本文详细解析了LIRE图像检索系统的三个核心过程：建立索引、提取特征向量及检索。介绍了ColorLayout特征提取方法及其在索引建立中的应用，并深入分析了检索过程中的关键技术。
上一篇文章我们介绍了LIRE的基本接口，这篇我们来看一看它建立索引，提取特征向量和检索的过程。
一、建立索引(DocumentBuilder)
不同的特征向量提取方法的建立索引的类各不相同，它们都位于“net.semanticmetadata.lire.impl”中，如下图所示：
由图可见，每一种方法对应一个DocumentBuilder和一个ImageSearcher，类的数量非常的多，无法一一分析。在这里仅分析一个比较有代表性的：颜色布局。
颜色布局建立索引的类的名称是ColorLayoutDocumentBuilder，该类继承了AbstractDocumentBuilder，它的源代码如下所示：
package net.semanticmetadata.lire.impl;
import net.semanticmetadata.lire.AbstractDocumentBuilder;
import net.semanticmetadata.lire.DocumentBuilder;
import net.semanticmetadata.lire.imageanalysis.ColorLayout;
import net.semanticmetadata.lire.utils.ImageUtils;
import org.apache.lucene.document.Document;
import org.apache.lucene.document.Field;
import java.awt.image.BufferedImage;
import java.util.logging.Logger;
/**
* Provides a faster way of searching based on byte arrays instead of Strings. The method
* {@link net.semanticmetadata.lire.imageanalysis.ColorLayout#getByteArrayRepresentation()} is used
* to generate the signature of the descriptor much faster.
* User: Mathias Lux, mathias@juggle.at
* Date: 30.06.2011
*/
public class ColorLayoutDocumentBuilder extends AbstractDocumentBuilder {
private Logger logger = Logger.getLogger(getClass().getName());
public static final int MAX_IMAGE_DIMENSION = 1024;
public Document createDocument(BufferedImage image, String identifier) {
assert (image != null);
BufferedImage bimg = image;
// Scaling image is especially with the correlogram features very important!
// All images are scaled to guarantee a certain upper limit for indexing.
if (Math.max(image.getHeight(), image.getWidth()) > MAX_IMAGE_DIMENSION) {
bimg = ImageUtils.scaleImage(image, MAX_IMAGE_DIMENSION);
}
Document doc = null;
logger.finer("Starting extraction from image [ColorLayout - fast].");
ColorLayout vd = new ColorLayout();
vd.extract(bimg);
logger.fine("Extraction finished [ColorLayout - fast].");
doc = new Document();
doc.add(new Field(DocumentBuilder.FIELD_NAME_COLORLAYOUT_FAST, vd.getByteArrayRepresentation()));
if (identifier != null)
doc.add(new Field(DocumentBuilder.FIELD_NAME_IDENTIFIER, identifier, Field.Store.YES, Field.Index.NOT_ANALYZED));
return doc;
}
}1.如果输入的图像分辨率过大（在这里是大于1024），则将图像缩小。
2.新建一个LireFeature类型的对象vd。
3.调用vd.extract()提取特征向量。
4.调用vd.getByteArrayRepresentation()获得特征向量。
5.将获得的特征向量加入Document，返回Document。
二、提取特征向量
在ColorLayoutDocumentBuilder中，使用了一个类型为ColorLayout的对象vd，并且调用了vd的extract()方法：
ColorLayout vd = new ColorLayout();
vd.extract(bimg); new Field(DocumentBuilder.FIELD_NAME_COLORLAYOUT_FAST, vd.getByteArrayRepresentation()) 
由图可见，这个包中有很多的类。这些类都是以检索方法的名字命名的。我们要找的ColorLayout类也在其中。看看它的代码吧：
package net.semanticmetadata.lire.imageanalysis;
import net.semanticmetadata.lire.imageanalysis.mpeg7.ColorLayoutImpl;
import net.semanticmetadata.lire.utils.SerializationUtils;
/**
* Just a wrapper for the use of LireFeature.
* Date: 27.08.2008
* Time: 12:07:38
*
* @author Mathias Lux, mathias@juggle.at
*/
public class ColorLayout extends ColorLayoutImpl implements LireFeature {
/*
public String getStringRepresentation() {
StringBuilder sb = new StringBuilder(256);
StringBuilder sbtmp = new StringBuilder(256);
for (int i = 0; i < numYCoeff; i++) {
sb.append(YCoeff[i]);
if (i + 1 < numYCoeff) sb.append(' ');
}
sb.append("z");
for (int i = 0; i < numCCoeff; i++) {
sb.append(CbCoeff[i]);
if (i + 1 < numCCoeff) sb.append(' ');
sbtmp.append(CrCoeff[i]);
if (i + 1 < numCCoeff) sbtmp.append(' ');
}
sb.append("z");
sb.append(sbtmp);
return sb.toString();
}
public void setStringRepresentation(String descriptor) {
String[] coeffs = descriptor.split("z");
String[] y = coeffs[0].split(" ");
String[] cb = coeffs[1].split(" ");
String[] cr = coeffs[2].split(" ");
numYCoeff = y.length;
numCCoeff = Math.min(cb.length, cr.length);
YCoeff = new int[numYCoeff];
CbCoeff = new int[numCCoeff];
CrCoeff = new int[numCCoeff];
for (int i = 0; i < numYCoeff; i++) {
YCoeff[i] = Integer.parseInt(y[i]);
}
for (int i = 0; i < numCCoeff; i++) {
CbCoeff[i] = Integer.parseInt(cb[i]);
CrCoeff[i] = Integer.parseInt(cr[i]);
}
}
*/
/**
* Provides a much faster way of serialization.
*
* @return a byte array that can be read with the corresponding method.
* @see net.semanticmetadata.lire.imageanalysis.CEDD#setByteArrayRepresentation(byte[])
*/
public byte[] getByteArrayRepresentation() {
byte[] result = new byte[2 * 4 + numYCoeff * 4 + 2 * numCCoeff * 4];
System.arraycopy(SerializationUtils.toBytes(numYCoeff), 0, result, 0, 4);
System.arraycopy(SerializationUtils.toBytes(numCCoeff), 0, result, 4, 4);
System.arraycopy(SerializationUtils.toByteArray(YCoeff), 0, result, 8, numYCoeff * 4);
System.arraycopy(SerializationUtils.toByteArray(CbCoeff), 0, result, numYCoeff * 4 + 8, numCCoeff * 4);
System.arraycopy(SerializationUtils.toByteArray(CrCoeff), 0, result, numYCoeff * 4 + numCCoeff * 4 + 8, numCCoeff * 4);
return result;
}
/**
* Reads descriptor from a byte array. Much faster than the String based method.
*
* @param in byte array from corresponding method
* @see net.semanticmetadata.lire.imageanalysis.CEDD#getByteArrayRepresentation
*/
public void setByteArrayRepresentation(byte[] in) {
int[] data = SerializationUtils.toIntArray(in);
numYCoeff = data[0];
numCCoeff = data[1];
YCoeff = new int[numYCoeff];
CbCoeff = new int[numCCoeff];
CrCoeff = new int[numCCoeff];
System.arraycopy(data, 2, YCoeff, 0, numYCoeff);
System.arraycopy(data, 2 + numYCoeff, CbCoeff, 0, numCCoeff);
System.arraycopy(data, 2 + numYCoeff + numCCoeff, CrCoeff, 0, numCCoeff);
}
public double[] getDoubleHistogram() {
double[] result = new double[numYCoeff + numCCoeff * 2];
for (int i = 0; i < numYCoeff; i++) {
result[i] = YCoeff[i];
}
for (int i = 0; i < numCCoeff; i++) {
result[i + numYCoeff] = CbCoeff[i];
result[i + numCCoeff + numYCoeff] = CrCoeff[i];
}
return result;
}
/**
* Compares one descriptor to another.
*
* @param descriptor
* @return the distance from [0,infinite) or -1 if descriptor type does not match
*/
public float getDistance(LireFeature descriptor) {
if (!(descriptor instanceof ColorLayoutImpl)) return -1f;
ColorLayoutImpl cl = (ColorLayoutImpl) descriptor;
return (float) ColorLayoutImpl.getSimilarity(YCoeff, CbCoeff, CrCoeff, cl.YCoeff, cl.CbCoeff, cl.CrCoeff);
}
}/**
* This is the basic interface for all content based features. It is needed for GenericDocumentBuilder etc.
* Date: 28.05.2008
* Time: 14:44:16
*
* @author Mathias Lux, mathias@juggle.at
*/
public interface LireFeature {
public void extract(BufferedImage bimg);
public byte[] getByteArrayRepresentation();
public void setByteArrayRepresentation(byte[] in);
public double[] getDoubleHistogram();
float getDistance(LireFeature feature);
java.lang.String getStringRepresentation();
void setStringRepresentation(java.lang.String s);
}注：这里没有注释了，仅能靠自己的理解了。
我简要概括一下自己对这些接口函数的理解：
1.extract(BufferedImage bimg)：提取特征向量
2.getByteArrayRepresentation()：获取特征向量（返回byte[]类型）
3.setByteArrayRepresentation(byte[] in)：设置特征向量（byte[]类型）
4.getDoubleHistogram()：
5.getDistance(LireFeature feature)：
6.getStringRepresentation()：获取特征向量（返回String类型）
7.setStringRepresentation(Java.lang.String s)：设置特征向量（String类型）
其中红色的是建立索引的过程中会用到的。
仔细看代码可以发现，所有的算法都实现了LireFeature接口，如下图所示：
三、检索(ImageSearcher)
我们继续看一下检索部分（ImageSearcher）。不同的方法的检索功能的类各不相同，它们都位于“net.semanticmetadata.lire.impl”中，如下图所示：
在这里仅分析一个比较有代表性的：颜色布局。前文已经分析过ColorLayoutDocumentBuilder，在这里我们分析一下ColorLayoutImageSearcher。源代码如下：
package net.semanticmetadata.lire.impl;
import net.semanticmetadata.lire.DocumentBuilder;
import net.semanticmetadata.lire.ImageDuplicates;
import net.semanticmetadata.lire.ImageSearchHits;
import net.semanticmetadata.lire.imageanalysis.ColorLayout;
import net.semanticmetadata.lire.imageanalysis.LireFeature;
import org.apache.lucene.document.Document;
import org.apache.lucene.index.IndexReader;
import java.io.FileNotFoundException;
import java.io.IOException;
import java.util.HashMap;
import java.util.LinkedList;
import java.util.List;
import java.util.logging.Level;
/**
* Provides a faster way of searching based on byte arrays instead of Strings. The method
* {@link net.semanticmetadata.lire.imageanalysis.ColorLayout#getByteArrayRepresentation()} is used
* to generate the signature of the descriptor much faster. First tests have shown that this
* implementation is up to 4 times faster than the implementation based on strings
* (for 120,000 images)
* <p/>
* User: Mathias Lux, mathias@juggle.at
* Date: 30.06 2011
*/
public class ColorLayoutImageSearcher extends GenericImageSearcher {
public ColorLayoutImageSearcher(int maxHits) {
super(maxHits, ColorLayout.class, DocumentBuilder.FIELD_NAME_COLORLAYOUT_FAST);
}
protected float getDistance(Document d, LireFeature lireFeature) {
float distance = 0f;
ColorLayout lf;
try {
lf = (ColorLayout) descriptorClass.newInstance();
byte[] cls = d.getBinaryValue(fieldName);
if (cls != null && cls.length > 0) {
lf.setByteArrayRepresentation(cls);
distance = lireFeature.getDistance(lf);
} else {
logger.warning("No feature stored in this document ...");
}
} catch (InstantiationException e) {
logger.log(Level.SEVERE, "Error instantiating class for generic image searcher: " + e.getMessage());
} catch (IllegalAccessException e) {
logger.log(Level.SEVERE, "Error instantiating class for generic image searcher: " + e.getMessage());
}
return distance;
}
public ImageSearchHits search(Document doc, IndexReader reader) throws IOException {
SimpleImageSearchHits searchHits = null;
try {
ColorLayout lireFeature = (ColorLayout) descriptorClass.newInstance();
byte[] cls = doc.getBinaryValue(fieldName);
if (cls != null && cls.length > 0)
lireFeature.setByteArrayRepresentation(cls);
float maxDistance = findSimilar(reader, lireFeature);
searchHits = new SimpleImageSearchHits(this.docs, maxDistance);
} catch (InstantiationException e) {
logger.log(Level.SEVERE, "Error instantiating class for generic image searcher: " + e.getMessage());
} catch (IllegalAccessException e) {
logger.log(Level.SEVERE, "Error instantiating class for generic image searcher: " + e.getMessage());
}
return searchHits;
}
public ImageDuplicates findDuplicates(IndexReader reader) throws IOException {
// get the first document:
SimpleImageDuplicates simpleImageDuplicates = null;
try {
if (!IndexReader.indexExists(reader.directory()))
throw new FileNotFoundException("No index found at this specific location.");
Document doc = reader.document(0);
ColorLayout lireFeature = (ColorLayout) descriptorClass.newInstance();
byte[] cls = doc.getBinaryValue(fieldName);
if (cls != null && cls.length > 0)
lireFeature.setByteArrayRepresentation(cls);
HashMap<Float, List<String>> duplicates = new HashMap<Float, List<String>>();
// find duplicates ...
boolean hasDeletions = reader.hasDeletions();
int docs = reader.numDocs();
int numDuplicates = 0;
for (int i = 0; i < docs; i++) {
if (hasDeletions && reader.isDeleted(i)) {
continue;
}
Document d = reader.document(i);
float distance = getDistance(d, lireFeature);
if (!duplicates.containsKey(distance)) {
duplicates.put(distance, new LinkedList<String>());
} else {
numDuplicates++;
}
duplicates.get(distance).add(d.getFieldable(DocumentBuilder.FIELD_NAME_IDENTIFIER).stringValue());
}
if (numDuplicates == 0) return null;
LinkedList<List<String>> results = new LinkedList<List<String>>();
for (float f : duplicates.keySet()) {
if (duplicates.get(f).size() > 1) {
results.add(duplicates.get(f));
}
}
simpleImageDuplicates = new SimpleImageDuplicates(results);
} catch (InstantiationException e) {
logger.log(Level.SEVERE, "Error instantiating class for generic image searcher: " + e.getMessage());
} catch (IllegalAccessException e) {
logger.log(Level.SEVERE, "Error instantiating class for generic image searcher: " + e.getMessage());
}
return simpleImageDuplicates;
}
}float getDistance(Document d, LireFeature lireFeature)：
ImageSearchHits search(Document doc, IndexReader reader)：检索。最核心函数。
ImageDuplicates findDuplicates(IndexReader reader)：目前还没研究。
ColorLayoutImageSearcher继承了一个类——GenericImageSearcher，看一下GenericImageSearcher的源代码：package net.semanticmetadata.lire.impl;
import net.semanticmetadata.lire.AbstractImageSearcher;
import net.semanticmetadata.lire.DocumentBuilder;
import net.semanticmetadata.lire.ImageDuplicates;
import net.semanticmetadata.lire.ImageSearchHits;
import net.semanticmetadata.lire.imageanalysis.LireFeature;
import net.semanticmetadata.lire.utils.ImageUtils;
import org.apache.lucene.document.Document;
import org.apache.lucene.index.IndexReader;
import java.awt.image.BufferedImage;
import java.io.FileNotFoundException;
import java.io.IOException;
import java.util.HashMap;
import java.util.LinkedList;
import java.util.List;
import java.util.TreeSet;
import java.util.logging.Level;
import java.util.logging.Logger;
/**
* This file is part of the Caliph and Emir project: http://www.SemanticMetadata.net
* <br>Date: 01.02.2006
* <br>Time: 00:17:02
*
* @author Mathias Lux, mathias@juggle.at
*/
public class GenericImageSearcher extends AbstractImageSearcher {
protected Logger logger = Logger.getLogger(getClass().getName());
Class<?> descriptorClass;
String fieldName;
private int maxHits = 10;
protected TreeSet<SimpleResult> docs;
public GenericImageSearcher(int maxHits, Class<?> descriptorClass, String fieldName) {
this.maxHits = maxHits;
docs = new TreeSet<SimpleResult>();
this.descriptorClass = descriptorClass;
this.fieldName = fieldName;
}
public ImageSearchHits search(BufferedImage image, IndexReader reader) throws IOException {
logger.finer("Starting extraction.");
LireFeature lireFeature = null;
SimpleImageSearchHits searchHits = null;
try {
lireFeature = (LireFeature) descriptorClass.newInstance();
// Scaling image is especially with the correlogram features very important!
BufferedImage bimg = image;
if (Math.max(image.getHeight(), image.getWidth()) > GenericDocumentBuilder.MAX_IMAGE_DIMENSION) {
bimg = ImageUtils.scaleImage(image, GenericDocumentBuilder.MAX_IMAGE_DIMENSION);
}
lireFeature.extract(bimg);
logger.fine("Extraction from image finished");
float maxDistance = findSimilar(reader, lireFeature);
searchHits = new SimpleImageSearchHits(this.docs, maxDistance);
} catch (InstantiationException e) {
logger.log(Level.SEVERE, "Error instantiating class for generic image searcher: " + e.getMessage());
} catch (IllegalAccessException e) {
logger.log(Level.SEVERE, "Error instantiating class for generic image searcher: " + e.getMessage());
}
return searchHits;
}
/**
* @param reader
* @param lireFeature
* @return the maximum distance found for normalizing.
* @throws java.io.IOException
*/
protected float findSimilar(IndexReader reader, LireFeature lireFeature) throws IOException {
float maxDistance = -1f, overallMaxDistance = -1f;
boolean hasDeletions = reader.hasDeletions();
// clear result set ...
docs.clear();
int docs = reader.numDocs();
for (int i = 0; i < docs; i++) {
// bugfix by Roman Kern
if (hasDeletions && reader.isDeleted(i)) {
continue;
}
Document d = reader.document(i);
float distance = getDistance(d, lireFeature);
assert (distance >= 0);
// calculate the overall max distance to normalize score afterwards
if (overallMaxDistance < distance) {
overallMaxDistance = distance;
}
// if it is the first document:
if (maxDistance < 0) {
maxDistance = distance;
}
// if the array is not full yet:
if (this.docs.size() < maxHits) {
this.docs.add(new SimpleResult(distance, d));
if (distance > maxDistance) maxDistance = distance;
} else if (distance < maxDistance) {
// if it is nearer to the sample than at least on of the current set:
// remove the last one ...
this.docs.remove(this.docs.last());
// add the new one ...
this.docs.add(new SimpleResult(distance, d));
// and set our new distance border ...
maxDistance = this.docs.last().getDistance();
}
}
return maxDistance;
}
protected float getDistance(Document d, LireFeature lireFeature) {
float distance = 0f;
LireFeature lf;
try {
lf = (LireFeature) descriptorClass.newInstance();
String[] cls = d.getValues(fieldName);
if (cls != null && cls.length > 0) {
lf.setStringRepresentation(cls[0]);
distance = lireFeature.getDistance(lf);
} else {
logger.warning("No feature stored in this document!");
}
} catch (InstantiationException e) {
logger.log(Level.SEVERE, "Error instantiating class for generic image searcher: " + e.getMessage());
} catch (IllegalAccessException e) {
logger.log(Level.SEVERE, "Error instantiating class for generic image searcher: " + e.getMessage());
}
return distance;
}
public ImageSearchHits search(Document doc, IndexReader reader) throws IOException {
SimpleImageSearchHits searchHits = null;
try {
LireFeature lireFeature = (LireFeature) descriptorClass.newInstance();
String[] cls = doc.getValues(fieldName);
if (cls != null && cls.length > 0)
lireFeature.setStringRepresentation(cls[0]);
float maxDistance = findSimilar(reader, lireFeature);
searchHits = new SimpleImageSearchHits(this.docs, maxDistance);
} catch (InstantiationException e) {
logger.log(Level.SEVERE, "Error instantiating class for generic image searcher: " + e.getMessage());
} catch (IllegalAccessException e) {
logger.log(Level.SEVERE, "Error instantiating class for generic image searcher: " + e.getMessage());
}
return searchHits;
}
public ImageDuplicates findDuplicates(IndexReader reader) throws IOException {
// get the first document:
SimpleImageDuplicates simpleImageDuplicates = null;
try {
if (!IndexReader.indexExists(reader.directory()))
throw new FileNotFoundException("No index found at this specific location.");
Document doc = reader.document(0);
LireFeature lireFeature = (LireFeature) descriptorClass.newInstance();
String[] cls = doc.getValues(fieldName);
if (cls != null && cls.length > 0)
lireFeature.setStringRepresentation(cls[0]);
HashMap<Float, List<String>> duplicates = new HashMap<Float, List<String>>();
// find duplicates ...
boolean hasDeletions = reader.hasDeletions();
int docs = reader.numDocs();
int numDuplicates = 0;
for (int i = 0; i < docs; i++) {
if (hasDeletions && reader.isDeleted(i)) {
continue;
}
Document d = reader.document(i);
float distance = getDistance(d, lireFeature);
if (!duplicates.containsKey(distance)) {
duplicates.put(distance, new LinkedList<String>());
} else {
numDuplicates++;
}
duplicates.get(distance).add(d.getFieldable(DocumentBuilder.FIELD_NAME_IDENTIFIER).stringValue());
}
if (numDuplicates == 0) return null;
LinkedList<List<String>> results = new LinkedList<List<String>>();
for (float f : duplicates.keySet()) {
if (duplicates.get(f).size() > 1) {
results.add(duplicates.get(f));
}
}
simpleImageDuplicates = new SimpleImageDuplicates(results);
} catch (InstantiationException e) {
logger.log(Level.SEVERE, "Error instantiating class for generic image searcher: " + e.getMessage());
} catch (IllegalAccessException e) {
logger.log(Level.SEVERE, "Error instantiating class for generic image searcher: " + e.getMessage());
}
return simpleImageDuplicates;
}
public String toString() {
return "GenericSearcher using " + descriptorClass.getName();
}
}1.输入图片如果尺寸过大（大于1024），则调整尺寸。
2.使用extract()提取输入图片的特征值。
3.根据提取的特征值，使用findSimilar()查找相似的图片。
4.新建一个ImageSearchHits用于存储查找的结果。
5.返回ImageSearchHits
在这里要注意一点：
GenericImageSearcher中创建特定方法的类的时候，使用了如下形式：
LireFeature lireFeature = (LireFeature) descriptorClass.newInstance(); AutoColorCorrelogram acc = new AutoColorCorrelogram(CorrelogramDocumentBuilder.MAXIMUM_DISTANCE) 相比而言，更具有通用型。
在search()函数中，调用了一个函数findSimilar()。这个函数的作用是查找相似图片的，分析了一下它的步骤：
1.使用IndexReader获取所有的记录
2.遍历所有的记录，和当前输入的图片进行比较，使用getDistance()函数
3.获取maxDistance并返回
在findSimilar()中，又调用了一个getDistance()，该函数调用了具体检索方法的getDistance()函数。
下面我们来看一下ColorLayout类中的getDistance()函数：
public float getDistance(LireFeature descriptor) {
if (!(descriptor instanceof ColorLayoutImpl)) return -1f;
ColorLayoutImpl cl = (ColorLayoutImpl) descriptor;
return (float) ColorLayoutImpl.getSimilarity(YCoeff, CbCoeff, CrCoeff, cl.YCoeff, cl.CbCoeff, cl.CrCoeff);
} public static double getSimilarity(int[] YCoeff1, int[] CbCoeff1, int[] CrCoeff1, int[] YCoeff2, int[] CbCoeff2, int[] CrCoeff2) {
int numYCoeff1, numYCoeff2, CCoeff1, CCoeff2, YCoeff, CCoeff;
//Numbers of the Coefficients of two descriptor values.
numYCoeff1 = YCoeff1.length;
numYCoeff2 = YCoeff2.length;
CCoeff1 = CbCoeff1.length;
CCoeff2 = CbCoeff2.length;
//take the minimal Coeff-number
YCoeff = Math.min(numYCoeff1, numYCoeff2);
CCoeff = Math.min(CCoeff1, CCoeff2);
setWeightingValues();
int j;
int[] sum = new int[3];
int diff;
sum[0] = 0;
for (j = 0; j < YCoeff; j++) {
diff = (YCoeff1[j] - YCoeff2[j]);
sum[0] += (weightMatrix[0][j] * diff * diff);
}
sum[1] = 0;
for (j = 0; j < CCoeff; j++) {
diff = (CbCoeff1[j] - CbCoeff2[j]);
sum[1] += (weightMatrix[1][j] * diff * diff);
}
sum[2] = 0;
for (j = 0; j < CCoeff; j++) {
diff = (CrCoeff1[j] - CrCoeff2[j]);
sum[2] += (weightMatrix[2][j] * diff * diff);
}
//returns the distance between the two desciptor values
return Math.sqrt(sum[0] * 1.0) + Math.sqrt(sum[1] * 1.0) + Math.sqrt(sum[2] * 1.0);
} 
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