Generating EigenFaces with Mahout SVD to recognize person faces

In this tutorial, we are going to describe how to generate and use eigenfaces to recognize people faces.
Eigenfaces are a set of eigenvectors derived from the covariance matrix of the probability distribution of the high-dimensional vector space of possible faces of human beings. It can be used to identify a face on a picture from a person face database very quickly. In this post, we’ll not give much details on the mathematical aspects but if you are interested on those, you can look at the excellent post Face Recognition using Eigenfaces and Distance Classifiers: A Tutorial from the Onionesque Reality Blog.

Requirements

To do this tutorial, you would need to have the following softwares installed on your machine:

• Java >= 1.6
• Hadoop
• Mahout
• Maven

You can find the instructions to install those from a previous post Playing with the Mahout recommendation engine on a Hadoop cluster.

Compiling the code

All the sourcecode, the training sets and testing sets are in the github repository at https://github.com/fredang/mahout-eigenface-example/

You can fetch the files from this repository by typing:

$ git clone https://github.com/fredang/mahout-eigenface-example.git

This repository is structured as follow:

• src/main/java/com/chimpler/example/eigenface/GenerateCovarianceMatrix.java: code to generate the covariance matrix from the images from the training set
• src/main/java/com/chimpler/example/eigenface/ComputeEigenFaces.java: code to compute the eigenfaces
• src/main/java/com/chimpler/example/eigenface/ComputeDistance.java: code to test the model with the testing set
• src/main/java/com/chimpler/example/eigenface/Helper.java: code used to do some matrix operations and image operations
• images/yalefaces-test: some additional images to add to the yalefaces testing set
• images/cats-train: training set for cat faces (not used in this example)
• images/cats-test: testing set for cat faces (not used in this example)

Once you fetched the project, you can compile it using maven:

$ mvn clean package assembly:single

It creates a jar file in the directory target which all the dependencies and the compiled class from the src/main/java directory.

Preparing the data set

You can download the yale face database by going to this page: http://vision.ucsd.edu/content/yale-face-database

Unzip the file:

$ unzip yalefaces.zip

Now we are going to split this file into two sets: a training set and a testing set:

$ mkdir training-set
$ mv yalefaces/* training-set/

For the testing set, we are removing the sad facial expression from the training set and move it to the testing set:

$ mkdir testing-set
$ mv training-set/*.sad testing-set/

We also add two non face images(hamburger and cat) and one person face unknown to the training set(Bruce Lee):

$ cp [MAHOUT EIGENFACE EXAMPLE DIRECTORY]/images/yalefaces-test/* testing-set

Training the model

The training is implemented in the class GenerateCovarianceMatrix to generate the covariance matrix.

The arguments of this class are:

• image width: it is used to scale down the image width so that the computation does not take too much memory
• image height
• training directory: directory containing the training face images
• output directory

public class Helper {
	public static void writeImage(String filename, double[] imagePixels,
			int width, int height) throws Exception {
		BufferedImage meanImage = new BufferedImage(width, height, BufferedImage.TYPE_BYTE_GRAY);
		WritableRaster raster = meanImage.getRaster();

		// convert byte array to byte array
		int[] pixels = new int[imagePixels.length];
		for(int i = 0 ; i < imagePixels.length ; i++) {
			pixels[i] = (int)imagePixels[i];
		}
		raster.setPixels(0, 0, width, height, pixels);

		ImageIO.write(meanImage, "gif", new File(filename));
	}

	public static double[] readImagePixels(String imageFileName, int width, int height) throws Exception {
		BufferedImage colorImage = ImageIO.read(new File(imageFileName));

		// convert to grayscale image
		BufferedImage greyImage = new BufferedImage(
				width,
				height,
			    BufferedImage.TYPE_BYTE_GRAY);
		greyImage.getGraphics().drawImage(colorImage, 0, 0, width, height, null);

		byte[] bytePixels = ((DataBufferByte)greyImage.getRaster().getDataBuffer()).getData();

		double[] doublePixels = new double[bytePixels.length];
		for(int i = 0 ; i < doublePixels.length ; i++) {
			doublePixels[i] = (double)(bytePixels[i] & 255);
		}
		return doublePixels;
	}

	public static double[][] computeDifferenceMatrixPixels(double[][] matrixPixels, double[] meanColumn) {
		int rowCount = matrixPixels.length;
		int columnCount = matrixPixels[0].length;

		double[][] diffMatrixPixels = new double[rowCount][columnCount];
		for(int i = 0 ; i < rowCount ; i++) {
			for(int j = 0 ; j < columnCount ; j++) {
				diffMatrixPixels[i][j] = matrixPixels[i][j] - meanColumn[i];
			}
		}

		return diffMatrixPixels;
	}

	public static double[] computeDifferencePixels(double[] pixels, double[] meanColumn) {
		int pixelCount = pixels.length;
		double[] diffPixels = new double[pixelCount];
		for(int i = 0 ; i < pixelCount ; i++) {
			diffPixels[i] = pixels[i] - meanColumn[i];
		}

		return diffPixels;
	}

	public static double[][] readMatrixSequenceFile(String fileName) throws Exception {
		Configuration configuration = new Configuration();
		FileSystem fs = FileSystem.get(configuration);
		Reader matrixReader = new SequenceFile.Reader(fs,
			new Path(fileName), configuration);

		List<double[]> rows = new ArrayList<double[]>();
		IntWritable key = new IntWritable();
		VectorWritable value = new VectorWritable();
		while(matrixReader.next(key, value)) {
			Vector vector = value.get();
			double[] row = new double[vector.size()];
			for(int i = 0 ; i < vector.getNumNondefaultElements() ; i++) {
				Element element = vector.getElement(i);
				row[element.index()] = element.get();
			}
			rows.add(row);
		}
		return rows.toArray(new double[rows.size()][]);
	}

	public static void writeMatrixSequenceFile(String matrixSeqFileName, double[][] covarianceMatrix) throws Exception{
		int rowCount = covarianceMatrix.length;
		int columnCount = covarianceMatrix[0].length;

		Configuration configuration = new Configuration();
		FileSystem fs = FileSystem.get(configuration);
		Writer matrixWriter = new SequenceFile.Writer(fs, configuration,
				new Path(matrixSeqFileName),
				IntWritable.class, VectorWritable.class);

		IntWritable key = new IntWritable();
		VectorWritable value = new VectorWritable();

		double[] doubleValues = new double[columnCount];
		for(int i = 0 ; i < rowCount ; i++) {
			key.set(i);
			for(int j = 0 ; j < columnCount ; j++) {
				doubleValues[j] = covarianceMatrix[i][j];
			}
			Vector vector = new DenseVector(doubleValues);
			value.set(vector);

			matrixWriter.append(key, value);
		}
		matrixWriter.close();
	}

	public static double[] computeWeights(double[] diffImagePixels,
			double[][] eigenFaces) {
		int pixelCount = eigenFaces.length;
		int eigenFaceCount = eigenFaces[0].length;
		double[] weights = new double[eigenFaceCount];
		for(int i = 0 ; i < eigenFaceCount ; i++) {
			for(int j = 0 ; j < pixelCount ; j++) {
				weights[i] += diffImagePixels[j] * eigenFaces[j][i];
			}
		}
		return weights;
	}

	public static double[] reconstructImageWithEigenFaces(
			double[] weights,
			double[][] eigenFaces,
			double[] meanImagePixels) throws Exception {
		int pixelCount = eigenFaces.length;
		int eigenFaceCount = eigenFaces[0].length;

		// reconstruct image from weight and eigenfaces
		double[] reconstructedPixels = new double[pixelCount];
		for(int i = 0 ; i < eigenFaceCount ; i++) {
			for(int j = 0 ; j < pixelCount ; j++) {
				reconstructedPixels[j] += weights[i] * eigenFaces[j][i];
			}
		}

		// add mean
		for(int i = 0 ; i < pixelCount ; i++) {
			reconstructedPixels[i] += meanImagePixels[i];
		}

		double min = Double.MAX_VALUE;
		double max = -Double.MAX_VALUE;
		for(int i = 0 ; i < reconstructedPixels.length ; i++) {
			min = Math.min(min, reconstructedPixels[i]);
			max = Math.max(max, reconstructedPixels[i]);
		}

		double[] normalizedReconstructedPixels = new double[pixelCount];
		for(int i = 0 ; i < reconstructedPixels.length ; i++) {
			normalizedReconstructedPixels[i] = (255.0 * (reconstructedPixels[i] - min)) / (max - min);
		}

		return normalizedReconstructedPixels;
	}

	public static double computeImageDistance(double[] pixelImage1, double[] pixelImage2) {
		double distance = 0;
		int pixelCount = pixelImage1.length;
		for(int i = 0 ; i < pixelCount ; i++) {
			double diff = pixelImage1[i] - pixelImage2[i];
			distance += diff * diff;
		}
		return Math.sqrt(distance / pixelCount);
	}

	public static List listImageFileNames(String directoryName) {
		File directory = new File(directoryName);
		List imageFileNames = new ArrayList();
		for(File imageFile: directory.listFiles()) {
//			if (imageFile.getName().endsWith(".gif")) {
				imageFileNames.add(imageFile.getAbsolutePath());
//			}
		}
		Collections.sort(imageFileNames);
		return imageFileNames;
	}

	public static String getShortFileName(String fullFileName) {
		return new File(fullFileName).getName();
	}
}

 

public class GenerateCovarianceMatrix {

	private static double[][] convertImagesToMatrix(Collection imageFileNames,
			int width, int height) throws Exception {
		int columnIndex = 0;
		double[][] pixelMatrix = new double[width * height][imageFileNames.size()];
		for(String fileName: imageFileNames) {
			System.out.println("Reading file " + fileName);
			double[] pixels = Helper.readImagePixels(fileName, width, height);
			for(int i = 0 ; i < pixels.length ; i++) {
				pixelMatrix[i][columnIndex] = pixels[i];
			}
			columnIndex++;
		}

		return pixelMatrix;
	}

	private static double[] computeMeanColumn(double[][] pixelMatrix) {
		int pixelCount = pixelMatrix.length;
		double[] meanColumn = new double[pixelCount];
		int columnCount = pixelMatrix[0].length;
		for(int i = 0 ; i < pixelCount ; i++) {
			int sum = 0;
			for(int j = 0 ; j < columnCount ; j++) {
				sum += pixelMatrix[i][j];
			}
			meanColumn[i] = sum / columnCount;
		}
		return meanColumn;
	}

	private static double[][] computeCovarianceMatrix(double[][] diffMatrixPixels) {
		int rowCount = diffMatrixPixels.length;
		int columnCount = diffMatrixPixels[0].length;

		double[][] covarianceMatrix = new double[columnCount][columnCount];
		for(int i = 0 ; i < columnCount ; i++) {
			for(int j = 0 ; j < columnCount ; j++) {
				int sum = 0;
				for(int k = 0 ; k < rowCount ; k++) {
					sum += diffMatrixPixels[k][i] * diffMatrixPixels[k][j];
				}
				covarianceMatrix[i][j] = sum;
			}
		}

		return covarianceMatrix;
	}

	public static void main(String args[]) throws Exception {
		if (args.length != 4) {
			System.out.println("Arguments: width height trainingDirectory outputDirectory");
			System.exit(1);
		}

		int width = Integer.parseInt(args[0]);
		int height = Integer.parseInt(args[1]);
		String imageDirectory = args[2];
		String outputDirectory = args[3];

		File outputDirectoryFile = new File(outputDirectory);
		if (!outputDirectoryFile.exists()) {
			outputDirectoryFile.mkdir();
		}

		List imageFileNames = Helper.listImageFileNames(imageDirectory);

		System.out.println("Reading " + imageFileNames.size() + " images...");
		double[][] pixelMatrix = convertImagesToMatrix(imageFileNames, width, height);
		double[] meanColumn = computeMeanColumn(pixelMatrix);
		Helper.writeImage(outputDirectory + "/mean-image.gif", meanColumn, width, height);
		double[][] diffMatrixPixels = Helper.computeDifferenceMatrixPixels(pixelMatrix, meanColumn);
		Helper.writeMatrixSequenceFile(outputDirectory + "/diffmatrix.seq", diffMatrixPixels);

		double[][] covarianceMatrix = computeCovarianceMatrix(diffMatrixPixels);
		Helper.writeMatrixSequenceFile(outputDirectory + "/covariance.seq", covarianceMatrix);
	}
}

$ java -cp target/mahout-eigenface-example-1.0-jar-with-dependencies.jar com.chimpler.example.eigenface.GenerateCovarianceMatrix 80 60 [TRAINING_SET_DIRECTORY] output
This program:

1. reads all the n image files from the training directory
2. convert each image to greyscale and scale down the image
   
3. create a matrix M with each column representing an image. The column has a length of w x h and each of its element represents a shade of grey with a value between 0(black) and 255(white).
4. compute the mean image and write it in output/mean-image.gif. It is computed by averaging each pixel of the images
   
5. compute the diff matrix DM by substracting the mean image to M
6. Compute the covariance matrix transpose(DM) x DM. It gives the matrix of size n x n
7. write the diff matrix DM to output/diffmatrix.seq
8. write the covariance matrix to output/covariance.seq

Now we need to compute the eigenvectors of the covariance matrix. It can be done using the Mahout Singular Value Decomposition(SVD).

To use it, first copy the file covariance.seq to HDFS:

$ hadoop fs -put output/covariance.seq covariance.seq

Then run the Mahout SVD:

$ mahout svd --input covariance.seq --numRows 150 --numCols 150 --rank 50 --output output

We set the –numRows and –numCols to the size of the covariance matrix (150 x 150) and the rank to 50 (we usually set it to one third of the number of images).

The computed eigen vectors might contain extra eigenvectors with invalid eigenvalues. To fix this, we can run mahout cleansvd:

$ mahout cleansvd -ci covariance.seq -ei output -o output2

We can now copy the clean eigen vector to the local filesystem:

$ hadoop fs -get output2/cleanEigenvectors output/cleanEigenvectors

Then execute the java class ComputeEigenFaces to create the eigenfaces.

public class ComputeEigenFaces {
	private static void writeEigenFaceImage(String filename, double[][] eigenFacePixels,
			int width, int height, int columnIndex) throws Exception {
		BufferedImage meanImage = new BufferedImage(width, height, BufferedImage.TYPE_BYTE_GRAY);
		WritableRaster raster = meanImage.getRaster();

		double min = Double.MAX_VALUE;
		double max = -Double.MAX_VALUE;
		for(int i = 0 ; i < eigenFacePixels.length ; i++) {
			min = Math.min(min, eigenFacePixels[i][columnIndex]);
			max = Math.max(max, eigenFacePixels[i][columnIndex]);
		}

		int[] pixels = new int[eigenFacePixels.length];
		for(int i = 0 ; i < eigenFacePixels.length ; i++) {
			pixels[i] = (int)(255.0 * (eigenFacePixels[i][columnIndex] - min) / (max - min));
		}
		raster.setPixels(0, 0, width, height, pixels);

		ImageIO.write(meanImage, "gif", new File(filename));
	}

	private static double[][] computeEigenFaces(double[][] diffMatrix, double[][] eigenVectors) {
		int pixelCount = diffMatrix.length;
		int imageCount = eigenVectors[0].length;
		int rank = eigenVectors.length;
		double[][] eigenFaces = new double[pixelCount][rank];

		for(int i = 0 ; i < rank ; i++) {
			double sumSquare = 0;
			for(int j = 0 ; j < pixelCount ; j++) {
				for(int k = 0 ; k < imageCount ; k++) {
					eigenFaces[j][i] += diffMatrix[j][k] * eigenVectors[i][k];
				}
				sumSquare += eigenFaces[j][i] * eigenFaces[j][i];
			}
			double norm = Math.sqrt(sumSquare);
			for(int j = 0 ; j < pixelCount ; j++) {
				eigenFaces[j][i] /= norm;
			}
		}
		return eigenFaces;
	}

	public static void main(String args[]) throws Exception {
		if (args.length != 7) {
			System.out.println("Arguments: eigenVectorFileName diffMatrixFileName meanImageFileName width height trainingDirectory outputDirectory");
			System.exit(1);
		}

		String eigenVectorsFileName = args[0];
		String diffMatrixFileName = args[1];
		String meanImageFilename = args[2];

		int width = Integer.parseInt(args[3]);
		int height = Integer.parseInt(args[4]);
		String trainingDirectory = args[5];
		String outputDirectory = args[6];

		File outputDirectoryFile = new File(outputDirectory);
		if (!outputDirectoryFile.exists()) {
			outputDirectoryFile.mkdir();
		}

		double[] meanPixels = Helper.readImagePixels(meanImageFilename, width, height);
		double[][] eigenVectors = Helper.readMatrixSequenceFile(eigenVectorsFileName);
		double[][] diffMatrix = Helper.readMatrixSequenceFile(diffMatrixFileName);
		double[][] eigenFaces = computeEigenFaces(diffMatrix, eigenVectors);

		int rank = eigenVectors.length;
		for(int i = 0 ; i < rank ; i++) {
			writeEigenFaceImage(outputDirectory + "/eigenface-" + i + ".gif", eigenFaces, width, height, i);
		}

		double minDistance = Double.MAX_VALUE;
		double maxDistance = -Double.MAX_VALUE;
		Helper.writeMatrixSequenceFile(outputDirectory + "/eigenfaces.seq", eigenFaces);

		List imageFileNames = Helper.listImageFileNames(trainingDirectory);
		int imageCount = diffMatrix[0].length;
		int pixelCount = width * height;
		double[][] weightMatrix = new double[imageCount][];
		for(int i = 0 ; i < imageCount ; i++) {
			double[] diffImagePixels = new double[pixelCount];
			for(int j = 0 ; j < pixelCount ; j++) {
				diffImagePixels[j] = diffMatrix[j][i];
			}
			double[] weights = Helper.computeWeights(diffImagePixels, eigenFaces);
			double[] reconstructedImagePixels = Helper.reconstructImageWithEigenFaces(
				weights, eigenFaces, meanPixels);
			String shortFileName = Helper.getShortFileName(imageFileNames.get(i));
			Helper.writeImage(outputDirectory + "/ef-" + shortFileName, reconstructedImagePixels, width, height);

			double[] imagePixels = Helper.readImagePixels(imageFileNames.get(i), width, height);
			double distance = Helper.computeImageDistance(imagePixels, reconstructedImagePixels);
			minDistance = Math.min(minDistance, distance);
			maxDistance = Math.max(maxDistance, distance);
			System.out.printf("Reconstructed Image distance for %1$s: %2$f\n", shortFileName, distance);

			weightMatrix[i] = weights;
		}
		Helper.writeMatrixSequenceFile(outputDirectory + "/weights.seq", weightMatrix);
		System.out.println("Min distance = " + minDistance);
		System.out.println("Max distance = " + maxDistance);
	}

}

To run the program:

$ java -cp target/mahout-eigenface-example-1.0-jar-with-dependencies.jar com.chimpler.example.eigenface.ComputeEigenFaces output/cleanEigenvectors output/diffmatrix.seq output/mean-image.gif 80 60 [TRAINING_SET_DIRECTORY] output

It creates the eigenfaces matrix in output/eigenfaces.seq and the images representing those eigenfaces in the output directory:

It also tries to reconstruct the faces of the training sets using the eigenfaces. To do that it computes the weight of each eigenface by doing a scalar product of the image pixel column with each eigenface column and then normalize it. Then it sums up each pixel of the eigenfaces weighted by those weights. You can think of this process as superposing the eigenfaces layers and give them a different transparency value (can be negative) to try to reconstruct the original image.

After having reconstructed the image, it computes the distance between the original image and the reconstructed image (using euclidian distance between the pixels):

Reconstructed Image distance for subject01.centerlight: 37.395691
Reconstructed Image distance for subject01.glasses: 32.350212
Reconstructed Image distance for subject01.happy: 27.559056
Reconstructed Image distance for subject01.leftlight: 28.008936
Reconstructed Image distance for subject01.noglasses: 47.047757
Reconstructed Image distance for subject01.normal: 32.627928
Reconstructed Image distance for subject01.rightlight: 25.465009
Reconstructed Image distance for subject01.sleepy: 23.635308
Reconstructed Image distance for subject01.surprised: 45.947206
Reconstructed Image distance for subject01.wink: 32.132286
[...]
Min distance = 14.470855648264822
Max distance = 47.047756576566904

These distances are quite small which means that our eigenfaces allows to efficiently represent faces.

Testing the Model

Now that we have trained our model, we are going to test it.
In the training set, we have some of the same people than in the training set but with a different facial expression. We also have two images with are not person face(hamburger and cat) and one image of a new person(Bruce Lee).

The class ComputeDistance tests if the images in the testing directory can be recognized as a person face and find the most similar image in the training set.

public class ComputeDistance {

	public static Object[] findClosestImage(double[] weights, double[][] weightMatrix) {
		int imageCount = weightMatrix.length;
		int eigenFaceCount = weightMatrix[0].length;

		int closestImageIndex = -1;
		double minWeightSquareDistance = Double.MAX_VALUE;
		for(int i = 0 ; i < imageCount ; i++) {
			double distance = 0;
			for(int j = 0 ; j < eigenFaceCount ; j++) {
				distance += (weightMatrix[i][j] - weights[j]) * (weightMatrix[i][j] - weights[j]);
			}

			if (distance < minWeightSquareDistance) {
				minWeightSquareDistance = distance;
				closestImageIndex = i;
			}
		}
		return new Object[]{closestImageIndex, Math.sqrt(minWeightSquareDistance / eigenFaceCount)};
	}

	public static void main(String args[]) throws Exception {
		if (args.length != 8) {
			System.out.println("Arguments: eigenFacesFileName meanImageFileName weightSeqFilename width height trainingDirectory testImageDirectory outputDirectory");
			System.exit(1);
		}

		String eigenFacesFilename = args[0];
		String meanImageFilename = args[1];
		String weightSeqFileName = args[2];

		int width = Integer.parseInt(args[3]);
		int height = Integer.parseInt(args[4]);
		String trainImageDirectory = args[5];
		String testImageDirectory = args[6];
		String outputDirectory = args[7];

		double[] meanPixels = Helper.readImagePixels(meanImageFilename, width, height);
		double[][] eigenFaces = Helper.readMatrixSequenceFile(eigenFacesFilename);
		double[][] weightMatrix = Helper.readMatrixSequenceFile(weightSeqFileName);

		List testImageFileNames = Helper.listImageFileNames(testImageDirectory);
		List trainImageFileNames = Helper.listImageFileNames(trainImageDirectory);

		for(String testImageFileName: testImageFileNames) {
			double[] imagePixels = Helper.readImagePixels(testImageFileName, width, height);
			double[] diffImagePixels = Helper.computeDifferencePixels(imagePixels, meanPixels);
			double[] weights = Helper.computeWeights(diffImagePixels, eigenFaces);

			double[] reconstructedImagePixels = Helper.reconstructImageWithEigenFaces(
					weights, eigenFaces, meanPixels);

			double distance = Helper.computeImageDistance(imagePixels, reconstructedImagePixels);
			String shortTestFileName = Helper.getShortFileName(testImageFileName);
			System.out.printf("Reconstructed Image distance for %1$s: %2$f\n", shortTestFileName, distance);

			Helper.writeImage(outputDirectory + "/test-ef-" + shortTestFileName, reconstructedImagePixels, width, height);

			Object[] closestImageInfo = findClosestImage(weights, weightMatrix);
			int closestImageIndex = (Integer)closestImageInfo[0];
			double closestImageSimilarity = (Double)closestImageInfo[1];
			System.out.printf("Image %1$s is most similar to %2$s: %3$f\n",
					shortTestFileName,
					Helper.getShortFileName(trainImageFileNames.get(closestImageIndex)),
					closestImageSimilarity);
		}
	}
}

To run the program:

$ java -cp target/mahout-eigenface-example-1.0-jar-with-dependencies.jar com.chimpler.example.eigenface.ComputeDistance output/eigenfaces.seq output/mean-image.gif output/weights.seq  68 68 [TRAINING_SET_DIRECTORY] [TESTING_SET_DIRECTORY] output

For each image of the testing set, it computes the  weight that needs to be applied on each eigenface to reconstruct the image and generate the reconstructed image in the output directory:

As expected, the images representing the face of people from the training set are well reconstructed but not the cat and the hamburger images. The reconstructed face of Bruce Lee is not recognizable but we can see that it is still a face. The program also computes the distance between the original image and the reconstructed image. It also tries for each test image, to find the most similar image in the training set by comparing the eigenfaces weight using euclidian distance:

Reconstructed Image distance for brucelee.gif: 51.404904
Image brucelee.gif is most similar to subject03.surprised: 447.574353
Reconstructed Image distance for cat.gif: 65.154281
Image cat.gif is most similar to subject05.centerlight: 638.072675
Reconstructed Image distance for hamburger.gif: 52.313601
Image hamburger.gif is most similar to subject01.rightlight: 684.214467
Reconstructed Image distance for subject01.sad: 32.473280
Image subject01.sad is most similar to subject01.sleepy: 101.895815
Reconstructed Image distance for subject02.sad: 22.418869
Image subject02.sad is most similar to subject02.noglasses: 104.859642
Reconstructed Image distance for subject03.sad: 35.468822
Image subject03.sad is most similar to subject03.noglasses: 120.972063
Reconstructed Image distance for subject04.sad: 30.370102
Image subject04.sad is most similar to subject04.normal: 0.000000
[...]

Those results confirm the visual interpretations we made previously: the distance between the reconstructed image and the original image of the hamburger and the cat are pretty high, also the weight distance with the images from the training set is pretty high. The image of Bruce Lee is reconstructed fairly but the weight distance is low.

For the other people faces, this distance is pretty small and it successfully associates them to the face of the same person from the training set.

Using the weight distance, we can define two thresholds:

• T1: threshold at which the images represent a face
• T2: weight threshold at which the image represents a face from the training set

So if the weight distance is above T1, then the image does not represent a face. Between T1 and T2, it represents an unknown face. And below T2, it represents a face from the training set. Choosing those thresholds is done heuristically.

Conclusion

We show in this post how to generate the eigenfaces from a training set and then uses those eigentafces to recognize person’s face. We also introduce some metrics to to determine if an image represents a person face or not and if it is similar to a face from the training set.

If you are trying this tutorial with other images make sure that:

• the faces are in the same position in the image
• the faces have the same scale/rotation angle
• the faces have the same brightness/contrasts

Some techniques were developed to alleviate those constraints. You can find several papers about this on the web.