Image Acquisition and Representation
A Simple Image Formation Model
Image Sampling and Quantization
Image Interpolation
Image quantization
Nearest Neighbor Interpolation
Image Processing is any form of signal processing for which our input is an image, such as photographs or frames of videos and our output can be either an image or a set of characterstics related to the image
This one is from image processing where i have explained how erosion and dilation works well i dint explained in detail but it will be helpful to understand what erosion and dilation are.
Image Acquisition and Representation
A Simple Image Formation Model
Image Sampling and Quantization
Image Interpolation
Image quantization
Nearest Neighbor Interpolation
Image Processing is any form of signal processing for which our input is an image, such as photographs or frames of videos and our output can be either an image or a set of characterstics related to the image
This one is from image processing where i have explained how erosion and dilation works well i dint explained in detail but it will be helpful to understand what erosion and dilation are.
Introduction to digital image processing, image processing, digital image, analog image, formation of digital image, level of digital image processing, components of a digital image processing system, advantages of digital image processing, limitations of digital image processing, fields of digital image processing, ultrasound imaging, x-ray imaging, SEM, PET, TEM
Introduction to digital image processing, image processing, digital image, analog image, formation of digital image, level of digital image processing, components of a digital image processing system, advantages of digital image processing, limitations of digital image processing, fields of digital image processing, ultrasound imaging, x-ray imaging, SEM, PET, TEM
Microscopic Digital Image Segmentation And feature Extraction of Acute LeukemiaEditor IJCATR
The goal of this paper is to identify and show the
differences in the properties of Acute lymphoblastic leukemia (ALL)
and normal white blood cells. This paper was conducted on a set of
microscopic digital images of blood samples that got it from the
“Oncology Center - Faculty of Medicine - Mansoura University
Hospital - Egypt” is made up of 50 microscope image samples of
Cancerous blood and 50 microscope image of the blood samples is
not Cancerous (normal blood). The microscope blood images are
undergo to chain of pre-processing steps which include resizing
image such as (512*512, 256*256, 128*128) and contrast
enhancement. By executing K-means clustering on the resultant
images, the cell's nucleus under consideration is obtained then these
segmented images enter sub-image stage. The next step is Extracted
Features that included: Shape features (Area, Perimeter,
Compactness, Solidity, Eccentricity, and Elongation); texture features
(Homogeneity, Energy, Correlation, Entropy, and Contrast); color
features and Fractal Dimension
ALEXANDER FRACTIONAL INTEGRAL FILTERING OF WAVELET COEFFICIENTS FOR IMAGE DEN...sipij
The present paper, proposes an efficient denoising algorithm which works well for images corrupted with
Gaussian and speckle noise. The denoising algorithm utilizes the alexander fractional integral filter which
works by the construction of fractional masks window computed using alexander polynomial. Prior to the
application of the designed filter, the corrupted image is decomposed using symlet wavelet from which only
the horizontal, vertical and diagonal components are denoised using the alexander integral filter.
Significant increase in the reconstruction quality was noticed when the approach was applied on the
wavelet decomposed image rather than applying it directly on the noisy image. Quantitatively the results
are evaluated using the peak signal to noise ratio (PSNR) which was 30.8059 on an average for images
corrupted with Gaussian noise and 36.52 for images corrupted with speckle noise, which clearly
outperforms the existing methods.
Fast Object Recognition from 3D Depth Data with Extreme Learning MachineSoma Boubou
Object recognition from RGB-D sensors has recently emerged as a renowned and challenging research topic. The current systems often require large amounts of time to train the models and to classify new data. We proposed an effective and fast object recognition approach from 3D data acquired from depth sensors such as Structure or Kinect sensors.
Our contribution in this work} is to present a novel fast and effective approach for real-time object recognition from 3D depth data:
- First, we extract simple but effective frame-level features, which we name as differential frames, from the raw depth data.
- Second, we build a recognition system based on Extreme Learning Machine classifier with a Local Receptive Field (ELM-LRF).
SAMPLE QUESTIONExercise 1 Consider the functionf (x,C).docxanhlodge
SAMPLE QUESTION:
Exercise 1: Consider the function
f (x,C)=
sin(C x)
Cx
(a) Create a vector x with 100 elements from -3*pi to 3*pi. Write f as an inline or anonymous function
and generate the vectors y1 = f(x,C1), y2 = f(x,C2) and y3 = f(x,C3), where C1 = 1, C2 = 2 and
C3 = 3. Make sure you suppress the output of x and y's vectors. Plot the function f (for the three
C's above), name the axis, give a title to the plot and include a legend to identify the plots. Add a
grid to the plot.
(b) Without using inline or anonymous functions write a function+function structure m-file that does
the same job as in part (a)
SAMPLE LAB WRITEUP:
MAT 275 MATLAB LAB 1 NAME: __________________________
LAB DAY and TIME:______________
Instructor: _______________________
Exercise 1
(a)
x = linspace(-3*pi,3*pi); % generating x vector - default value for number
% of pts linspace is 100
f= @(x,C) sin(C*x)./(C*x) % C will be just a constant, no need for ".*"
C1 = 1, C2 = 2, C3 = 3 % Using commans to separate commands
y1 = f(x,C1); y2 = f(x,C2); y3 = f(x,C3); % supressing the y's
plot(x,y1,'b.-', x,y2,'ro-', x,y3,'ks-') % using different markers for
% black and white plots
xlabel('x'), ylabel('y') % labeling the axis
title('f(x,C) = sin(Cx)/(Cx)') % adding a title
legend('C = 1','C = 2','C = 3') % adding a legend
grid on
Command window output:
f =
@(x,C)sin(C*x)./(C*x)
C1 =
1
C2 =
2
C3 =
3
(b)
M-file of structure function+function
function ex1
x = linspace(-3*pi,3*pi); % generating x vector - default value for number
% of pts linspace is 100
C1 = 1, C2 = 2, C3 = 3 % Using commans to separate commands
y1 = f(x,C1); y2 = f(x,C2); y3 = f(x,C3); % function f is defined below
plot(x,y1,'b.-', x,y2,'ro-', x,y3,'ks-') % using different markers for
% black and white plots
xlabel('x'), ylabel('y') % labeling the axis
title('f(x,C) = sin(Cx)/(Cx)') % adding a title
legend('C = 1','C = 2','C = 3') % adding a legend
grid on
end
function y = f(x,C)
y = sin(C*x)./(C*x);
end
Command window output:
C1 =
1
C2 =
2
C3 =
3
More instructions for the lab write-up:
1) You are not obligated to use the 'diary' function. It was presented only for you convenience. You
should be copying and pasting your code, plots, and results into some sort of "Word" type editor that
will allow you to import graphs and such. Make sure you always include the commands to generate
what is been asked and include the outputs (from command window and plots), unless the pr.
We study an elliptic eigenvalue problem, with a random coefficient that can be parametrised by infinitely-many stochastic parameters. The physical motivation is the criticality problem for a nuclear reactor: in steady state the fission reaction can be modeled by an elliptic eigenvalue
problem, and the smallest eigenvalue provides a measure of how close the reaction is to equilibrium -- in terms of production/absorption of neutrons. The coefficients are allowed to be random to model the uncertainty of the composition of materials inside the reactor, e.g., the
control rods, reactor structure, fuel rods etc.
The randomness in the coefficient also results in randomness in the eigenvalues and corresponding eigenfunctions. As such, our quantity of interest is the expected value, with
respect to the stochastic parameters, of the smallest eigenvalue, which we formulate as an integral over the infinite-dimensional parameter domain. Our approximation involves three steps: truncating the stochastic dimension, discretizing the spatial domain using finite elements and approximating the now finite but still high-dimensional integral.
To approximate the high-dimensional integral we use quasi-Monte Carlo (QMC) methods. These are deterministic or quasi-random quadrature rules that can be proven to be very efficient for the numerical integration of certain classes of high-dimensional functions. QMC methods have previously been applied to linear functionals of the solution of a similar elliptic source problem; however, because of the nonlinearity of eigenvalues the existing analysis of the integration error
does not hold in our case.
We show that the minimal eigenvalue belongs to the spaces required for QMC theory, outline the approximation algorithm and provide numerical results.
Computing the masses of hyperons and charmed baryons from Lattice QCDChristos Kallidonis
Poster presented at the Computational Sciences 2013 Conference (Winner of poster competition). We present results on the masses of all forty light, strange and charm baryons from Lattice QCD simulations, focusing particularly on the computational aspects and requirements of such calculations.
We examine the effectiveness of randomized quasi Monte Carlo (RQMC) to improve the convergence rate of the mean integrated square error, compared with crude Monte Carlo (MC), when estimating the density of a random variable X defined as a function over the s-dimensional unit cube (0,1)^s. We consider histograms and kernel density estimators. We show both theoretically and empirically that RQMC estimators can achieve faster convergence rates in
some situations.
This is joint work with Amal Ben Abdellah, Art B. Owen, and Florian Puchhammer.
this is presentation on " skewed plate problem " which is related to advanced mechanic of material of subject.here explain about principle,method and approximation function with also steps wise for solving problem in Ritz method and discuss about the skewed plate problem. included conclusion and references.
Similar to Snakes in Images (Active contour tutorial) (20)
Houston machine learning meetup, two papers are discussed:
- A Contextual-Bandit Approach to Personalized News Article Recommendation
http://rob.schapire.net/papers/www10.pdf
- An efficient bandit algorithm for realtime multivariate optimization
https://www.kdd.org/kdd2017/papers/view/an-efficient-bandit-algorithm-for-realtime-multivariate-optimization
Sr. Architect Pradeep Reddy, from Qubole, presents the state of Data Science in the enterprise industries today, followed by deep dive of an end-to-end real world machine learning use case. We'll explore the best practices and challenges of big data operations when developing new machine learning features and advanced analytics products at scale in the cloud.
UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Transcript: Selling digital books in 2024: Insights from industry leaders - T...BookNet Canada
The publishing industry has been selling digital audiobooks and ebooks for over a decade and has found its groove. What’s changed? What has stayed the same? Where do we go from here? Join a group of leading sales peers from across the industry for a conversation about the lessons learned since the popularization of digital books, best practices, digital book supply chain management, and more.
Link to video recording: https://bnctechforum.ca/sessions/selling-digital-books-in-2024-insights-from-industry-leaders/
Presented by BookNet Canada on May 28, 2024, with support from the Department of Canadian Heritage.
Removing Uninteresting Bytes in Software FuzzingAftab Hussain
Imagine a world where software fuzzing, the process of mutating bytes in test seeds to uncover hidden and erroneous program behaviors, becomes faster and more effective. A lot depends on the initial seeds, which can significantly dictate the trajectory of a fuzzing campaign, particularly in terms of how long it takes to uncover interesting behaviour in your code. We introduce DIAR, a technique designed to speedup fuzzing campaigns by pinpointing and eliminating those uninteresting bytes in the seeds. Picture this: instead of wasting valuable resources on meaningless mutations in large, bloated seeds, DIAR removes the unnecessary bytes, streamlining the entire process.
In this work, we equipped AFL, a popular fuzzer, with DIAR and examined two critical Linux libraries -- Libxml's xmllint, a tool for parsing xml documents, and Binutil's readelf, an essential debugging and security analysis command-line tool used to display detailed information about ELF (Executable and Linkable Format). Our preliminary results show that AFL+DIAR does not only discover new paths more quickly but also achieves higher coverage overall. This work thus showcases how starting with lean and optimized seeds can lead to faster, more comprehensive fuzzing campaigns -- and DIAR helps you find such seeds.
- These are slides of the talk given at IEEE International Conference on Software Testing Verification and Validation Workshop, ICSTW 2022.
Dev Dives: Train smarter, not harder – active learning and UiPath LLMs for do...UiPathCommunity
💥 Speed, accuracy, and scaling – discover the superpowers of GenAI in action with UiPath Document Understanding and Communications Mining™:
See how to accelerate model training and optimize model performance with active learning
Learn about the latest enhancements to out-of-the-box document processing – with little to no training required
Get an exclusive demo of the new family of UiPath LLMs – GenAI models specialized for processing different types of documents and messages
This is a hands-on session specifically designed for automation developers and AI enthusiasts seeking to enhance their knowledge in leveraging the latest intelligent document processing capabilities offered by UiPath.
Speakers:
👨🏫 Andras Palfi, Senior Product Manager, UiPath
👩🏫 Lenka Dulovicova, Product Program Manager, UiPath
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
PHP Frameworks: I want to break free (IPC Berlin 2024)Ralf Eggert
In this presentation, we examine the challenges and limitations of relying too heavily on PHP frameworks in web development. We discuss the history of PHP and its frameworks to understand how this dependence has evolved. The focus will be on providing concrete tips and strategies to reduce reliance on these frameworks, based on real-world examples and practical considerations. The goal is to equip developers with the skills and knowledge to create more flexible and future-proof web applications. We'll explore the importance of maintaining autonomy in a rapidly changing tech landscape and how to make informed decisions in PHP development.
This talk is aimed at encouraging a more independent approach to using PHP frameworks, moving towards a more flexible and future-proof approach to PHP development.
Essentials of Automations: The Art of Triggers and Actions in FMESafe Software
In this second installment of our Essentials of Automations webinar series, we’ll explore the landscape of triggers and actions, guiding you through the nuances of authoring and adapting workspaces for seamless automations. Gain an understanding of the full spectrum of triggers and actions available in FME, empowering you to enhance your workspaces for efficient automation.
We’ll kick things off by showcasing the most commonly used event-based triggers, introducing you to various automation workflows like manual triggers, schedules, directory watchers, and more. Plus, see how these elements play out in real scenarios.
Whether you’re tweaking your current setup or building from the ground up, this session will arm you with the tools and insights needed to transform your FME usage into a powerhouse of productivity. Join us to discover effective strategies that simplify complex processes, enhancing your productivity and transforming your data management practices with FME. Let’s turn complexity into clarity and make your workspaces work wonders!
Epistemic Interaction - tuning interfaces to provide information for AI supportAlan Dix
Paper presented at SYNERGY workshop at AVI 2024, Genoa, Italy. 3rd June 2024
https://alandix.com/academic/papers/synergy2024-epistemic/
As machine learning integrates deeper into human-computer interactions, the concept of epistemic interaction emerges, aiming to refine these interactions to enhance system adaptability. This approach encourages minor, intentional adjustments in user behaviour to enrich the data available for system learning. This paper introduces epistemic interaction within the context of human-system communication, illustrating how deliberate interaction design can improve system understanding and adaptation. Through concrete examples, we demonstrate the potential of epistemic interaction to significantly advance human-computer interaction by leveraging intuitive human communication strategies to inform system design and functionality, offering a novel pathway for enriching user-system engagements.
Welcome to the first live UiPath Community Day Dubai! Join us for this unique occasion to meet our local and global UiPath Community and leaders. You will get a full view of the MEA region's automation landscape and the AI Powered automation technology capabilities of UiPath. Also, hosted by our local partners Marc Ellis, you will enjoy a half-day packed with industry insights and automation peers networking.
📕 Curious on our agenda? Wait no more!
10:00 Welcome note - UiPath Community in Dubai
Lovely Sinha, UiPath Community Chapter Leader, UiPath MVPx3, Hyper-automation Consultant, First Abu Dhabi Bank
10:20 A UiPath cross-region MEA overview
Ashraf El Zarka, VP and Managing Director MEA, UiPath
10:35: Customer Success Journey
Deepthi Deepak, Head of Intelligent Automation CoE, First Abu Dhabi Bank
11:15 The UiPath approach to GenAI with our three principles: improve accuracy, supercharge productivity, and automate more
Boris Krumrey, Global VP, Automation Innovation, UiPath
12:15 To discover how Marc Ellis leverages tech-driven solutions in recruitment and managed services.
Brendan Lingam, Director of Sales and Business Development, Marc Ellis
Le nuove frontiere dell'AI nell'RPA con UiPath Autopilot™UiPathCommunity
In questo evento online gratuito, organizzato dalla Community Italiana di UiPath, potrai esplorare le nuove funzionalità di Autopilot, il tool che integra l'Intelligenza Artificiale nei processi di sviluppo e utilizzo delle Automazioni.
📕 Vedremo insieme alcuni esempi dell'utilizzo di Autopilot in diversi tool della Suite UiPath:
Autopilot per Studio Web
Autopilot per Studio
Autopilot per Apps
Clipboard AI
GenAI applicata alla Document Understanding
👨🏫👨💻 Speakers:
Stefano Negro, UiPath MVPx3, RPA Tech Lead @ BSP Consultant
Flavio Martinelli, UiPath MVP 2023, Technical Account Manager @UiPath
Andrei Tasca, RPA Solutions Team Lead @NTT Data
Accelerate your Kubernetes clusters with Varnish CachingThijs Feryn
A presentation about the usage and availability of Varnish on Kubernetes. This talk explores the capabilities of Varnish caching and shows how to use the Varnish Helm chart to deploy it to Kubernetes.
This presentation was delivered at K8SUG Singapore. See https://feryn.eu/presentations/accelerate-your-kubernetes-clusters-with-varnish-caching-k8sug-singapore-28-2024 for more details.
Pushing the limits of ePRTC: 100ns holdover for 100 daysAdtran
At WSTS 2024, Alon Stern explored the topic of parametric holdover and explained how recent research findings can be implemented in real-world PNT networks to achieve 100 nanoseconds of accuracy for up to 100 days.
Enhancing Performance with Globus and the Science DMZGlobus
ESnet has led the way in helping national facilities—and many other institutions in the research community—configure Science DMZs and troubleshoot network issues to maximize data transfer performance. In this talk we will present a summary of approaches and tips for getting the most out of your network infrastructure using Globus Connect Server.
Securing your Kubernetes cluster_ a step-by-step guide to success !KatiaHIMEUR1
Today, after several years of existence, an extremely active community and an ultra-dynamic ecosystem, Kubernetes has established itself as the de facto standard in container orchestration. Thanks to a wide range of managed services, it has never been so easy to set up a ready-to-use Kubernetes cluster.
However, this ease of use means that the subject of security in Kubernetes is often left for later, or even neglected. This exposes companies to significant risks.
In this talk, I'll show you step-by-step how to secure your Kubernetes cluster for greater peace of mind and reliability.
zkStudyClub - Reef: Fast Succinct Non-Interactive Zero-Knowledge Regex ProofsAlex Pruden
This paper presents Reef, a system for generating publicly verifiable succinct non-interactive zero-knowledge proofs that a committed document matches or does not match a regular expression. We describe applications such as proving the strength of passwords, the provenance of email despite redactions, the validity of oblivious DNS queries, and the existence of mutations in DNA. Reef supports the Perl Compatible Regular Expression syntax, including wildcards, alternation, ranges, capture groups, Kleene star, negations, and lookarounds. Reef introduces a new type of automata, Skipping Alternating Finite Automata (SAFA), that skips irrelevant parts of a document when producing proofs without undermining soundness, and instantiates SAFA with a lookup argument. Our experimental evaluation confirms that Reef can generate proofs for documents with 32M characters; the proofs are small and cheap to verify (under a second).
Paper: https://eprint.iacr.org/2023/1886
SAP Sapphire 2024 - ASUG301 building better apps with SAP Fiori.pdfPeter Spielvogel
Building better applications for business users with SAP Fiori.
• What is SAP Fiori and why it matters to you
• How a better user experience drives measurable business benefits
• How to get started with SAP Fiori today
• How SAP Fiori elements accelerates application development
• How SAP Build Code includes SAP Fiori tools and other generative artificial intelligence capabilities
• How SAP Fiori paves the way for using AI in SAP apps
Generative AI Deep Dive: Advancing from Proof of Concept to ProductionAggregage
Join Maher Hanafi, VP of Engineering at Betterworks, in this new session where he'll share a practical framework to transform Gen AI prototypes into impactful products! He'll delve into the complexities of data collection and management, model selection and optimization, and ensuring security, scalability, and responsible use.
2. How snake works in images
http://www.iacl.ece.jhu.edu/~chenyang/research/levset/movie/index.html
3. How snake works in images
Step 1: Model the shape by curve or surface
Step 2: Define the energy function E along the curve
E = internal energy (curvature) + external energy (images)
Step 3: Derive the curve to minimize the energy (calculus
of variations)
Step 4: Propagate the curves (gradient descent) to reach
the minimum using level set (easier to implement in reality).
Keywords:
Curve
Energy
Level set
4. Snake -> Curves
Consider a closed planar curve C ( p) {x( p), y( p)}, p [0,1]
C(0.1)
S(p) is the arclength
C(0.2)
C(0.7)
C(0)
C(0.4)
C(0.95)
Cp =tangent
C(0.8)
Css
| Cp |
Css
N
N
1
C(0.9)
Cs
Cp
C
The geometric trace of the curve can be alternatively
represented implicitly as C {( x, y ) | ( x, y ) 0}
1
0
0
1
5. Properties of level sets
T
The level set normal
N
|
N
|
Proof. Along the level sets (equal 0) we have zero change,
that is s 0 , but by the chain rule
So,
s
|
( x, y )
|
,T
x
0
xs
|
y
|
,T
ys
T
N
|
|
6. Properties of Level Sets
The level set curvature
Css =div
|
|
Proof. zero change along the level sets,
ss
So,
( x, y )
d
( x xs
ds
,
|
|
y
|
ys )
|
[
,T
d
ds
x
xx s
xy
ys ,
ss
,T
d
ds
x
xy s
yy
ys ],
|
0 , also
, N
|
Numerical Geometry of Images:
http://webcourse.cs.technion.ac.il/236861/Winter2012-2013/en/ho_Tutorials.html
7. Constant Flow
Ct
N
Propagation of Curves
Change in topology
Curvature Flow
Ct
Grayson
N
Vanish at a
Circular
point
First becomes
convex
Gage-Hamilton
8. Curve propagation in Level Sets
dC
dt
VN
C(t)
d
dt
y
V
x
x, y, t
( x, y; t )
t
0
t
N
x
xt
y
yt
x xt
|
t
C(t) level set
, VN
, Ct
V
|
y
y yt
,N
,N
V
V
0
,
|
x
|
V|
|
9. Curve propagation in Level Sets
Handles changes in topology
Numeric grid points never collide
or drift apart.
Parametric Curve -> Level set in plane -> add time axis for propagation
10. Curve propagation in Level Sets
Some flows
Ct
VN
Ct
N
Ct
g
t
t
g, N N
V
2
xx y
div
t
2
xy x y
2
2
x
y
2
yy x
div g
g
2
xx y
2
xy x y
2
2
x
y
2
yy x
g,
11. Snake Model
1
arg min
C
Cp
0
2
C pp
tension
2
rigidity
2 g (C ) dp
external energy from the image
Energy
g (C )
|
[G (C ) * I (C )] |2
A snake that minimize E must satisfy the Euler equation
Cpp
Cpppp
g C
0
To find a solution
dC
dt
C pp
C pppp
g C
When C stabilizes and the term Ct vanishes, a solution is achieved.
Snakes: Active Contour Models, 1988
12. Geodesic Active Contour
arg min
C
L C
0
g C ds
ds is the Euclidean arc-length.
Trying to find minimum weighted distance
I(x)
Euler Lagrange gradient descent
Ct
g (C) n
g (C), n n
x
g(x)
d
dt
div g x, y
Geodesic Active Contours, 1997
x
13. GVF Snakes
Replace the external force with gradient vector flow (GVF) force v.
dC
dt
C pp
C pppp
g C
V(x,y) = [u(x,y),v(x,y)]
Why?
f generally have large magnitudes only in the vicinity of the edges, the
capture range would be small.
It’s nearly zero in homogeneous regions.
V(x,y) = [u(x,y), v(x,y)] is defined as the vector field that minimizes
the energy functional:
http://www.iacl.ece.jhu.edu/static/gvf/
18. Calculus of Variations
Generalization of Calculus that seeks to find the
path, curve, surface, etc., for which a given Functional has
a minimum or maximum.
Goal: find extrema values of integrals of the form
F (u, u x )dx
It has an extremum only if the Euler-Lagrange
Differential Equation is satisfied,
u
d
F ( u, u x )
dx u x
0
19. Calculus of Variations
Example: Find the shape of the curve {x,y(x)} with
shortest length:
x1
y ( x1 )
2
1 y x dx ,
y ( x0 )
x0
given y(x0),y(x1)
x0
Solution: a differential equation that y(x) must
satisfy,
y xx
0
yx a
y ( x ) ax b
3/ 2
2
1 yx
x1
