LETTER IMAGE RECOGNITION

1. Introduction.

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1. Introduction.• Objective:

– design classifiers for letter image recognition. – consider accuracy and time in taking the decision.

• 20,000 samples: – Starting set: images based on 20 different fonts (20x26 samples)– Data set: each letter was randomly distorted to produce our data

set (the 20,000 samples)– we did not have this initial set free of noise.

• 16 numerical features: – statistical moments and edge counts– scaled to fit into a range of integer values from 0 through 15.

• We use H, R or L method to estimate the error of the classifier.

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1. Introduction.

– Attribute Information:• Capital Letter: (26 Values From A To Z)• X-Box: Horizontal Position Of Box• Y-Box: Vertical Position Of Box• Width: Width Of Box• High: Height Of Box• Onpix: Total # On Pixels• X-Bar: Mean X Of On Pixels In Box• Y-Bar: Mean Y Of On Pixels In Box

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1. Introduction.• Y2bar: Mean X Variance• Y2bar: Mean Y Variance• Xybar: Mean X Y Correlation• X2ybr: Mean Of X * X * Y• Xy2br: Mean Of X * Y * Y• X-Ege: Mean Edge Count Left To Right• Xegvy: Correlation Of X-Ege With Y• Y-Ege: Mean Edge Count Bottom To Top• Yegvx: Correlation Of Y-Ege With X

2. Euclidean distance classifier.

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2. Euclidean distance classifier.

• The decision rule:

• Estimate the means for each category:

ijxxx jii ≠∀−<−⇔∈ µµω

∑=

=in

kk

ii xn 1

^ 1µ

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2. Euclidean distance classifier.

• Estimate the error with R method:

42.175057.8250.676

Error(%)

NC (%)

Accuracy(%)

Average Decision Time (ms)

3. Gaussian classifier.

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3. Gaussian classifier.

• Assume Gaussian distribution. • Estimate the mean and covariance matrix

for each class, with these estimators:

∑=

−−−

=in

k

tikik

ii XXn

C1

^^^))((

11 µµ

∑=

=in

kk

ii xn 1

^ 1µ

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3. Gaussian classifier.

• Decision rule:

• Where gi(x) are the discriminant functions:

ijxgxgx jii ≠∀>⇔∈ )()(ω

iiiitii

tii

ti LnPLnxxxg 22)( 111 +Σ−Σ−Σ+Σ−= −−− µµµ

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3. Gaussian classifier.• We can estimate the error of the classifier with

the R method. The result:

10.245089.7553.156

ErrorNCAccuracyAverage Decision Time

4. KNN classifier.

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4. KNN classifier.

• We will use the KNN rule, for each test-sample we find K nearest neighbors:

• The decision rule:

∑=i

iKK

ijKKx jii ≠∀>⇔∈ω

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4. KNN classifier.

• 1st approach: – compute the distance to all the training-

samples for each test-sample. – not optimum in the sense of decision time for

each sample.

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4. KNN classifier.• 2nd Approach:

– The features are numbers from 0 to 15. – We can order the training-samples by their distance

to the origin. – Given a test-sample, we measure its distance to the

origin and look for its knn only in training-samples with a similar distance.

– Suppose that the samples will be equally distributed in the 16D space

• use more training samples for furthest samples and less for closest samples (a smallest window for close samples and a big window for far samples).

• Optimum window: linearly from 50 to 1000.

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4. KNN classifier.

• 26 classes: we will get ties whether K is odd or even. Two options: – not to take a decision – break the tie

• give more importance to the closer samples (k votes for the nn, 1 vote for the knn).

• Example 4nn:– AABB -> A:1+1, B:1+1– AABB -> A:4+3, B:2+1– ABBA -> A:5, B:5

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4. KNN classifier.

11.095088.90538.009OW

6.86093.1468.475W=1000

12.705087.29527.805W=400

39.375060.6253.5115W=50

ErrorNCAccuracyAverage Decision TimeK=1

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4. KNN classifier.

11.095088.90547.515TB,OW

7.785092.21584.3111000,TB

1108954.595600,TB

44.075055.9256.468350,TB

5.8554.1889.96580.3561000

7.7756.79585.4349.28460022.1435.6142.2555.616650

ErrorNCAccuracyAverage Decision TimeK=3

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4. KNN classifier.

12.581.3486.0847.807TB,OW

7.4651.1191.42582.9631000,TB

10.971.27587.75551.378600,TB

46.051.72552.2256.640650,TB

5.9955.95588.0585.4851000

8.6858.2383.08553.822600

29.70530.4139.8857.11350

ErrorNCAccuracyAverage Decision TimeK=4

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4. KNN classifier.

8.960.4390.6184.588TB,OW (50, 2000)

14.210.61585.1847.263TB,OW4.470.2795.27233.093000,TB8.580.40591.0285.6851000,TB12.540.5786.8952.906600,TB49.21.58549.227.581450,TB7.194.36588.4584.513100010.256.64583.1154.20760034.1227.4838.47.504850

ErrorNCAccDecision TimeK=5

5. Neural Network classifier.

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5. Neural Network classifier.• Multilayer neural network with backpropagation

algorithm.• We used the resilient backpropagation training

algorithm. Faster.• Many parameters of the networks and training

method where changed to find the optimum classifier:– Number of neurons in the hidden layer.– Number of hidden layers.– Functions in the layers: hyperbolic tangent, logistic,

lineal.– Learning rate.

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5. Neural Network classifier.– Number of training samples.– Preprocessing of the input data: mean and SD

normalization, principal components analysis (take out the components that contribute less than 2% in the total variation of the data set).

– Training algorithms.– Targets vectors: 0..1 (logistic), -1..1 (hyperbolic

tangent), -0.9..0.9 (hyperbolic tangent), 0..10 (lineal).– Performance functions.

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5. Neural Network classifier.

• The network that had better performance was the following one:

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5. Neural Network classifier.

• The hidden layer has 15 neurons, and both layers use the logistic function.

• Rule of thumb: around 30 neurons– the performance was better with 15 neurons

• The output layer had 26 neurons as the number of classes

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5. Neural Network classifier.

• We did not preprocess the inputs: – no scaling (actually the data was already

normalized)– no principal components analysis.

:ar• The target vector was

jj

ii

papa

ωω∈⇔=

∉⇔=r

r

9.00

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5. Neural Network classifier.• 5000 training-samples: randomly distributed

according to their class.– the performance was not better using more samples

• 1000 validation-samples – early stop of 50 (if in 50 iterations the performance

measured with the validation data was worse, then we stop to avoid overfitting)

• 2000 iterations maximum• Learning rate: η=0.1

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5. Neural Network classifier.

• We could compare this to the performance of other network:– Input preprocessing: scaled and principal

components (11 over 16)– Training data: 2000– Hidden neurons: 10– Hidden layer function: hyperbolic tangent– Learning rate: 0.2

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5. Neural Network classifier.

• H method: test the performance of the classifier with 5000 new samples – 2000 from the validation set and 3000

different one.• The decision rule was:

ijaax jii ≠∀>⇔∈ω

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5. Neural Network classifier.• The results:

– Probably better if we could train with the set without noise.

24.42075.584.744

ErrorNCAccuracyAverage Decision Time

6. Summary and conclusion.

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6. Summary and conclusion.• The best accuracy was achieved with the 5nn

classifier• If we consider the time in taking the decision, the

best classifier is the Gaussian.

12.705 087.295 27.805 1NN4.4650.2795.265233.095NN24.42075.584.744N. Network

10.245089.7553.156Gaussian42.175057.8250.676Euclidean

ErrorNCAccuracyTimeClassifier