The document describes a proposed patient positioning system for maskless head and neck radiotherapy using a soft robot. The system uses a Kinect camera for vision-based sensing of patient head position. A soft robot consisting of an inflatable air bladder and pneumatic valves would manipulate the patient's head to correct for any motion during treatment. Preliminary results show the system was able to control 1 degree of freedom of motion (flexion/extension) of a mannequin head using proportional valve control and Kinect vision feedback to a control system. Further work is needed to validate the system for actual use in radiotherapy treatment.
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PhD Qualifying Exam Slides
1. A PATIENT
POSITIONING
SYSTEM FOR
MASKLESS HEAD
AND NECK
RADIOTHERAPY
Electrical Engineering Department, UT Dallas, TX
4/23/2015
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Presented by Olalekan Ogunmolu. April 23,
2015
2. Olalekan Ogunmolu: Bio Snippet
o Semester of PhD Enrolment: Fall 2014 GPA:
4.00/4.00
o Advisor: Nicholas Gans, Assistant Professor
o MSc (in Engineering) in Control Systems 2011 -
2012
The University of Sheffield, England, UK
MS Thesis: “Autonomous Navigation of a Rotorcraft Unmanned Aerial Vehicle Using
Machine Vision”, The University of Sheffield, South Yorkshire, England, Sept. 2012.
Advisor: Tony J. Dodd, Professor of Autonomous Systems Engineering, ACSE Dept.,
UoS.
o Research Thrusts
o Vision-based Control: Features tracking, Active Appearance Models, Neural
Networks
o Control Systems and Automation: Classical Control, Non-linear Systems,
Switched Systems
o Publication
o Olalekan Ogunmolu et. al, “A Real-Time Soft Robotic Patient Positioning System for
Maskless Head and Neck Cancer Radiotherapy”, IEEE Conference on Automation
and Systems Engineering [Submitted], Gothenburg, Sweden, August 2015Electrical Engineering Department, UT Dallas, TX 4/23/2015
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3. Overview of Presentation
A first-stage systematic examination of a novel method
at automating patient positioning systems during cancer
head and neck cancer radiotherapy (RT)
Visual-servoing control using a radio-transparent soft
robot on the flexion/extension cranial motion of a patient
during mask-less radiotherapy
Proof-of-concept validation of one degree of freedom
patient’s motion control with designed soft robot system
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4. Background and Motivation
Head and Neck (H&N) Cancers contribute nearly 3% to all cancer
developments in the United States [Jemal, A.]
Very dangerous to benign tissues/organs nearby malign cancerous formations in
H&N region
Treatment often involves intensity modulated radiotherapy (IMRT)
Clinical studies have shown that small perturbations cause high
sensitivity to IMRT treatment dose [Xing., L]
State-of-the-art robotic positioning treatments assume immobilized
patient motion on a 6D couch which is unrealistic
Electrical Engineering Department, UT Dallas, TX 4/23/2015
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5. Recent Research Results
Frameless and Maskless Cranial SRS, Cervino et al
Anthropomorphic head phantoms employed in checking the
accuracy of a 3D surface imaging system (AlignRT System)
Compared results from an infra-red optical tracking system with
the AlignRT software system
For different couch angles, the difference between phantom
positions recorded by the two systems were within 1mm
displacement and 1° rotation
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6. Recent Research Approaches
Recent Research Thrusts – 6D Motion Frameless SRS
Approach, Wiersma et al
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8. Research Overview
Proof-of-concept study and experiment demonstrating a 1-DOF
flexion/extension control of patient cranial motion during H&N Cancer RT
Testbed is a Mannequin head lying in a supine position on an inflatable air
bladder (IAB)
Soft-robot consists of the IAM, two two-port SMC Pnematics Co.
proportional valves, and silicone tubes for conveying air from a pressurized
air canister
A Kinect RGB-D camera is employed for head motion sensing and feedback
to a classical control network implemented on an NI myRIO hardware
Work in partnership with the Drs. Xuejun Gu and Steve Jiang of the
Radiation Oncology Department of UT Southwestern, Dallas, TX, USA
Electrical Engineering Department, UT Dallas, TX 4/23/2015
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9. Recent Research Results
Average max. volunteer head motion in the head mold during the
20 minutes interval in any direction was 0.7mm (range 0.4 –
1.1mm)
Patient motion due to couch motion was less than 0.2mm
Drawbacks
Minimal immobilization provided by head mold results in poor
positioning
Relies heavily on patient cooperation to achieve immobilization
Inter-fractional motions often ignored during pre-treatments
Electrical Engineering Department, UT Dallas, TX 4/23/2015
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10. Research Aims and Objectives
Aims
Accurate and automatic patient positioning system (pre-
treatment)
In-treatment automatic and accurate patient positioning with
patient drift compensation
Objectives
Surface-image control of the cranial flexion/extension motion of a
patient during simulated H&N RT (pre-treatment)
Use of radio-transparent soft robot system for
positioning/manipulation tasks
Electrical Engineering Department, UT Dallas, TX 4/23/2015
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11. Current System Set-up
Electrical Engineering Department, UT Dallas, TX 4/23/2015
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Kinect RGBD Camera Sensor
Hair packed with medic neck
pillow to reduce effect of
infrared wavelengths
scattering and minimize
dropped pixels
Inflatable Air Bladder (IAB)
Mannequin
Head
NI myRIO microcontroller
Current Source (Air Flow)
Regulator Circuit
Inlet/Outlet Silicone
Tubes
Torso Ball Joint
Simulator
Inlet Proportional Flow
Control Valve
Data Processing System
24V DC Power
Supply
12. Vision Sensing System
Kinect RGB-D Camera employed for position-based visual
servoing: Better depth image and alignment; Skeleton tracking
Real-time Human Pose Recognition in Parts from Single Depth
Images. Jamie Shotton, et. al, CVPR 2011, Best Paper Award
Real-time 3D face-tracking based on active appearance model
constrained by depth data. Nikolai Smolyanski et. al, Image and
Vision Computing, 2014, MS SDK v1.5.2
Online Resources: OpenNI, ofxKinect
OpenNI Process
Generates 640 × 480 image at 30 fps with depth resolution of 40
centimeters
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13. Kinect Depth Imaging
Works with low light levels; Color and texture invariant
Subtraction of background simplified
Easy synthesis of realistic depth images of objects
Cheap computational cost of building a large training dataset
3 trees, 20 deep, 300k training images per tree, 2000 training example
pixels per image, 2000 candidate features θ, and 50 candidate
thresholds τ
Electrical Engineering Department, UT Dallas, TX 4/23/2015
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14. Depth Constrained 3D Face Tracking
MS Kinect SDK v1.5.2
Uses depth data from the Kinect RGB-D Commodity camera to
enable 3D tracking using an active appearance method
image difference patterns corresponding to changes in each
model parameter are learned and used to modify a model
estimate
Resolves the monocular tracking problem by fitting an energy
term into the 2D+3D AAM fitting that minimizes a distance
between 3D face model vertices and depth data coming from a
RGBD camera.
Electrical Engineering Department, UT Dallas, TX 4/23/2015
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15. Depth Constrained 3D Face Tracking
Process Flow in Face Tracking System
Find a face rectangle in a video frame using a face detector
Use a neural network to find five points inside the face area – eye
centers, mouth corners, and tip of the nose
Precompute scale of tracked face from the five points un-
projected to 3D camera space and scale 3D camera space
appropriately.
Initialize next frame’s 2D face shape based on the
correspondences found by a robust local feature matching
between that frame and the previous frame.
Results
With the depth constrained 2D+3D AAM fitting, we found good
position-estimation results on a human subject when object is at a
distance of 1 to 2.5m from the Kinect System
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16. Face Tracking Results
Generalization errors and hence incorrect position estimation errors with
respect to the mannequin head due to inconsistency in depth data
An ongoing investigation
So we mostly relied on the OpenNI depth map centroids which we
computed for our position estimation
Electrical Engineering Department, UT Dallas, TX 4/23/2015
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Tracking of a Human Face Tracking of a Mannequin Head
17. Modeling of Soft Robot System
Modeling procedure: Overview
Collect Data Set, 𝒁 𝑵
, of input signal, 𝑢 𝑡 , and output measurement,
𝑦(𝑡), respectively where
𝒁 𝑵
= {𝑢 1 , 𝑦 1 , ⋯ , 𝑢 𝑁 , 𝑦 𝑁 }, 1 ≤ 𝑡 ≤ 𝑁 (1)
Obtain a continuous-time parametric model structure similar to a one-
step ahead predictor
𝑦 𝑛
= −𝑎 𝑛−1 𝑦 𝑛−1
− ⋯ − −𝑎0 𝑦 + 𝑏 𝑚 𝑢 𝑚
+ 𝑏 𝑚−1 𝑢 𝑚−1
+ ⋯ + 𝑏1 𝑢 + 𝑏0 𝑢 (2)
Introduce vectors
𝜽 = 𝑎 𝑛−1, ⋯ , 𝑎0 𝑏 𝑚−1, ⋯ , 𝑏0 and 𝝓 𝒕 = −𝑦 𝑛−1
⋯ − 𝑦 − 𝑦 𝑢 𝑚
⋯ 𝑢
(3)
Since the model output depends on past data (2), we call the estimated value
𝒚 𝑡 𝜃 . Therefore, 𝒚 𝑡 𝜃 = 𝝓 𝑻
𝑡 𝜽 .
Identification Goal: identify the best model, χ, in the set guided by the frequency
distribution analysis
Choice of Excitation Input Signal: Sawtooth Waveform
Integral and differential of a sawtooth waveform preserves the sawtooth
waveform with only phase and amplitude shifts
Electrical Engineering Department, UT Dallas, TX 4/23/2015
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18. Modeling of Soft Robot System
Spectrum contains both even and odd harmonics of the fundamental
frequency i.e. it contains all integer harmonics
𝑥 𝑠𝑡ℎ 𝑡 =
𝐴
2
−
𝐴
𝜋 𝑘=1
8800 sin 2𝜋𝑘𝑓𝑡
𝑘
(4)
where A is the amplitude, A = 180mA and f is the signal frequency, 𝑓 =
10𝐻𝑧
Fig 1. Excitation Signal and Corresponding Head Position (Kinect
Measurement)
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19. Data Pre-Processing
Data Pre-processing
We removed means and linear trends in collected data to minimize
disturbances above interest to desired system dynamics, mitigate
measurement outliers and non-continuous records in collected data
Means removal
𝒖 𝒂𝒗𝒆(𝒕) = 𝒖 𝑡 − 𝒖(𝑡) ; 𝒚 𝒂𝒗𝒆 𝑡 = 𝒚 𝑡 − 𝒚(𝑡); (5)
where 𝒖 𝒂𝒗𝒆(𝒕) and 𝒚 𝒂𝒗𝒆(𝒕) are the respective averages of the input and output
signals;
𝒖(𝒕) =
1
𝑁
𝑡=1
𝑛
𝑢 𝑡 𝑎𝑛𝑑 𝒚 (𝒕) =
1
𝑁
𝑡=1
𝑛
𝑦 𝑡 ;
n is the discrete data length and N is the total data length
Data Pre-processing
Data detrending
𝒖 𝒅 = 𝒖 𝒂𝒗𝒆 − 𝑨𝜽 𝒖, 𝒚 𝒅 = 𝒚 𝒂𝒗𝒆 − 𝑨𝜽 𝒚, (6)
𝜽 𝒖 𝑎𝑛𝑑 𝜽 𝒚 are solutions to the least square fit equations
𝑨 𝑻
𝑨 𝜽 𝑢 = 𝑨 𝑻
𝒖, 𝑨 𝑻
𝑨 𝜽 𝒚 = 𝑨 𝑇
𝒚
Electrical Engineering Department, UT Dallas, TX 4/23/2015
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20. Data Pre-Processing
and
𝑨 𝑻
=
1 1 ⋯ 1 1
1
𝑁
1
𝑁
⋯
1
𝑁
1
𝑁
(7)
Electrical Engineering Department, UT Dallas, TX 4/23/2015
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Fig 2. Data Preprocessing: Means Removal
21. Cross-Correlation Analysis
The cross-correlation function provides an estimate of the
system impulse response and is defined as:
𝜓 𝑡 = 𝑡=𝜏+1
𝑁 𝑢 𝑡 −𝜏 − 𝑢 𝑦 𝑡 −𝜏 −𝑦(𝑡)
√ 𝑡=1
𝑁 𝑢 𝑡 − 𝑢
2
√ 𝑡=1
𝑁 𝑦 𝑡 −𝑦(𝑡)
2, 𝜏 = 0, ±1, Λ, ± 𝑁 − 1 (8)
The cross-correlation function (CCF) is given by the convolution of the
system impulse response and the process auto-correlation function
(Wiener-Hopf equation)
𝜓 𝑢𝑦(𝜏)=∫ ℎ 𝜈 𝔼 [𝑢 𝑡 𝑢(𝑡 + 𝜏 − 𝜈)d𝜈 =∫ ℎ 𝜈 𝜓 𝑢𝑢(𝜏 − 𝜈)d𝜈
The cross correlation function between the output and test input is proportional to
the system impulse response when the input is white noise
𝒖 𝑡 = 𝒖 𝒘 𝑡
1
𝑭(𝑧−1)
where 𝒖 𝒘 𝑡 is a zero mean white input sequence and
𝑭 𝑧−1
is an autoregressive model filter thus defined: 𝑭 𝑧−1
= 1 + 𝜆1 𝑧−1
+ Λ𝜆 𝜆 𝑧−𝜆
Electrical Engineering Department, UT Dallas, TX 4/23/2015
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22. Cross-Correlation Analysis
We estimate the parameters 𝜆𝑗 𝑗 = 1, 2, Λ, 𝜆 by fitting an AR model to 𝑢(𝑡) using a
least squares algorithm.
The estimates of the filter, 𝑭 𝒛−𝟏
, are used to filter the input and output signals as
follows
𝑢 𝑤 𝑡 = 𝑭 𝒛−𝟏
𝑢 (𝑡)
𝑦 𝑤 𝑡 = 𝑭 𝒛−𝟏
𝑦 (𝑡)
(9)
By estimating the filter 𝑭 𝒛−𝟏 𝑤𝑖𝑡ℎ 𝜆 = 20, the estimate of the cross-
correlation function was generated. The auto-correlation function tells of the
quality of the filter 𝑭 𝒛−𝟏 and is defined as
𝜙11 𝜏 = 𝑡=𝜏+1
𝑁 𝒖 𝒘 𝑡 − 𝑢 𝑤 𝑡 [ 𝑢 𝑤 𝑡−𝜏 − 𝑢 𝑤]
𝑡=1
𝑁 [ 𝑢 𝑤 𝑡 − 𝑢 𝑤]2 , 𝜏 = 0, ±1, Λ, ± 𝑁 − 1
(10)
Electrical Engineering Department, UT Dallas, TX 4/23/2015
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25. Correlation of Residuals
To determine the model structure of the system, we used the original
detrended data
We chose a linear, second-order grey-box model set whose quality is
measurable by a mean-square error (MSE)
Our choice, garnered from erstwhile analysis, ensures cost of model is not
too high in solving for 𝜽 𝑁;
a high − order complex model is more difficult to use for simulation and
control design. If it is not marginally better than a simpler model, it may
not be worth the higher price [Llung, §16.8]
Electrical Engineering Department, UT Dallas, TX 4/23/2015
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26. Sub-model Selection & Model Validation
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A control system will perform well with an optimal linear sub-
model, tolerate disturbances and nonlinearities.
We pick the linear frequency range: 0.00232 rad/sec to 6.85
rad/sec to represented the model of the soft robot system
We carried out the canonical correlation to gain insight into the
desired model characteristics
The correlation is measured between the measured head
position, 𝒚(𝑡), and estimated position by the auto-regressive
model we chose, i.e. the residuals
𝝐 𝑡, 𝜽 𝑁 = 𝒚 𝑡 − 𝒚 𝑡 𝜽 𝑁
The prediction errors (model error model) in the obtained model are
computed as a frequency response from the input to the residuals
27. Model Validation
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Fig. 6. Correlation analysis of residuals between output, 𝒚(𝒕), and its estimate,
𝒚(𝒕)
Fig.5.Bodeplotofinputandoutput
Fig.6.Correlationfunctionofresiduals
Fig. 7. Estimates of standard variation of model from validation data set
28. Model Validation
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The confidence interval compares the estimate with the estimated standard deviation
from the validation dataset
A 99% confidence region (yellow bands) encloses the model response informing us
we have a reliable model [Llung (1999), §16.6]
Comparing Model Structure
Evaluation of different model structures and comparing quality of offered models
Best fit:
a second-order process model with delay and a RHP zero
𝐺 𝑠 =
−0.006(𝑠 −1.7137)
(𝑠+0.01)(𝑠+0.1028)
𝑒−2𝑠
(11)
The model has an 87.35% fit to original data with a mean square error of
0.05498𝑚𝑚2
and a final prediction error of 1.672.
The open-loop step response of the identified transfer function is shown in Fig. 8.
29. Soft Robot Step Response
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Fig. 8. Open Loop Step Response of Identified System
30. Control Analysis
We see that the system is non-minimum phase with very slow transient
response.
We require a controller that will increase the response time, guarantee
closed-loop stability whilst balancing robustness and controller
aggressiveness.
Approximating the delay with the second-order Pade function,
𝐻 𝑠 =
𝑠2 − 3𝑠 + 3
𝑠2 + 3𝑠 + 3
we introduce a PI controller,
𝐺𝑐 𝑠 = 3.79 +
0.0344
s
nested within a PID controller, 𝐺 𝑝𝑖𝑑 = 3.4993 +
0.054765
𝑠
+ 55.8988𝑠, as in Fig. 9.
Electrical Engineering Department, UT Dallas, TX 4/23/2015
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31. Close Loop Diagram and Step
Responses
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Fig 10. Closed Loop Step Response of Simulated
System
Fig. 9. Block Diagram of Control Network
Fig. 11. Experimental Results: Constant
Set-point
32. Experimental Results: Video
Video 1. Experimental Results: Set-points Tracking
Electrical Engineering Department, UT Dallas, TX 4/23/2015
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33. Conclusions and Future Work
Deviations from desired positions during H&N Cancer RT cause dose variations
and degenerate treatment efficacy
We have presented and demonstrated accurate control of cranial
flexion/extension motion of a patient during maskless H&N RT
Our soft robot system can accurately track desired trajectory within 14 seconds
after start-up with the aid of a PID/PI feedforward network
Future efforts include
Extending results to deformable motions of the upper torso, and H&N.
Improved bladder control: Adaptive Control, Gain Scheduling e.t.c.
Incorporation of multi-bladders to accommodate multi-axis positioning
Benefits
Comprehensive and accurate control of the patient’s position
Elimination of anatomical deformations as a result of positioning errors
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34. References
Cervino, L. I., et al. Frame-less and mask-less cranial stereotactic
radiosurgery: a feasibility study. 2010, Physics In Medicine And Biology
55(7): 1863-1873.
Jemal A, Siegel R, Xu J, Ward E. Cancer statistics, 2010. CA: A Cancer
Journal for Clinicians2010; 60(5):277–300.
L. Llung, System Identification Theory for the User, 2nd Edition, Upper
Saddle River, NJ, USA. Prentice Hall, 1999.
Xing, L. Dosimetric effects of patient displacement and collimator and
gantry angle misalignment on intensity modulated radiation therapy.
Radiotherapy & Oncology, 2000. 56(1): p. 97 - 108
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Editor's Notes
~the head ring locks the patient’s head into a precise 3D position suppressing both voluntary and involuntary physiological motions
~head ring may not necessarily fit into patients’ head with prior intracranial surgery or other anatomical inhibitions
A 3mm deviation of couch position in anterior-posterior position resulted in 38% decrease in minimum target dose or 41% in spinal cord dose [2]
Pixels in a depth image indicate calibrated depth in the scene rather than a measure of intensity or color.
Depth cameras offer several advantages over traditional intensity sensors, working in low light levels, giving a calibrated scale estimate, color and texture invariant, resolving silhouettes ambiguities in pose, simplifying background subtraction; straightforward to synthesize realistic depth images of people and cheap computational cost of building a large training dataset
Set Parameters in real-time human-body parts tracking: 3 trees, 20 deep, 300k training images per tree, 2000 training example pixels per image, 2000 candidate features θ, and 50 candidate thresholds τ
Face tracking algorithms belong in two broad categories: the first class consists of feature-based tracking algorithms which track local points from frame to frame to compute head pose and facial expressions based on locations of the points. Global regularization constraints are used to guarantee facial alignments. Local features matching make these algorithms less prone to generalization, illumination and occlusion problems. Cons: errors in features tracking lead to jittery and inaccurate results.
Second method involves algorithms that use appearance-based generative face models such as active appearance methods and 3D morphable models. Generally more accurate and produces better global alignment since they compute over the input data space. But they do have the difficulty of generalizing to unseen faces and may suffer from illumination changes and occlusions.
Both classes use a generative linear 3D facial alignment by either fitting a projected 3D model to video camera input or combine projected model fitting with 3D fitting to depth camera input.
2D + 3D AAM, introduced by Zhou et al, uses a projected linear 3D tracking parameters (head pose, expressions) but still suffers from illuminations, occlusions and generalization.
AAM: image difference patterns corresponding to changes in each model parameter are learned and used to modify a model estimate. AAMs use a training set of examples; AAMs learn what are valid shape, and intensity variations from their training set.
Cootes et al Paper
Generate models based on a model of shape variation and a model of the appearance variations in a shape-normalized frame. A training set of labelled images with points where key landmark points are marked on each example object.
Smolyanski’s Paper
Initializes 3D face model by computing realistic face shapes from a set of input RGBD frames. This improves tracking accuracy since the 3D model and its projection are closer to the input RGBD data.
Uses a linear 3D morphable model as a face shape model. It is computed by applying pca to a set of 3D human faces collected with the help of a high-resolution stereo color rig. At first, the face tracker uses an average face as the 3D face model. Once enough tracking data are collected, a personalized face model is computed.
Uses 2D facial landmarks with corresponding depth frames for face shape computation. The shape modeling algorithm deforms the 3D model so that to all the collected data.
After the computation of the personalized 3D face shape model, the face tracker accuracy is improved by using the 3D face model in the tracking runtime.
Resolves the monocular tracking problem by fitting a depth energy term into the 2D+3D AAM fitting that minimizes the Euclidean distance between 3D face model vertices and depth data coming from a RGBD camera. The term is formulated similar to the energy function used in Iterative Closest Point (ICP) algorithm. This addition reduces the solution space and finds better 3D alignments.
The extended energy function provides better accuracy in 3D space which is generally the issue with other 2D+3D AAM.
Captures 500 neural faces (no expressions) in 3D with the stereo capture rig and annotate them. Used PCA to build a statistical linear 3D face shape model. A 3D artist created a realistic set of facial expression deformations for this model. The 2D AAM model was trained from face images with various expressions. The AAM was trained by using PCA for a set of annotated 2D images. The training process was less expensive than full 3D AAM training for real-world applications that require tracking many facial types.
3D model has explicit face shape and facial animation parts to simplify creation of avatar animation systems. The 3D mean face topology was created by a 3D artist and includes 1347 vertices. Mean face vertices and shape basis were computed by the PCA process; the shape basis consists of 110 shape deformations. The animation basis consists of 17 animations. The resulting animation basis is transformed to form an orthonormal basis together with the shape basis.
In the event of the unavailability of depth data, user’s head is assumed to match the mean face size. Also align a 2D AAM shape to the five feature points.
In the event of the unavailability of depth data, user’s head is assumed to match the mean face size. Also align a 2D AAM shape to the five feature points.
With the regulated air compressor providing a standard pressure of 30 psi, we applied a persistently exciting input current in the form of a sawtooth waveform to deliver operational currents to the coil in the inlet PVQ valve.
Airflow out of the outlet valve was kept constant by opening it to it mid-position thus varying the head pose through an open-loop inflation and deflation process.
of the IAB.
With the regulated air compressor providing a standard pressure of 30 psi, we applied a persistently exciting input current in the form of a sawtooth waveform to deliver operational currents to the coil in the inlet PVQ valve.
Airflow out of the outlet valve was kept constant by opening it to it mid-position thus varying the head pose through an open-loop inflation and deflation process.
of the IAB.
For the identification of linear systems, there are three basic facts that govern the choices:
The asymptotic properties of the estimate (bias and variance) depend only on the input spectrum – not the actual waveform of the input
2. The input must have limited amplitude: u \leq u(t) \leq \bar{u}
3 Periodic inputs mat=y have certain advantages
The covariance matrix is typically inversely proportional to the input power. We want this to have as much input power as possible. In practice, the actual input limitation concerns amplitude constraints 𝑢 and 𝑢 .
A good signal waveform is one that has a small crest factor. The lower bound =d for C_r is 1 which is achieved for binary, symmetric signals: u(t) = \pm \bar(u). A binary input will not however have validation against non-linearity.
To achieve a large information matrix, we should spend the input power at frequencies where M(omega) is large, i.e., where the Bode plot is sensitive to parameter variations. Put differently, if a parameter is of special interest, then vary it and check where the Bode plot moves, and put the input power there. In many cases, this may give sufficient guidance for good input design.
For the identification of linear systems, there are three basic facts that govern the choices:
The asymptotic properties of the estimate (bias and variance) depend only on the input spectrum – not the actual waveform of the input
2. The input must have limited amplitude: u \leq u(t) \leq \bar{u}
3 Periodic inputs mat=y have certain advantages
The covariance matrix is typically inversely proportional to the input power. We want this to have as much input power as possible. In practice, the actual input limitation concerns amplitude constraints 𝑢 and 𝑢 .
A good signal waveform is one that has a small crest factor. The lower bound =d for C_r is 1 which is achieved for binary, symmetric signals: u(t) = \pm \bar(u). A binary input will not however have validation against non-linearity's.
To achieve a large information matrix, we should spend the input power at frequencies where M(omega) is large, i.e., where the Bode plot is sensitive to parameter variations. Put differently, if a parameter is of special interest, then vary it and check where the Bode plot moves, and put the input power there. In many cases, this may give sufficient guidance for good inout design.
Matlab, “detrend” removes the best straight-line fit from the vector x and returns in y. If x uis a matrix, detrend removes the trend from each column
Cross-correlation is similar to convolution in that it it a nofolding (time reversal),
In Matlab:
Conv(x,fliplr(y))
The correlation time musdt be finite. Second, the input signal must have a finite bandwidth and retain an autocorrelation function that is approximately a delta function. It should have the ff definitions:
It is binary and assumes values of plus or minus one with equal probability
It may (but not necessarily) change sign only every \delta t seconds
It is periodic with period T = N\deltat, N>>1
The autocorrelation function will look like a delta function in every peiod T. But will have side lobes because of the finite signal length. The side lobe binary amplitude distribution is the binary probability distribution , and for large N is nearly normal with \sigma = 1/sqrt(N). The side lobes introduce very large errors in the cross correlation result except N is very large.
The autocorrelation function is always symmetric with respect to its center, even for assymetric pulses
Auto-correlation function used to differentiate the presence of a like-signal e.g. a zero or one it’s the correlation of a signal with itself
The maximum value of the auto-correlartion function occurs at zero lag, i.e. when the signals are perfectly matched
The transfer function of the system can be obtained by taking the Laplace transform of the impulse response function
Use of autocorrelation to detect the presence of a periodic signal corrupted by noise
The correlation time musdt be finite. Second, the input signal must have a finite bandwidth and retain an autocorrelation function that is approximately a delta function. It should have the ff definitions:
It is binary and assumes values of plus or minus one with equal probability
It may (but not necessarily) change sign only every \delta t seconds
It is periodic with period T = N\deltat, N>>1
The autocorrelation function will look like a delta function in every peiod T. But will have side lobes because of the finite signal length. The side lobe binary amplitude distribution is the binary probability distribution , and for large N is nearly normal with \sigma = 1/sqrt(N). The side lobes introduce very large errors in the cross correlation result except N is very large.
MSE – 12.26, Lennart Llung
Akaike’s Final Prediction Error – Provides a measure of model quality by simulating the situation where the model is tested on a different data set. After computing several different models, one can compare them using this criterion. The most accurate model will hjave the lowest FPE.
Akaike's Final Prediction Error (FPE) is defined by the following equation:
FPE = 𝑉 ( 1+ 𝑑 𝑁 1− 𝑑 𝑁 ) where V is the loss function, d is the number of estimated parameters and N is the number of values in the estimation data set.
where V is the loss function, d is the number of estimated parameters, and N is the number of values in the estimation data set.
The Matlab toolbox assumes that the final prediction error is asymptotic for d<<N and uses the following approximation to compute FPE:
FPE = 𝑉 (1+ 2𝑑 𝑁 ) where the loss function V is defined by
V = det ├ 1 𝑁 𝑚𝑖𝑛 𝑚𝑎𝑥 𝜖 𝑡, 𝜃 𝑁 𝜖 𝑡, 𝜃 𝑁 𝑇