination of the reconstructed layout and the scene-aligned object shapes.

##  4 Experiments

###  4.1 Experiment Settings

####  4.1.1 Dataset

We test our method on ScanNetv2 [12] with ground truths from Scan2CAD [1] and
SceneCAD datasets [3] for holistic indoor scene understanding. (1) ScanNetv2
is a benchmark for indoor scene analysis with 1,513 scanned room point clouds;
(2) Scan2CAD is an alignment dataset that matches ShapeNet models to their
counterpart object instances in ScanNet with oriented 3D bounding boxes; and
(3) SceneCAD provides 3D layout annotations for the scans in ScanNetv2. We
follow [7] and [9] to pre-process the layout polygons and object meshes for
network training supervision. The train/test split is kept in line with the
previous works with eight challenging object categories considered in the
experiments.

####  4.1.2 Evaluation Metrics

The quality of semantic scene reconstruction is evaluated with the performance
of object detection, layout estimation, and scene-aligned shape
reconstruction. In line with previous works [28, 7], we use mean average
precision across all classes (mAP) with 0.5 IoU threshold for object detection
and F1-score for layout estimation. To evaluate the reconstruction quality, we
use the chamfer distance (CD) based mAP with thresholds 0.1 and 0.047, light
field distance (LFD) based mAP with thresholds 5000 and 2500, as well as the
mAP at 3D IoU thresholds 0.25 and 0.5.

####  4.1.3 Implementation Details

The training of our method is conducted on a Titan GPU with two stages, and
all parameters are updated by the Adam optimizer. In the first stage, we train
the dual-branch instance detector of AncRec for simultaneous optimization of
object detection and layout estimation modules. We set the batch size to 8,
initialize the learning rate to 1e-3, and adopt the _ReducerLrOnPlateu_
learning scheduler in PyTorch. In the second stage, we use the object
proposals predicted by the instance detector for mesh reconstruction. We train
the shape predictor with the BSPNet decoder pre-trained on the 8-category
ShapeNet data [6] for training stability and efficiency. The training process
lasts 100 epochs with batch size set to 32, learning rate initialized as 1e-4,
and the same training schedule. We found the number of shape anchors $N$ is an
insensitive hyper-parameter and thus set it to 18 for computational
efficiency.

###  4.2 Comparison and Analysis

---
GT |
RfD-Net |
DIMR |
Ours |
Figure 5: Visualization comparison of scene semantic reconstruction on
ScanNetv2. Figure 6:  Holistic scene understanding by our proposed AnchorRec.
(The comparison methods didn’t conduct layout estimation.)

Scene Reconstruction. We compare our method with the state-of-the-art works,
_i.e_., RfD-Net [28] and DIMR [33]. RfD-Net predicts object shapes with the
instance points segmented from proposals given by VoteNet, while DIMR infers
object models from the instance points provided by a complex instance
segmentation backbone. As shown in Fig. 5, our AncRec can robustly recognize
and reconstruct objects from noisy point clouds with accurate object
localization and high-fidelity shape details. With the target-focused proposal
feature learning offered by AncLearn in terms of object detection, our method
generates fewer false positive models than the compared approaches. With more
accurate detection results, our AncRec can leverage outlier-reduced instance
priors sampled by AncLearn to infer more detailed object structures, _e.g_.,
the folding chairs are recovered in the second column of Fig. 5. Tab. 1
presents the quantitative comparison of different methods. Our AncRec
outperforms other methods under all metrics. As the comparison methods didn’t
conduct layout estimation, we illustrate our holistic scene understanding
results in Fig. 6. Considering that semantic scene reconstruction is a
comprehensive problem, we further analyze the performance of our method on the
key sub-tasks, _i.e_., scene parsing and object reconstruction, as follows.

Table 1: Quantitative results on the ScanNetv2 dataset. We evaluate the
reconstruction quality with mAP at different thresholds. (higher is better)

|<EMAIL_ADDRESS>|<EMAIL_ADDRESS>|<EMAIL_ADDRESS>|<EMAIL_ADDRESS>| LFD@5000 | LFD@2500
---|---|---|---|---|---|---
RfD-Net | 42.5 | 16.9 | 45.7 | 19.1 | 28.6 | 7.8
DIMR | 46.3 | 12.5 | 51.9 | 25.7 | 29.5 | 8.6
AnchorRec | 52.9 | 18.9 | 56.8 | 29.4 | 30.3 | 9.9

---
Figure 7: Visualization comparison results of object detection on ScanNetv2 GT |  |  |
---|---|---|---
RfD-Net |  |  |
DIMR |  |  |
Ours |  |  |
Figure 8: The influence of different instance point sampling strategies on object reconstruction. Points from the same instance are painted in the same color. Without the reliance on segmentation like RfD-Net and DIMR, our method can correctly separate adjacent objects and predict shape models. Table 2: Quantitative results of object detection and layout estimation. The AP scores are measured with an IoU threshold of 0.5. Higher values mean better results. | Object | Layout
---|---|---
Method | table | sofa | chair | cabinet | bookshelf | display | bathtub | trash bin | mAP | F1-scores
VoteNet [14] | 44.5 | 47.5 | 76.3 | 28.8 | 20.1 | 8.0 | 26.0 | 14.6 | 33.2 | -
MLCVNet [35] | 50.5 | 46.6 | 81.8 | 25.5 | 19.2 | 7.8 | 25.3 | 19.2 | 34.5 | -
H3DNet [38] | 53.6 | 52.4 | 76.0 | 25.6 | 27.2 | 9.4 | 20.7 | 10.0 | 34.4 | -
BRNet [10] | 56.4 | 43.1 | 80.9 | 35.4 | 26.1 | 10.4 | 28.2 | 11.0 | 36.4 | -
PQ-Transformer [7] | 52.7 | 52.5 | 81.1 | 32.1 | 36.4 | 6.7 | 20.6 | 20.8 | 37.8 | 67.0
AnchorRec (Ours) | 57.3 | 56.1 | 84.5 | 36.9 | 29.8 | 17.0 | 34.0 | 31.5 | 43.4 | 70.5
Table 3: Numeric results on object reconstruction. The AP scores are measured with CD at a threshold of 0.1. (Higher is better.) Method | table | chair | bookshelf | sofa | trash bin | cabinet | display | bathtub | mAP
---|---|---|---|---|---|---|---|---|---
RfD-Net [28] | 25.51 | 82.11 | 32.53 | 44.21 | 44.74 | 28.37 | 65.51 | 42.57 | 45.70
DIMR [33] | 39.44 | 81.04 | 38.24 | 44.09 | 62.60 | 23.57 | 75.12 | 50.93 | 51.88
AnchorRec (Ours) | 53.60 | 86.23 | 36.82 | 50.92 | 59.59 | 39.56 | 73.68 | 54.32 | 56.84

Scene Parsing. We compare AncRec with the state-of-the-art methods regarding
8-category object detection and layout estimation. The quantitative results in
Tab. 2 show that our AnchorRec edges out the comparison methods by significant
performance gains. Especially for the categories of the display, bathtub, and
trash bin, AncRec exceeds the second best by 6.6%, 5.8%, and 10.7% in
precision measurement. The results indicate that, with adaptive grouping
feature areas, AncLearn works effectively for detecting objects shaped
irregularly, _e.g_., diverse-scaled bathtubs, thin displays, and tiny trash
bins. In Tab. 2, we also compare our method with PQ-Transofrmer [7], the
current state-of-the-art scene parsing approach, in terms of layout
estimation. The results demonstrate that our AncLearn is also applicable to
localizing walls that are particularly large and thin. Visualized object
detection result provided in Fig. 7 further manifests that our AnchorRec can
reliably delineate objects with compact bounding box prediction.

Object Mesh Reconstruction. In Tab. 3, we also evaluate the performance of
different methods with respect to class-wise object reconstruction. The
evaluation is based on how matchable the predicted object meshes are with the
scene-aligned ground truths. The numeric results in Tab. 3 show that AncRec
achieves the best performance on 5 categories and the overall chamfer
distance-based mAP score.

###  4.3 Ablation Studies

AncLearn for Scene Parsing. We study different settings of AncLearn for
instance detection and report the results in Tab. 4. Compared to the baseline
without AncLearn, the proposed AncLearn brings consistent improvement to
object detection and layout estimation. Especially, when cooperating with the
self-attention layer for layout estimation, AncLearn enables the dual-branch
instance detector of AncRec to obtain the best performance in both parsing
tasks.

Table 4: Ablation study of the proposed Anclearn<EMAIL_ADDRESS>and F1 score
respectively reflect the performance of object detection and layout
estimation. Higher values mean better results.

OD anchor | layout anchor | layout SA |<EMAIL_ADDRESS>| F1 score
---|---|---|---|---
|  |  | 41.12 | 60.62
$\checkmark$ |  |  | 43.33(+2.21) | 60.59(- 0.03)
| $\checkmark$ |  | 40.21(-0.91) | 62.61(+ 1.99)
$\checkmark$ | $\checkmark$ |  | 41.24(+0.12) | 64.00(+3.38)
| $\checkmark$ | $\checkmark$ | 41.16(+0.04) | 65.34(+4.72)
$\checkmark$ | $\checkmark$ | $\checkmark$ | 43.38(+2.26) | 70.45(+9.83)

AncLearn for Object Reconstruction. In the shape predictor of AncRec, AncLearn
works to sample instance points as geometry priors for reconstruction. With
the instance detector of AncRec, we compare AncLearn with the segmentation-
based strategy used in RfD-Net and a box cropping sampling method. The
quantitative comparison in Tab. 5 shows that AncLearn outperforms the other
two approaches by providing instructive geometry priors with shape anchors in
terms of all metrics.

Table 5: Comparison between different instance point sampling methods. Higher
values mean better results.

|<EMAIL_ADDRESS>|<EMAIL_ADDRESS>|<EMAIL_ADDRESS>
---|---|---|---
Segmentation [28] | 17.4 | 51.4 | 55.8
Box Cropping | 17.7 | 51.9 | 55.5
AncLearn (Ours) | 18.9 | 52.9 | 56.8
|<EMAIL_ADDRESS>| LFD@5000 | LFD@2500
Segmentation [28] | 29.1 | 29.1 | 7.8
Box Cropping | 28.8 | 29.9 | 9.8
AncLearn (Ours) | 29.4 | 30.3 | 9.9

###  4.4 Contributions of the Vote and Anchor Features

As mentioned in Sec. 3.1.1, we perform instance detection by taking both vote
and anchor features into account. To investigate the difference in their
contributions to the detection parameters, we visualize the two weight vectors
in Eq. 3 for perceptual comparison. As shown in Fig. 9, vote features have a
larger impact on scoring objectness, indicating that the context encoded in
vote features is useful in differentiating between object and non-object
regions. Compared to vote features, anchor features contribute significantly
more to angle prediction, which demonstrates the superiority of anchor-guided
strategy in providing shape clues for learning shape-sensitive parameters.
Considering the gap between the bounding box and the shape of objects, the
vote features contribute more to predicting the bounding box center and size
residuals.

Figure 9: The learned weights for combining vote and anchor features.

###  4.5 The Characteristics of the Learned Anchors

The supervision with surface points derived from complete 3D models enables
the anchors to depict object shapes, even with unobserved structures, which
assists in object detection and reconstruction when some parts of objects are
missing. There also exists a difference between anchor distribution
characteristics for objects and non-object areas. As shown in Fig. 10, the
anchors and sampled points depict object shapes, while those for non-object
areas scatter irregularly. This characteristic may play a role in
differentiating objects from non-object regions.

|
---|---
Object Anchors | Non-object Anchors

Figure 10: The characteristics of the learned anchors. The anchors and sampled
points are shown in red and purple respectively.

##  5 Conclusions

In this paper, we introduce a shape anchor guided learning strategy (AncLearn)
that is embedded into a reconstruction-from-detection learning system (AncRec)
to handle the issue of noise interference in point-based holistic scene
understanding. Extensive experiments demonstrate that AncRec achieves high-
quality indoor semantic scene reconstruction. The quantitative and qualitative
results show that AncRec outperforms current methods in terms of object
detection, layout estimation, and shape modeling. The ablation studies
convincingly verify that AncLearn can largely exclude noise from search space
for reliable feature grouping and robust instance point sampling. In the
future, it is promising to study the application of shape anchor guided
learning strategy in other point-based 3D vision tasks.

## Acknowledgment

This research was supported by the Fundamental Research Funds for the Central
Universities of China under Grant 2042022dx0001, NSFC-projects under Grant
42071370, Wuhan University-Huawei Geoinformatics Innovation Laboratory, and
the Open fund of Key Laboratory of Urban Land Resources Monitoring and
Simulation, Ministry of Natural Resources under Grant KF202106084.

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from the coadjoint action of $\mathfrak{sl_{2}(\ch@airxmathbbm{R})}$, which is
a trivial vector bundle with global frame $\\{e_{1},e_{2},e_{3}\\}$ and anchor

$\displaystyle\rho(e_{1})$
$\displaystyle=-x_{3}\partial_{2}-x_{2}\partial_{3},$ (6.31)
$\displaystyle\rho(e_{2})$
$\displaystyle=x_{3}\partial_{1}+x_{1}\partial_{3},$
$\displaystyle\rho(e_{3})$
$\displaystyle=x_{2}\partial_{1}-x_{1}\partial_{2}.$

The bracket can be expressed in terms of the global frame, but will not be
used in this section.

The orbits of this Lie algebroid are given by the connected components of the
level sets of the function $f=x_{1}^{2}+x_{2}^{2}-x_{3}^{2}$, where
$f^{-1}(\\{0\\})$ splits into three orbits $x_{3}>0$, $x_{3}<0$, and the
origin. Hence, the origin is the only leaf of this foliation that is not two
dimensional. In this section we show that by means of repeatedly blowing up
restrictions of $A$ (or its blowups) to orbits, one cannot construct a regular
Lie algebroid.

######  Proposition 6.12.

The restriction of the Lie algebroid $\tilde{A}=\mathrm{Blup}(A,A_{\\{0\\}})$
to $\ch@airxmathbbm{P}$ consists of three orbits:

  * •

A one-dimensional orbit

$Z=\\{[x_{1}:x_{2}:x_{3}]\in\ch@airxmathbbm{P}\colon
x_{1}^{2}+x_{2}^{2}=x_{3}^{2}\\};$ (6.32)

  * •

Two two-dimensional orbits given by the connected components of
$\ch@airxmathbbm{P}\setminus Z$, explicitly given by

$\\{[x_{1}:x_{2}:x_{3}]\in\ch@airxmathbbm{P}\colon
x_{1}^{2}+x_{2}^{2}<x_{3}^{2}\\}\quad\text{ and
}\quad\\{[x_{1}:x_{2}:x_{3}]\in\ch@airxmathbbm{P}\colon
x_{1}^{2}+x_{2}^{2}>x_{3}^{2}\\}.$ (6.33)

.

######  Proof:

In the charts of $\mathrm{Blup}(\ch@airxmathbbm{R}^{3},\\{0\\})$ from Remark
3.1 we can e.g. compute over $(U_{1},\tilde{x})$

$\displaystyle\rho_{\mathrm{Blup}}(p^{\sharp}e_{1})\big{|}_{U_{1}}$
$\displaystyle=-(\tilde{x}_{3}\tilde{\partial}_{2}+\tilde{x}_{2}\tilde{\partial}_{3}),$
$\displaystyle\rho_{\mathrm{Blup}}(p^{\sharp}e_{2})\big{|}_{U_{1}}$
$\displaystyle=\tilde{x}_{1}\tilde{x}_{3}\tilde{\partial}_{1}-\tilde{x}_{2}\tilde{x}_{3}\tilde{\partial}_{2}+(1-\tilde{x}_{3}^{2})\tilde{\partial}_{3},$
$\displaystyle\rho_{\mathrm{Blup}}(p^{\sharp}e_{3})\big{|}_{U_{1}}$
$\displaystyle=\tilde{x}_{1}\tilde{x}_{2}\tilde{\partial}_{1}-(1+\tilde{x}_{2}^{2})\tilde{\partial}_{2}-\tilde{x}_{2}\tilde{x}_{3}\tilde{\partial}_{3}.$

Thus, on the invariant submanifold $\ch@airxmathbbm{P}\cap U_{1}$, the orbit
foliation is spanned by

$\\{\underbrace{\tilde{x}_{3}\tilde{\partial}_{2}+\tilde{x}_{2}\tilde{\partial}_{3}}_{(I)},\underbrace{(1+\tilde{x}_{2}^{2})\tilde{\partial}_{2}+\tilde{x}_{2}\tilde{x}_{3}\tilde{\partial}_{3}}_{(II)},\underbrace{\tilde{x}_{2}\tilde{x}_{3}\tilde{\partial}_{2}-(1-\tilde{x}_{3}^{2})\tilde{\partial}_{3}}_{(III)}\\}.$

If $\tilde{x}_{3}=0$ then this equals the span of
$\\{\tilde{\partial}_{2},\tilde{\partial}_{3}\\}$, which is two-dimensional.
If $\tilde{x}_{3}\neq 0$ then $(I)$ and $(II)$ span a two-dimensional subspace
as long as $1+\tilde{x}_{2}^{2}-\tilde{x}_{3}^{2}\neq 0$ as

$\det\begin{pmatrix}\tilde{x}_{2}&\tilde{x}_{3}\\\
(1-\tilde{x}_{3}^{2})&-\tilde{x}_{2}\tilde{x}_{3}\end{pmatrix}=-\tilde{x}_{3}(1+\tilde{x}_{2}^{2}-\tilde{x}_{3}^{2}).$

If $\tilde{x}_{2}=0$ and $\tilde{x}_{3}=1$, i.e. it is a point in $Z$, the
span clearly is one-dimensional. Thus let $\tilde{x}_{2}\neq 0$ and
$1+\tilde{x}_{2}^{2}=\tilde{x}_{3}^{2}$. Then

$\displaystyle\tilde{x}_{3}(II)+\tilde{x}_{2}(I)$
$\displaystyle=\tilde{x}_{3}\tilde{\partial}_{2}+\tilde{x}_{2}\tilde{\partial}_{3},$
(6.34) $\displaystyle\tilde{x}_{3}(II)-\tilde{x}_{2}(I)$
$\displaystyle=\tilde{x}_{3}(\tilde{x}_{2}^{2}+\tilde{x}_{3}^{2})\tilde{\partial}_{2}+\tilde{x}_{2}(\tilde{x}_{2}^{2}+\tilde{x}_{3}^{2})\tilde{\partial}_{3},$

both of which are multiples of
$\tilde{x}_{3}\tilde{\partial}_{2}+\tilde{x}_{2}\tilde{\partial}_{3}$, which
shows that the image of $\rho$ over points in $Z\cap U_{1}$ is one-
dimensional. Similar computations for the other charts show that the orbit
foliation is indeed as stated. $\ch@irxboxempty$

Thus by blowing up the foliation has become less singular. Since $Z$ is
completely contained in $U=U_{3}\setminus\\{[0:0:1]\\}$, in showing that the
remaining singularity cannot be resolved in the sense that always at least one
orbit will be one-dimensional, we can restrict the discussion to this open
subset. Firstly, in this chart let us introduce polar coordinates for
$(\tilde{x}_{1},\tilde{x}_{2})$ by

$(\tilde{x}_{1},\tilde{x}_{2},\tilde{x}_{3})=((r+1)\cos\phi,(r+1)\sin\phi,\tilde{x}_{3}),$
(6.35)

where $r\in(-1,\infty)$.

######  Lemma 6.13.

Over $U$ the image of the anchor map is generated by

$\\{\partial_{\phi},r(r+2)\partial_{r}-(r-1)\tilde{x}_{3}\tilde{\partial}_{3}\\}.$
(6.36)

######  Proof:

Over $U_{3}$, the anchor of the blowup maps

$\displaystyle\rho_{\mathrm{Blup}}(p^{\sharp}e_{1})\big{|}_{U_{3}}$
$\displaystyle=\tilde{x}_{1}\tilde{x}_{2}\tilde{\partial}_{1}+(\tilde{x}_{2}^{2}-1)\tilde{\partial}_{2}-\tilde{x}_{2}\tilde{x}_{3}\tilde{\partial}_{3},$
$\displaystyle\rho_{\mathrm{Blup}}(p^{\sharp}e_{2})\big{|}_{U_{3}}$
$\displaystyle=(1-\tilde{x}_{1}^{2})\tilde{\partial}_{1}-\tilde{x}_{1}\tilde{x}_{2}\tilde{\partial}_{2}+\tilde{x}_{1}\tilde{x}_{3}\tilde{\partial}_{3},$
$\displaystyle\rho_{\mathrm{Blup}}(p^{\sharp}e_{3})\big{|}_{U_{3}}$
$\displaystyle=\tilde{x}_{2}\tilde{\partial}_{1}-\tilde{x}_{1}\tilde{\partial}_{2}.$

The change of coordinates leads to

$\displaystyle\rho_{\mathrm{Blup}}(p^{\sharp}e_{1})\big{|}_{U}$
$\displaystyle=r(r+2)\sin\phi\partial_{r}-\tfrac{1}{r+1}\cos\phi\partial_{\phi}-(r+1)\tilde{x}_{3}\sin\phi\tilde{\partial}_{3},$
$\displaystyle\rho_{\mathrm{Blup}}(p^{\sharp}e_{2})\big{|}_{U}$
$\displaystyle=-r(r+2)\cos\phi\partial_{r}-\tfrac{1}{r+1}\sin\phi\partial_{\phi}+(r+1)\tilde{x}_{3}\cos\phi\tilde{\partial}_{3},$
$\displaystyle\rho_{\mathrm{Blup}}(p^{\sharp}e_{3})\big{|}_{U}$
$\displaystyle=-\partial_{\phi},$

which implies the statement. Indeed, the collection $\\{f_{1},f_{2},f_{3}\\}$
with $f_{1}=\sin\phi p^{\sharp}e_{1}-\cos\phi p^{\sharp}e_{2}$,
$f_{2}=\cos\phi p^{\sharp}e_{1}+\sin\phi p^{\sharp}e_{2}$ and
$f_{3}=p^{\sharp}e_{3}$ constitutes a frame over $U$, where $f_{3}$ and
$f_{1}$ are mapped to (6.36), while $\rho_{\mathrm{Blup}}(f_{2})$ is a
multiple of $\partial_{\phi}$. $\ch@irxboxempty$

Setting $s=\frac{r}{(r+2)^{3}}$ one can simplify (6.36) in the sense that the
image of the anchor is then spanned by

$\\{\partial_{\phi},2s\partial_{s}+\tilde{x}_{3}\tilde{\partial}_{3}\\}.$
(6.37)

######  Remark 6.14.

From Lemma 6.2 we also find that $\tilde{A}_{\ch@airxmathbbm{P}}$ is an
abelian extension

$0$$L$
$\tilde{A}_{\ch@airxmathbbm{P}}$$\mathrm{Blup}(T\ch@airxmathbbm{P},TZ)$$0$
(6.38)

where $(L\subseteq\ker\rho_{\mathrm{Blup}})_{\ch@airxmathbbm{P}}$ is an
abelian Lie algebroid of rank $1$.

However, the generating sections (6.37) of
$\rho_{\mathrm{Blup}}(\tilde{A}_{U})$ show that we cannot remove the
singularity by blowing up further. There will always remain at least one leaf
of dimension $1$.

######  Lemma 6.15.

Let $U\subseteq\ch@airxmathbbm{R}^{n}$ be open with $0\in U$ and
$a_{1},\dots,a_{n}\in\mathscr{C}^{\infty}(U)$. Then the section

$\sum_{i}x_{i}a_{i}\partial_{i}\in\mathfrak{X}(U)$ (6.39)

induces a section $s\in\Gamma^{\infty}(\mathrm{Blup}(TU,\\{0\\}))$ of the Lie
algebroid blowup $\mathrm{Blup}(TU,\\{0\\})$, and $\rho_{\mathrm{Blup}}(s)$
vanishes on the subset $Z=\\{[1:0:\dots:0],\dots,[0:\dots:0:1]\\}$.

Moreover, around points in $Z$ the vector field $\rho_{\mathrm{Blup}}(s)$ is
again of the form (6.39).

######  Proof:

Fix $i\in\\{1,\dots,n\\}$. Writing $p\colon\mathrm{Blup}(U,\\{0\\})\to U$ for
the blow-down map, we have inside the chart $(U_{i},\tilde{x})$

$\rho_{\mathrm{Blup}}(s)\big{|}_{U_{i}}=\tilde{x}_{i}p^{\ast}(a_{i})\tilde{\partial}_{i}+\sum_{j\neq
i}\tilde{x}_{j}(p^{\ast}(a_{j})-p^{\ast}(a_{i}))\tilde{\partial}_{j}.$

This section vanishes in the origin of $U_{i}$, i.e. the point
$[0:,\dots,:\underbrace{1}_{i}:\dots:0]$, and is of the form (6.39).
$\ch@irxboxempty$

######  Corollary 6.16.

Repeatedly blowing up the one-dimensional orbits will not result in a Lie
algebroid that has a $2$-dimensional orbit foliation. Every blowup along a
single $1$-dimensional orbit will increase the number of $1$-dimensional
orbits by $1$.

######  Proof:

After the first blowup the statement follows inductively from Lemma 6.2.
Indeed, considering the sections (6.37) that generate the image of the anchor
map around the one-dimensional orbit, the section
$2s\partial_{s}+\tilde{x}_{3}\tilde{\partial}_{3}$ satisfies the requirements
of Lemma 6.2. $\ch@irxboxempty$

In conclusion, blowing up further will only increase the number of
singularities.

##  Appendix A A Gysin sequence for Lie algebroids

We develope a Gysin sequence for Lie algebroids, which we made use of in
Section 5. To be able to formulate the result we first introduce a notion of
fibre integration for Lie algebroids.

###  A.1 Integration along fibres

Throughout this section let $\mathrm{pr}\colon F\to N$ denote a locally
trivial fibre bundle with orientable and connected fibres and $B\Rightarrow N$
a Lie algebroid. We aim to define a notion of integrating along fibres

$(\pi^{!})_{\ast}\colon\mathrm{H}^{\bullet}_{\mathrm{cv}}(\pi^{!}B)\to\mathrm{H}^{\bullet-\operatorname{\mathrm{rank}}F}(B,o(F)),$
(A.1)

where $\mathrm{H}^{\bullet}_{\mathrm{cv}}(\pi^{!}B)$ denotes compact vertical
cohomology, see Definition 5. Here, $o(F)\to N$ denotes the orientation bundle
of $F$. This is standard for vector bundles, but can also be made sense of in
our more general situation. Since the fibres are orientable we can construct a
double cover $\tilde{N}$ of $N$ by

$\tilde{N}_{p}=\\{\lambda\colon\lambda\text{ is an orientation on the manifold
}F_{p}\\}$ (A.2)

with the obvious smooth structure, which we call the orientation double cover.
On the trivial bundle $\tilde{N}\times\ch@airxmathbbm{R}\to\tilde{N}$ we have
a $\ch@airxmathbbm{Z}_{2}$ action given by

$(-1).(\lambda,t)=(\bar{\lambda},-t),$ (A.3)

where $\bar{\lambda}$ denotes the opposite orientation. Then

$o(F)=\tilde{N}\times\ch@airxmathbbm{R}/\ch@airxmathbbm{Z}_{2}\to N.$ (A.4)

The bundle (A.4) is trivial if and only if there is a globally consistent way
of choosing orientations for the fibres of $F$. In this case, we call the
fibre bundle $F\to N$ orientable.

######  Remark A.1.

If $F\to N$ is a fibre bundle such that $o(F)$ is not the trivial bundle,
pulling back $F$ to $\tilde{N}$ will result in an orientable fibre bundle.

The orientation bundle carries a canonical flat connection of $TN$ (induced by
the trivial $T\tilde{N}$-connection on $\tilde{N}\times\ch@airxmathbbm{R}$)
analogous to [5, Chapter 7] in the case of vector bundles, and the
representation of $B$ on $o(F)$ is induced by this connection using the anchor
map.

To define fibre integration we first consider the case $B=0$, i.e.
$\pi^{!}B=\mathcal{F}(\pi)$ is the Lie algebroid corresponding to the
foliation of $F$ into the fibres of $\pi$.

######  Definition A.2.

Let $\pi\colon F\to N$ be a locally trivial fibre bundle of rank $k$ with
orientable and connected fibres. Then fibre integration on
$\Omega^{\bullet}_{\mathrm{cv}}(\mathcal{F}(\pi))$ is defined by

$\int_{\mathcal{F}(\pi)}\omega=\begin{cases}\Big{(}N\ni
p\mapsto\int_{F_{p}}\iota_{p}^{\ast}\omega\Big{)}\in\Gamma^{\infty}(o(F))&\text{
if }\omega\in\Omega^{k}_{\mathrm{cv}}(\mathcal{F}(\pi))\\\ 0&\text{
otherwise}.\end{cases}$ (A.5)

Here, $\iota_{p}\colon F_{p}\to F$ denotes the inclusion of the fibre at $p\in
N$.

######  Remark A.3.

Fibre integration is well defined and coincides with the standard notion of
fibre integration in the following sense: Since
$\mathcal{F}(\pi)\hookrightarrow TF$ we can choose a vector bundle complement
(a horizontal subbundle) $C$, i.e. $TF=\mathcal{F}\oplus C$, to extend
foliated forms to forms on $TF$. The result of the ordinary fibre integration
will not depend on the chosen complement (since it is a top-degree form of
$\mathcal{F}(\pi)$) and coincides with (A.5).

######  Proposition A.4.

Fibre integration descends to a map

$\int_{\mathcal{F}(\pi)}\colon\mathrm{H}^{k}_{\mathrm{cv}}(\mathcal{F}(\pi))\to\Gamma^{\infty}(o(F)).$
(A.6)

######  Proof:

We only need to check that integration vanishes on exact forms. But, since we
have for foliated forms $\omega\in\Omega^{\bullet}(\mathcal{F}(\pi))$ that

$\mathop{}\\!\mathrm{d}_{F_{p}}\iota^{\ast}_{p}\omega=\iota^{\ast}_{p}\mathop{}\\!\mathrm{d}_{\mathcal{F}}\omega,$

this follows from Stoke’s theorem. $\ch@irxboxempty$

Recall that in case of an orientable vector bundle $F\to N$, ordinary fibre
integration of differential forms yields an isomorphism

$\mathrm{H}_{\mathrm{cv}}^{\bullet}(F)\simeq\mathrm{H}^{\bullet-k}(N),$ (A.7)

where the pre-image of $1\in\mathrm{H}^{0}(N)$ is the Thom class
$\theta\in\mathrm{H}^{k}_{\mathrm{cv}}(F)$ (see [5, (12.2.1)]). In our setting
we obtain a similar statement in Lemma A.1, which in the current situation is
the following.

######  Lemma A.5.

Let $\pi\colon F\to N$ be a vector bundle of rank $k$. Then

$\mathrm{H}^{n}_{\mathrm{cv}}(\mathcal{F}(\pi))=\begin{cases}\Gamma^{\infty}(o(F))&\text{
if }n=k\\\ 0&\text{ otherwise},\end{cases}$ (A.8)

where we identify the $\mathscr{C}^{\infty}(N)$-modules

$\int_{\mathcal{F}(\pi)}\colon\mathrm{H}^{k}_{\mathrm{cv}}(\mathcal{F}(\pi))\xrightarrow{\sim}\Gamma^{\infty}(o(F))$
(A.9)

If $F$ is oriented then for
$1\in\mathscr{C}^{\infty}(N)=\Gamma^{\infty}(o(F))$ we have

$\mathopen{}\mathclose{{}\left(\int_{\mathcal{F}(\pi)}}\right)^{-1}(1)=\rho_{\mathcal{F}}^{\ast}\theta,$
(A.10)

where $\theta\in\mathrm{H}^{k}_{\mathrm{cv}}(F)$ denotes the Thom class and
$\rho_{\mathcal{F}}$ denotes the anchor of $\mathcal{F}(\pi)$.

######  Proof:

We first show (A.9). Note that integrating is indeed a map of
$\mathscr{C}^{\infty}(N)$-modules, where the module structure on
$\mathrm{H}^{\bullet}_{\mathrm{cv}}(\mathcal{F}(\pi))$ is given by multiplying
with pullbacks. The statement in the non-orientable case follows from the
orientable one by pulling everything back to a double cover $\tilde{N}\to N$
which trivialises $o(F)$. Thus, suppose $F$ is orientable and let
$\theta\in\mathrm{H}^{k}_{\mathrm{cv}}(F)$ denote the Thom class of the
orientable vector bundle. We can pick any representative of the Thom class to
compute its integral, and for the computation we only need the contributions
that are $k$-tangent to the foliation since the rest is mapped to zero anyway.
But those are given by $\rho_{\mathcal{F}}^{\ast}\theta$, so we get

$\int_{\mathcal{F}(\pi)}\rho_{\mathcal{F}}^{\ast}\theta=1\in\mathscr{C}^{\infty}(N)$

immediately. Since integrating is a module morphism, this implies
surjectivity. For injectivity it is enough to argue locally, since we can
exploit the $\mathscr{C}^{\infty}(N)$-module structure and the existence of a
partition of unity on $N$. Thus, let $F=N\times\ch@airxmathbbm{R}^{k}$ with
coordinates $(x,y)$ and

$\omega=f\mathop{}\\!\mathrm{d}y_{1}\wedge\cdots\wedge\mathop{}\\!\mathrm{d}y_{k}\in\Omega^{k}_{\mathrm{cv}}(\mathcal{F}(\pi))$

be given, where $f\in\mathscr{C}^{\infty}_{\mathrm{cv}}(F)$. Suppose that
$\int_{\mathcal{F}(\pi)}\omega=0$, i.e.

$\int f(x,y)dy=0$

for every $x\in N$. One can adapt the proof of [16, Lemma 2.4] to show that
there exist functions
$g_{1},\ldots,g_{k}\in\mathscr{C}^{\infty}_{\mathrm{cv}}(F)$ such that

$f=\sum_{i=1}^{k}\frac{\partial g_{i}}{\partial y_{i}}.$

Then $\eta\in\Omega^{k-1}_{\mathrm{cv}}(\mathcal{F}(\pi))$ defined by

$\eta=\sum_{i=1}^{k}(-1)^{i+1}g_{i}\mathop{}\\!\mathrm{d}y_{1}\wedge\cdots\mathop{}\\!\mathrm{d}y_{i-1}\wedge\mathop{}\\!\mathrm{d}y_{i+1}\wedge\cdots\wedge\mathop{}\\!\mathrm{d}y_{k}$

is a primitive for $\omega$.

To show that $\mathrm{H}^{n}_{\mathrm{cv}}(\mathcal{F}(\pi))=0$ if $n\neq k$,
first note that for $n=0$ the statement is clear. For $1\leq n<k$ note that
again $\mathrm{H}^{n}_{\mathrm{cv}}(\mathcal{F}(\pi))$ is a
$\mathscr{C}^{\infty}(N)$-module. Hence it is enough to show that the
cohomology vanishes over a small enough open subsets of $N$, where we utilise
the proof of the well-known Poincaré-Lemma for compact support. Let $U\subset
N$ be relatively compact such that $F$ is trivial over an open neighbourhood
of the closure of $U$. Let
$\omega\in\Omega^{n}_{\mathrm{cv}}(\mathcal{F}(\pi))$ be closed. By definition
of $\Omega^{n}_{\mathrm{cv}}(\mathcal{F}(\pi))$ and by compactness of $K$
there exists a compact subset $A\subseteq\ch@airxmathbbm{R}^{k}$ such that

$\operatorname{\mathrm{supp}}\omega\big{|}_{\pi^{-1}(U)}\subseteq A\times U.$

Denote by $i\colon\ch@airxmathbbm{R}^{k}\hookrightarrow\ch@airxmathbbm{S}^{k}$
the embedding of $\ch@airxmathbbm{R}^{k}$ into the $k$-sphere via the
stereographic projection of the north pole $\mathcal{N}$, and by

$i_{\ast}\colon\Omega^{n}_{\mathrm{cv}}(\mathcal{(}\pi)_{\pi^{-1}(U)})\to\Omega_{\mathcal{F}}^{n}(\ch@airxmathbbm{S}^{k}\times
U)$

the extension by $0$ as a foliated form on $S^{k}\times U\to U$. Then, since
$\mathrm{H}^{n}_{\mathcal{F}}(\ch@airxmathbbm{S}^{k}\times U)=0$ by [28, Lemma
4.8], and $\mathop{}\\!\mathrm{d}i_{\ast}\omega=0$, there exists
$\eta\in\Omega^{n-1}_{\mathcal{F}}(\ch@airxmathbbm{S}^{k}\times U)$ with
$\mathop{}\\!\mathrm{d}_{\mathcal{F}}\eta=i_{\ast}\omega$. Let
$O\subset\ch@airxmathbbm{S}^{k}\setminus i(A)$ be a contractible neighbourhood
of $\mathcal{N}$ and $\chi$ a bump function which is $1$ in a neighbourhood of
$\mathcal{N}$ and supported on $O$. On $O\times U$, we have

$\mathop{}\\!\mathrm{d}_{\mathcal{F}}\eta\big{|}_{O\times
U}=i_{\ast}\omega\big{|}_{O\times U}=0.$ ($\ast$)

Thus, if $n>1$, there exists $\tilde{\eta}\in\Omega^{n-2}(O\times U)$ with
$\mathop{}\\!\mathrm{d}_{\mathcal{F}}\tilde{\eta}=\eta\big{|}_{O\times U}$,
again by [28, Lemma 4.8]. Then

$i^{\ast}(\eta-\mathop{}\\!\mathrm{d}_{\mathcal{F}}(\chi\tilde{\eta}))\in\Omega_{\mathrm{cv}}^{n-1}(\ch@airxmathbbm{R}^{k}\times
U)$

is a well-defined primitive of $\omega$.

If $n=1$, then $(\ast)$ implies that $\eta\big{|}_{O\times U}$ is the pullback
of a function $f\in\mathscr{C}^{\infty}(U)$. Hence, in this case
$i^{\ast}(\eta-\pi_{\ch@airxmathbbm{S}^{k}\times
U}^{\ast}f)\in\mathscr{C}^{\infty}_{\mathrm{cv}}(\ch@airxmathbbm{R}^{k}\times
U)$ is a primitive of $\omega$. $\ch@irxboxempty$

If the fibres of $\pi\colon F\to N$ are compact, orientable, and connected the
situation, we obtain a similar statement for the top degree foliated
cohomology.

######  Lemma A.6.

Let $\pi\colon F\to N$ be a locally trivial fibre bundle with typical fibre a
compact, orientable, and connected manifold of dimension $k$. Then fibre
integration yields an isomorphism

$\int_{\mathcal{F}(\pi)}\colon\mathrm{H}^{k}(\mathcal{F}(\pi))\xrightarrow{\sim}\Gamma^{\infty}(o(F))$
(A.11)

######  Proof:

By [28, Lemma 4.8], the foliated cohomology $\mathrm{H}^{k}(\mathcal{F}(\pi))$
are sections of a line bundle over $N$ with fibres
$\mathrm{H}^{k}(\pi^{-1}(x))$ for all $x\in N$, which can be readily
identified with $o(F)$ via fibre integration. $\ch@irxboxempty$

This concludes the discussion for $\pi^{!}B$, where $B\Rightarrow N$ is the
zero Lie algebroid. For a general Lie algebroid we can first decompose the
forms on $\pi^{!}B$ according to their vertical part.

######  Lemma A.7.

Let $F\to N$ be a locally trivial fibre bundle and $B\Rightarrow N$ a Lie
algebroid. Picking a connection on $F$ leads to a decomposition

$\pi^{!}B=\operatorname{\mathrm{Ver}}(F)\oplus\pi^{\sharp}B$ (A.12)

with the property that the anchor is given by the identity on vertical vectors
and maps $\pi^{\sharp}a\mapsto\rho(a)^{\mathrm{hor}}$ for
$a\in\Gamma^{\infty}(B)$, and $\pi^{!}\colon\pi^{!}B\to B$ is given by
$\pi^{!}=\pi^{\sharp}\circ(\mathrm{pr}_{\pi^{!}B\to\pi^{\sharp}B})$.

######  Proof:

Given a connection on $E$ and local frames $\\{s_{\alpha}\\}_{\alpha}$ of
$F_{U}$ and $\\{a_{i}\\}_{i}$ of $B_{U}$ for some open $U\subseteq N$, the
collection

$\\{(0,s_{\alpha}^{\mathrm{ver}})\\}_{\alpha}\cup\\{(\pi^{\sharp}a,\rho(a_{i})^{\mathrm{hor}})\\}_{i}$

yields a local frame for $\pi^{!}B_{F_{U}}$. Then the statements follow
immediately. $\ch@irxboxempty$

Thus, by choosing a connection on $F$ we obtain a decomposition

$\Omega_{\mathrm{cv}}^{\bullet}(\pi^{!}B)=\bigoplus_{p+q=\bullet}\Omega^{p}_{\mathrm{cv}}(\mathcal{F}(\pi),\pi^{\sharp}\Lambda^{q}B^{\ast}).$
(A.13)

This allows to define fibre integration for forms on $\pi^{!}B$ by just
integrating out the fibre components. More precisely, for
$\omega=\sum_{i,j}\eta_{i,j}\mathbin{\otimes}\pi^{\sharp}\alpha_{i,j}$, where
$\eta_{i,j}\in\Omega^{j}_{\mathrm{cv}}(\mathcal{F}(\pi))$ and
$\alpha_{i,j}\in\Omega^{\bullet}(B)$, we define

$(\pi^{!})_{\ast}\Big{(}\sum_{i,j}\eta_{i,j}\mathbin{\otimes}\pi^{\sharp}\alpha_{i,j}\Big{)}=\sum_{i}\Big{(}\int_{\mathcal{F}(\pi)}\eta_{i,k}\Big{)}\alpha_{i,k}.$
(A.14)

It is clear that (A.14) yields a map

$(\pi^{!})_{\ast}\colon\Omega_{\mathrm{cv}}^{\bullet}(\pi^{!}B)\to\Omega^{\bullet-k}(B,o(F)).$
(A.15)

Since only contributions of $\Omega^{k}(\mathcal{F}(\pi))$ (i.e. of top
degree) matter in computing the integral, it does not depend on the chosen
connection, which is also emphasised by the following description. Consider
$\omega\in\Omega^{\bullet}_{\mathrm{cv}}(\pi^{!}B)$ and fix $p\in N$. Then we
can define the $k$-fold restriction
$\omega\big{|}_{F_{p}}^{k}\in\Omega^{k}_{c}(F_{p},\Lambda^{\bullet-k}B_{p}^{\ast})$
of $\omega$ to $F_{p}$ in the following way: For
$X_{1},\dots,X_{k}\in\Gamma^{\infty}(TF_{p})$ we define

$\omega\big{|}_{F_{p}}^{k}(X_{1},\dots,X_{k})\colon
F_{p}\to\Lambda^{\bullet-k}(\pi^{\sharp}B^{\ast})\big{|}_{F_{p}}=\Lambda^{\bullet-k}(B^{\ast}_{p})$
(A.16)

in the obvious way, as $\pi^{!}B/\mathcal{F}(\pi)=\pi^{\sharp}B$. Then one
integrates this form along $F_{p}$, which yields the same result as (A.14)
(note that the integral will vanish if $\omega$ is not of foliated degree
$k$).

######  Lemma A.8.

Let $F\to N$ be a locally trivial fibre bundle of rank $k$ with orientable and
connected fibres with trivial orientation bundle, and $B\Rightarrow N$ a Lie
algebroid. Then
$(\pi^{!})_{\ast}\colon\Omega^{\bullet}_{\mathrm{cv}}(\pi^{!}B)\to\Omega^{\bullet-k}(B)$
is, up to a sign depending on the degree, a morphism of cochain complexes and
induces a map

$(\pi^{!})_{\ast}\colon\mathrm{H}^{\bullet}_{\mathrm{cv}}(\pi^{!}B)\to\mathrm{H}^{\bullet-k}(B).$
(A.17)

######  Proof:

Since $(\pi^{!})_{\ast}$, $\mathop{}\\!\mathrm{d}_{\pi^{!}B}$ and
$\mathop{}\\!\mathrm{d}_{B}$ are local in the sense that to calculate them in
a point $p\in N$ we only need information about the form on $F_{U}$ and $U$,
respectively, where $U$ is a neighbourhood of $p$. Thus, suppose that
$\pi\colon F=\tilde{F}\times N\to N\tilde{F}$ is a product bundle. Then the
canonical flat connection induces a decomposition

$\Omega^{n}_{\mathrm{cv}}(\pi^{!}B)\simeq\bigoplus_{i+j=n}\Omega^{i}_{\mathrm{cv}}(\mathcal{F}(\pi))\otimes_{\mathscr{C}^{\infty}(B)}\Omega^{j}(B).$

Under this decomposition, we obtain the following.

  1. 1.

By flatness of the connection the differential on
$\Omega^{n}_{\mathrm{cv}}(\pi^{!}B)$ splits into

$\mathop{}\\!\mathrm{d}_{\pi^{!}B}=\mathop{}\\!\mathrm{d}_{\mathcal{F}(\pi)}\otimes\operatorname{\mathrm{id}}+(-1)^{j}\operatorname{\mathrm{id}}\otimes\mathop{}\\!\mathrm{d}_{B}.$

  2. 2.

By definition of $(\pi^{!})_{\ast}$ we have for
$\omega\in\Omega^{\bullet}_{\mathrm{cv}}(\mathcal{F}(\pi))$ and
$\alpha\in\Omega^{\bullet}(B)$ that

$(\pi^{!})_{\ast}(\omega\mathbin{\otimes}(\pi^{!})^{\alpha})=((\pi^{!})_{\ast}\omega)\otimes(\pi^{!})^{\ast}\alpha.$

  3. 3.

By Lemma A.1 we have

$(\pi^{!})_{\ast}\mathop{}\\!\mathrm{d}_{\mathcal{F}(\pi)}\Omega^{\bullet}_{\mathrm{cv}}(\mathcal{F}(\pi))=0.$

Together, this implies the statement. $\ch@irxboxempty$

With the fibre integration on cohomology defined, we get a Thom isomorphism
for Lie algebroids as a consequence of Lemma A.1.

######  Lemma A.9 (Thom isomorphism for Lie algebroids).

Let $\pi\colon E\to N$ be a vector bundle of rank $k$ with orientation bundle
$o(E)$ and $B\Rightarrow N$ a Lie algebroid. Then fibre integration

$(\pi^{!})_{\ast}\colon\mathrm{H}_{\mathrm{cv}}^{\bullet}(\pi^{!}B)\to\mathrm{H}^{\bullet-k}(B,o(E))$
(A.18)

is an isomorphism. If
$\theta\in\mathrm{H}^{k}_{\mathrm{cv}}(E,\pi^{\sharp}o(E)^{\ast})$ denotes the
Thom class of the vector bundle, the inverse of $(\pi^{!})_{\ast}$ is given by

$\mathrm{H}^{\bullet-k}(B,o(E))\ni\omega\mapsto\rho_{\pi^{!}B}^{\ast}\theta\wedge(\pi^{!})^{\ast}\omega\in\mathrm{H}_{\mathrm{cv}}^{\bullet}(\pi^{!}B).$
(A.19)

######  Proof:

The statement follows from a spectral sequence argument. Consider the
filtration on $\Omega^{\bullet}_{\mathrm{cv}}(\pi^{!}B)$ given by

$\mathcal{F}^{p}\Omega^{\bullet}_{\mathrm{cv}}(\pi^{!}B)=(\Omega_{\mathrm{cv}}(\mathcal{F}(\pi))^{\wedge
p}\wedge\Omega_{\mathrm{cv}}(\pi^{!}B))^{\bullet}$

And the induced spectral sequence $\\{E_{r}^{p,q}\\}_{r\geq 0}$. By standard
arguments we obtain

$E_{1}^{p,q}\simeq\mathrm{H}^{q}(B,\mathrm{H}^{p}_{\mathrm{cv}}(\mathcal{F}(\pi))).$

Then by Lemma A.1 the statement follows. $\ch@irxboxempty$

###  A.2 The Gysin sequence

We prove a Gysin-like sequence, which we have used in Section 5, for the
cohomology of $\pi^{!}B$, where $B\Rightarrow N$ is a Lie algebroid and
$\pi\colon\ch@airxmathbbm{S}\to N$ is a sphere bundle of rank $k$, i.e. a
locally trivial fibre bundle with a $k$-dimensional sphere as typical fibre.
In this case, by the Serre spectral sequence for fibre bundles [28, Theorem
5.7] we obtain a spectral sequence converging to the Lie algebroid cohomology
of $\pi^{!}B$ with second page given by

$E_{2}^{p,q}=\mathrm{H}^{p}(B,\mathcal{H}^{q}(\ch@airxmathbbm{S})).$ (A.20)

Here, $\mathcal{H}^{q}(\ch@airxmathbbm{S})\to N$ denotes a smooth vector
bundle with fibres given by
$\mathcal{H}^{q}(\ch@airxmathbbm{S})_{x}=\mathrm{H}^{q}(\pi^{-1}(x))$ for
$x\in N$. Thus, $E_{2}^{p,q}$ has nontrivial entries only if $q=0$ or $q=k$.
Therefore the next (and last) nontrivial differential is
$\mathop{}\\!\mathrm{d}_{k+1}$. Recall that in case of $B=TN$ and trivial
orientation bundle it is given by $\mathop{}\\!\mathrm{d}_{k+1}=\wedge e$,
where $e\in\mathrm{H}^{k+1}(N)$ is the Euler class of the sphere bundle [5,
Chapter 11]. If $o(\ch@airxmathbbm{S})$ is nontrivial, we can consider the
pullback to a trivialising double cover $\tilde{N}\to N$ and find that
$e\in\mathrm{H}^{k+1}(\tilde{N})_{-}=\mathrm{H}^{k+1}(N,o(\ch@airxmathbbm{S}))$
instead.

######  Theorem A.10.

Let $B\Rightarrow N$ be a Lie algebroid with anchor $\rho$ and
$\pi\colon\ch@airxmathbbm{S}\to N$ a sphere bundle of rank $k$. Then there is
a long exact sequence

$\dots$$\mathrm{H}^{\bullet}(B)$$\mathrm{H}^{\bullet}(\pi^{!}B)$$\mathrm{H}^{\bullet-k}(B,o(\ch@airxmathbbm{S}))$$\mathrm{H}^{\bullet+1}(B)$$\dots$$(\pi^{!})^{\ast}$$(\pi^{!})_{\ast}$$\wedge\rho^{\ast}e$
(A.21)

Here, $(\pi^{!})_{\ast}$ denotes fibre integration and
$e\in\mathrm{H}^{k+1}(N,o(\ch@airxmathbbm{S}))$ is the Euler class of the
sphere bundle.

######  Proof:

If $o(\ch@airxmathbbm{S})$ is nontrivial we can pull everything back to a
trivialising double cover. Using Lemma 2.2 and noting that integration and
$\wedge e$ map $\ch@airxmathbbm{Z}_{2}$-invariant classes to anti-invariant
ones and vice versa, one then obtains the result in general. Thus, let
$o(\ch@airxmathbbm{S})$ be trivial. The differential
$\mathop{}\\!\mathrm{d}_{k+1}\colon E_{2}^{0,k}\to E_{2}^{k+1,0}$ can be
computed by evaluating on a generator $s$. We can write
$s=\sum_{i}\chi_{i}[\omega_{i}]$, where
$\omega_{i}\in\Omega^{k}(\ch@airxmathbbm{S}_{U_{i}})$ are closed with respect
to the de Rham differential and $\chi_{i}\in\mathscr{C}^{\infty}(N)$. Let
$\gamma\colon\pi^{\sharp}B\to\pi^{!}B$ be a splitting,
$\omega=\sum_{i}\chi_{i}\omega_{i}$, and
$a_{0},\dots,a_{k}\in\Gamma^{\infty}(B)$. Using that for $B=TN$ we have
$\mathop{}\\!\mathrm{d}_{\mathrm{dR}}\omega=\pi^{\ast}e$, by a standard
calculation we obtain

$\displaystyle\mathop{}\\!\mathrm{d}_{k+1}\omega(\gamma$
$\displaystyle\pi^{\sharp}a_{0},\dots,\gamma\pi^{\sharp}a_{k})=\sum_{i=0}^{k}(-1)^{i}\rho_{\pi^{!}A}(\gamma\pi^{\sharp}a_{i})\omega(\gamma\pi^{\sharp}a_{0},\dots,\stackrel{{\scriptstyle
i}}{{\wedge}},\dots,\gamma\pi^{\sharp}a_{k})$
$\displaystyle\phantom{\pi^{\sharp}a_{0},\dots,\gamma\pi^{\sharp}a_{k})=}+\sum_{\mathclap{0\leq
i<j\leq
k}}(-1)^{i+j}\omega([\gamma\pi^{\sharp}a_{i},\gamma\pi^{\sharp}a_{j}],\gamma\pi^{\sharp}a_{0},\dots,\stackrel{{\scriptstyle
i}}{{\wedge}},\dots,\stackrel{{\scriptstyle
j}}{{\wedge}},\dots,\gamma\pi^{\sharp}a_{k})$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{k}(-1)^{i}\rho_{\pi^{!}A}(\gamma\pi^{\sharp}a_{i})\omega(\rho_{\pi^{!}A}(\gamma\pi^{\sharp}a_{0}),\dots,\stackrel{{\scriptstyle
i}}{{\wedge}},\dots,\rho_{\pi^{!}A}(\gamma\pi^{\sharp}a_{k}))$
$\displaystyle+\sum_{\mathclap{0\leq i<j\leq
k}}(-1)^{i+j}\omega(\rho_{\pi^{!}A}([\gamma\pi^{\sharp}a_{i},\gamma\pi^{\sharp}a_{j}]),\rho_{\pi^{!}A}(\gamma\pi^{\sharp}a_{0}),\dots,\stackrel{{\scriptstyle
i}}{{\wedge}},\dots,\stackrel{{\scriptstyle
j}}{{\wedge}},\dots,\rho_{\pi^{!}A}(\gamma\pi^{\sharp}a_{k}))$
$\displaystyle=$
$\displaystyle\mathop{}\\!\mathrm{d}_{\mathrm{dR}}\omega(\rho_{\pi^{!}A}(\gamma\pi^{\sharp}a_{0}),\dots,\rho_{\pi^{!}A}(\gamma\pi^{\sharp}a_{k}))$
$\displaystyle=$
$\displaystyle\pi^{\ast}e(\rho_{\pi^{!}A}(\gamma\pi^{\sharp}a_{0}),\dots,\rho_{\pi^{!}A}(\gamma\pi^{\sharp}a_{k}))$
$\displaystyle=$
$\displaystyle\pi^{\ast}(\rho_{A}^{\ast}e(a_{0},\dots,a_{k})).$

The rest then follows analogously to the proof of [5, Proposition 14.33].
$\ch@irxboxempty$

If $\ch@airxmathbbm{S}\to N$ actually comes from a vector bundle
$\mathrm{pr}\colon V\to N$, we can rewrite the Gysin sequence using the Thom
isomorphism on $\mathrm{H}^{\bullet-(k-1)}(B,o(\ch@airxmathbbm{S}))$.

######  Lemma A.11.

Let $B\Rightarrow N$ be a Lie algebroid with anchor $\rho$ and
$\mathrm{pr}\colon V\to N$ a vector bundle of rank $k$. Then, under the
isomorphism $\mathrm{H}^{\bullet}(B)=\mathrm{H}^{\bullet}(\mathrm{pr}^{!}B)$
induced by $(\iota^{!})^{\ast}$, the diagram

$\mathrm{H}^{\bullet-k}(B,o(\ch@airxmathbbm{S}))$$\mathrm{H}^{\bullet}(B)$$\mathrm{H}_{\mathrm{cv}}^{\bullet}(\mathrm{pr}^{!}B)$$\mathrm{H}^{\bullet}(\mathrm{pr}^{!}B)$$\rho^{\ast}e\wedge$$\Phi$$\cong$$i$$(\iota^{!})^{\ast}$$\cong$

commutes, where
$\Phi\colon\mathrm{H}^{\bullet-k}(B,o(\ch@airxmathbbm{S}))\to\mathrm{H}^{\bullet}_{\mathrm{cv}}(\mathrm{pr}^{!}B)$
denotes the Thom isomorphism from Lemma A.1, $\iota^{!}\colon
B\to\mathrm{pr}^{!}B$ is the inclusion of Lie algebroids over the zero section
$\iota\colon N\to V$, and $i$ denotes the natural map of considering a compact
vertical form as just a form on $\mathrm{pr}^{!}B$.

######  Proof:

First note that $o(V)=o(\ch@airxmathbbm{S})$ since the sphere bundle is
associated to $V$ [5, Proposition 11.2]. Again, we will only proof the
statement for orientable vector bundles. Writing
$\theta\in\mathrm{H}_{\mathrm{cv}}^{k}(V)$ for the Thom class, for any
$\omega\in\mathrm{H}^{\bullet}(B)$ we have

$(\iota^{!})^{\ast}\Phi(\omega)=(\iota^{!})^{\ast}\rho_{\mathrm{pr}^{!}B}^{\ast}\theta\wedge\omega=(\rho_{\mathrm{pr}^{!}B}\circ\iota^{!})^{\ast}\theta\wedge\omega=(T\iota\circ\rho)^{\ast}\theta\wedge\omega=\rho^{\ast}e\wedge\omega$

as the pullback of $\theta$ by the zero section is the Euler class.
$\ch@irxboxempty$

######  Corollary A.12.

Let $B\Rightarrow N$ be a Lie algebroid, and $\mathrm{pr}\colon V\to N$ a
vector bundle with induced sphere bundle $\pi\colon\ch@airxmathbbm{S}\to N$.
Then there is a long exact sequence

$\dots$$\mathrm{H}^{\bullet}(\mathrm{pr}^{!}B)$$\mathrm{H}^{\bullet}(\pi^{!}B)$$\mathrm{H}^{\bullet+1}_{\mathrm{cv}}(\mathrm{pr}^{!}B)$$\mathrm{H}^{\bullet+1}(\mathrm{pr}^{!}B)$$\dots$$(\pi^{!})^{\ast}$$(\pi^{!})_{\ast}$$i$
(A.22)

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A. Schüßler, KU Leuven, Department of Mathematics, Celestijnenlaan 200B box
2400, BE-3001 Leuven, Belgium, email<EMAIL_ADDRESS>