Polyhedral Classifier for Target Detection A Case Study: Colorectal Cancer

Copyright © Siemens Medical Solutions, USA, Inc.; 2008. All rights reserved.

Polyhedral Classifier for Target DetectionA Case Study: Colorectal Cancer

Murat Dundar, Matthias Wolf, Sarang Lakare, Marcos Salganicoff, Vikas C. Raykar

Siemens Medical Solutions, Inc. USAMalvern, PA 19355

Page 2 July-08Copyright © Siemens Medical Solutions, USA, Inc.; 2008. All rights reserved.

CAD & Knowledge Solutions / Malvern, USA / IKMDundar et al.

Computer Aided Diagnosis (CAD) for Colon Cancer

Identify suspicious regions

(candidates)

Extract features for each

candidate

Classify candidates as a polyp or

non-polyp



Multi-mode nature of CAD data

The only ground truth

available is the location of the

polyp.

All other candidates that are

not pointing to a known polyp

are pooled into the negative

class.

Variation among the different

negatives is large.



A CAD Example: Colorectal CancerPolyps vs. common false positives

Fold

Stool Noise

Rectal tube

Sessile polyp

Pedunculated polyp



Model class distribution by a mixture model, one mode for

each subclass, then design a maximum a posteriori or

maximum likelihood classifier

Too few positives, too many features with redundancy!

Robust estimation of model parameters for positive class is

very difficult, if not impractical

State-of-the-Art – Finite Mixture Models



State-of-the-Art – Discriminative Techniques

Pool all negative candidates into a single class and

learn a binary classifier, i.e. polyps vs. negatives

A kernel-based discriminative technique (SVM, RVM,

KFD) can yield nonlinear decision boundaries

suitable for classifying multi-mode data.

Too few positive candidates, too many features with

redundancy! Data can be easily overfit by a nonlinear

classifier



State-of-the-Art – One-Class Classifiers

Omits the negative class, learns a model with

positive samples only.

Kernel-based and neural network implementation

yield nonlinear decision boundaries suitable for

classifying multi-mode data.

Like other nonlinear classifiers susceptible to

overfitting



State-of-the-art in a Nutshell

Linear classifiers

less prone to overfitting

not enough capacity to deal with multi-mode data

Finite mixture models

Parameter estimation is an issue!

Discriminative & One-class Classifiers

good capacity

more prone to overfitting



A Viable solution

A series of linear classifiers one for each subclass of the

negatives

More capacity than a linear classifier, yet less prone to

overfitting than a nonlinear classifier

An unseen sample is classified as positive if all the classifier

classifies it as positive



Training Multiple Linear Classifiers

Train each classifier independently: Negative subclass k vs.

Positives, for k=1,…,K.

Inefficient! Potentially excessive penalization due to a

misclassified positive sample



Proposed Approach

Optimize classifiers jointly

One classifier for each subclass of negative data

Objective function is penalized once due to a

misclassified positive sample

Yields a polyhedral decision surface



A Toy Example



Hyperplane Classifiers with Hinge Loss

}xα1 ,0max{ iT

ii y

TP+

FP-

ξ

0xα T

ξ



i-th Positive example: -- “AND”

i-th Negative example:

Polyhedral Classifier with AND Framework

If the hinge loss = 0, the example is correctly classified,If the hinge loss > 0, the example is mis-classified

Let be the hinge loss of i-th example induced by the classifier k

),0max( ik

),,max(0, 21 iKii ξξ

ik



Objective Function with the AND Framework

K

k

iKiCi

i

k CiikK

P

ξξ

Jk

1k

212

121

)α(

),,max(0,

),0max()α,α,α(

Error on Negative Examples

Error on Positive Examples

Regularization to Control Complexity

Convex Problem!



Incomplete Ground Truth for Subclasses

AND algorithm assumes the subclass membership is known for all samples. Not Realistic!

Annotate a small portion of the negatives

identify potential subclasses

pool training samples for each subgroup.

Three different types of samples in the training data

Positives

Negatives with known and unknown subclass membership



Objective Function with the AND-OR Framework

K

k

iKiCi

i

Ci kik

k CiikK

P

ξξ

Jk

1k

212

ˆ1

121

)α(

),,max(0,

),0max(

),0max()α,α,α(

Error on Negative Examples with known subclasses

Error on Positive ExamplesAND operation

Regularization to Control Complexity

Error on Negative Examples with unknown subclasses, OR operation

Not Convex!



Each iteration contains K steps, and each step optimizes a single classifier

Alternating Optimization Iterative Algorithm

At the k-th step,Fix all classifiers (α’s) but the classifier kMinimize J(α1,…, αk ,… αK) for optimal αk



Cascaded Design

1 2 KT1 ….

rejected candidates

T2 TK-1 TK

F1F2 FK

Candidates

Training Sets: T1 T2 TK….

TP



Cascade Design with Sparse Linear Classifiers

Setting P(k)=| k | yields K sparse classifiers, each with varying

number of non-zero coefficients

Run-time order does not change the outcome

Start with the classifier that has the least number of nonzero

coefficients

Classify the sample, if negative reject, if positive pass it to the next

classifier that requires computation of least number of additional

features. Continue until all K classifiers are run



Experiments – Automatic Polyp Detection

Data

Volumes Polyps Negative candidatesTraining 316 226 1,249Test 385 245 1,920

98 numerical image features are computed,out of 1249 negatives, 177 are annotated, 9 subclasses are identified



ROC plots



Run-time Performance

25 % gain in execution time over SVDD and RBF-SVM

Classifiers Sens (at 3fp/vol) Time (t)

Polyhedral 84 452

SVDD 80 595

Rbf-SVM 60 595

Linear-SVM 45 437



Conclusions

Polyhedral classifier for multi-mode data

AND framework when subclass information is fully available

AND-OR framework when subclass information is partially available

Cascade design as a by-product to speed-up online execution

Thank you! Questions and Comments

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Polyhedral Classifier for Target Detection A Case Study: Colorectal Cancer