CS 556: Computer Vision

Lecture 8

Prof. Sinisa Todorovic

sinisa@eecs.oregonstate.edu

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Outline

• Object Recognition

• Image Classification

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Object Recognition and Image Classification

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Car/non-car

Classifier

Feature

extraction

Training examples

How to Classify?

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ximagefeature

class

discriminantfunction

{+1, �1}F (x, y)

y

Linear Classifier

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ximagefeature

class

discriminantfunction

{+1, �1}F (x, y)

y

F (x, y) = sign(w.x + b)

Example Application: Pedestrian Detection

• Multiscale scanning windows

• For each window compute the wavelet transform

• Classify the window using SVM

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“A general framework for object detection,” C. Papageorgiou, M. Oren and T. Poggio -- CVPR 98

Example Application: Pedestrian Detection

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“A general framework for object detection,” C. Papageorgiou, M. Oren and T. Poggio -- CVPR 98

Support Vector Machine — MATLAB

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SVMModel = fitcsvm(X,Y,’KernelFunction’,’rbf’,... 'Standardize',true,... 'ClassNames',{'negClass','posClass'});

[label,score] = predict(SVMModel,newX);

training:

classification:

SVM — MATLAB: Training

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rng(1); % For reproducibility size_data1 = 100; r = sqrt(rand(size_data1,1)); % Radius t = 2*pi*rand(size_data1,1); % Angle data1 = [r.*cos(t), r.*sin(t)]; % Points

size_data2 = 100; r2 = sqrt(3*rand(size_data2,1)+1); % Radius t2 = 2*pi*rand(size_data2,1); % Angle data2 = [r2.*cos(t2), r2.*sin(t2)]; % Points

figure; plot(data1(:,1),data1(:,2),'r.','MarkerSize',15) hold on plot(data2(:,1),data2(:,2),'b.','MarkerSize',15) ezpolar(@(x)1);ezpolar(@(x)2); axis equal hold off

Specify training data:

SVM — MATLAB: Training

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SVMModel = fitcsvm(X,Y,’KernelFunction’,’rbf’,... 'Standardize',true,... ‘ClassNames’,[-1,1]);

X = [data1;data2]; % Matrix, where each row is one feature size_data = size_data1 + size_data2; Y = ones(size_data,1); % Vector of classes for each feature Y(1:size_data1) = -1;

Specify training data:

Conduct training:

MATLAB centers and scales each column of the data Xby the weighted column mean and standard deviation, respectively

SVM — MATLAB: Prediction

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XTest = ; % Matrix, where each row is one feature YTest = ; % Vector of ground-truth classes for each feature

Specify new data:

Classification:

[YTestHat,score] = predict(SVMModel,XTest);

confidence in classification = signed distance to the SVM’s decision boundary

sigmoid function

normalized confidencein the positive class “1”:

normalized confidencein the negative class “-1”:

Evaluation — MATLAB: Confusion Matrix

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[C, order] = confusionmat(YTest,YTestHat)

ground truth labels predicted labelsconfusion matrix with rows = ground truthcolumns = prediction

g1 = [1 1 2 2 3 3]'; % Ground truth labels g2 = [1 1 2 3 4 NaN]'; % Predicted labels, NaN is not counted

[C,order] = confusionmat(g1,g2) C = 2 0 0 0 0 1 1 0 0 0 0 1 0 0 0 0 order = 1 2 3 4

Example:

Evaluation — MATLAB: Confusion Matrix (Alternative)

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[c,cm,ind,per] = confusion(targets,outputs)

binary matrix ofground truth labels

binary matrix ofpredicted labelsrows = labels

columns = test features

Evaluation — MATLAB: Example

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g1 = [1 1 2 2 3 3 3 4 3 1]'; % Ground truth labels for 10 features g2 = [1 1 2 3 4 2 3 4 2 4]'; % Predicted labels for 10 features

targets = zeros(4,10); % 4 labels and 10 features for i=1:10 targets(g1(i),i)=1; end

outputs = zeros(4,10); % 4 labels and 10 features for i=1:10 outputs(g2(i),i)=1; end

[c,cm,ind,per] = confusion(targets,outputs);

Evaluation — MATLAB: Example

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>> cm % confusion matrix

cm =

2 0 0 1 0 1 1 0 0 2 1 1 0 0 0 1

>> per

per =

0.1250 0 1.0000 0.8750 0.1429 0.6667 0.3333 0.8571 0.3750 0.5000 0.5000 0.6250 0 0.6667 0.3333 1.0000

rows = ground truth

columns = predictions

Evaluation — Precision & Recall

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Recall = TP / P

Precision = TP / (TP + FP)

F1 = 2 * Recall * Precision / (Recall + Precision)

Evaluation — MATLAB: Example — Precision & Recall

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>> for i=1:4 Recall(i)=sum(outputs(i,:).*targets(i,:))/sum(targets(i,:)); end

>> Recall % recall per each class

Recall =

0.6667 0.5000 0.2500 1.0000

>> for i=1:4 Precision(i)=sum(outputs(i,:).*targets(i,:))/sum(outputs(i,:)); end

>> Precision % precision per each class

Precision =

1.0000 0.3333 0.5000 0.3333

>> F1= (2*Precision(:).*Recall(:)./(Precision(:)+Recall(:)))'

F1 =

0.8000 0.4000 0.3333 0.5000