SIFT特征点匹配中KD-tree与Ransac算法的使用

转自：http://blog.csdn.net/ijuliet/article/details/4471311

Step1:BBF算法，在KD-tree上找KNN。第一步做匹配咯~

1.什么是KD-tree（fromwiki）

K-Dimension tree，实际上是一棵平衡二叉树。

一般的KD-tree构造过程：

functionkdtree (list of points pointList, int depth)

{

ifpointList is empty

returnnil;

else {

// Select axis based on depth so thataxis cycles through all valid values

varint axis := depth mod k;

// Sort point list and choose medianas pivot element

selectmedian by axis from pointList;

// Create node and construct subtrees

vartree_node node;

node.location:= median;

node.leftChild:= kdtree(points in pointList before median, depth+1);

node.rightChild:= kdtree(points in pointList after median, depth+1);

returnnode;

}

}

2.BBF算法，在KD-tree上找KNN ( K-nearest neighbor)

BBF(BestBin First)算法，借助优先队列（这里用最小堆）实现。从根开始，在KD-tree上找路子的时候，错过的点先塞到优先队列里，自己先一个劲儿扫到leaf；然后再从队列里取出目前key值最小的（这里是是ki维上的距离最小者），重复上述过程，一个劲儿扫到leaf；直到队列找空了，或者已经重复了200遍了停止。

Step1:将img2的features建KD-tree; kd_root = kdtree_build( feat2,n2 );。在这里，ki是选取均方差最大的那个维度，kv是各特征点在那个维度上的median值，features是你率领的整个儿子孙子特征大军，n是你儿子孙子个数。

/** a node in a k-d tree */

struct kd_node{

int ki; /**< partition key index */

double kv; /**< partition key value */

int leaf; /**< 1 if node is a leaf, 0 otherwise */

struct feature* features; /**< features at this node */

int n; /**< number of features */

struct kd_node* kd_left; /**< left child */

struct kd_node* kd_right; /**< right child */

};

Step2: 将img1的每个feat到KD-tree里找k个最近邻，这里k=2。

k= kdtree_bbf_knn( kd_root, feat, 2, &nbrs, KDTREE_BBF_MAX_NN_CHKS );

min_pq = minpq_init();

minpq_insert( min_pq, kd_root, 0 );

while( min_pq->n > 0 && t < max_nn_chks ) //队列里有东西就继续搜，同时控制在t<200（即200步内）

{

expl = (struct kd_node*)minpq_extract_min( min_pq ); //取出最小的，front & pop

expl = explore_to_leaf( expl, feat, min_pq ); //从该点开始，explore到leaf，路过的“有意义的点”就塞到最小队列min_pq中。

for( i = 0; i < expl->n; i++ ) //

{

tree_feat = &expl->features[i];

bbf_data->old_data = tree_feat->feature_data;

bbf_data->d = descr_dist_sq(feat, tree_feat); //两feat均方差

tree_feat->feature_data = bbf_data;

n += insert_into_nbr_array( tree_feat, _nbrs, n, k ); //按从小到大塞到neighbor数组里，到时候取前k个就是 KNN 咯~ n 每次加1或0，表示目前已有的元素个数

}

t++;

}

对“有意义的点”的解释：

struct kd_node* explore_to_leaf( struct kd_node* kd_node, struct feature* feat,

struct min_pq* min_pq )//expl, feat, min_pq

{

struct kd_node* unexpl, * expl = kd_node;

double kv;

int ki;

while( expl && ! expl->leaf )

{

ki = expl->ki;

kv = expl->kv;

if( feat->descr[ki] <= kv ) {

unexpl = expl->kd_right;

expl = expl->kd_left; //走左边，右边点将被记下来

}

else{

unexpl = expl->kd_left;

expl = expl->kd_right; //走右边，左边点将被记下来

}

minpq_insert( min_pq, unexpl, ABS( kv - feat->descr[ki] ) ) ;//将这些点插入进来，key键值为|kv - feat->descr[ki]| 即第ki维上的差值

}

return expl;

}

Step3: 如果k近邻找到了(k=2)，那么判断是否能作为有效特征，d0/d1<0.49就算是咯~

d0 = descr_dist_sq( feat, nbrs[0] );//计算两特征间squared Euclidian distance

d1 = descr_dist_sq( feat, nbrs[1] );

if( d0 < d1 * NN_SQ_DIST_RATIO_THR )//如果d0/d1小于阈值0.49

{

pt1 = cvPoint( cvRound( feat->x ), cvRound( feat->y ) );

pt2 = cvPoint( cvRound( nbrs[0]->x ), cvRound( nbrs[0]->y ) );

pt2.y += img1->height;

cvLine( stacked, pt1, pt2, CV_RGB(255,0,255), 1, 8, 0 );//画线

m++;//matches个数

feat1[i].fwd_match = nbrs[0];

}

Step2:通过RANSAC算法来消除错配，什么是RANSAC先？

1.RANSAC(Random Sample Consensus, 随机抽样一致)(from wiki)

该算法做什么呢？呵呵，用一堆数据去搞定一个待定模型，这里所谓的搞定就是一反复测试、迭代的过程，找出一个error最小的模型及其对应的同盟军（consensusset）。用在我们的SIFT特征匹配里，就是说找一个变换矩阵出来，使得尽量多的特征点间都符合这个变换关系。

算法思想：

input:

data - a set of observations

model - a model that can be fitted todata

n - the minimum number of datarequired to fit the model

k - the maximum number of iterationsallowed in the algorithm

t - a threshold value for determiningwhen a datum fits a model

d - the number of close data valuesrequired to assert that a model fits well to data

output:

best_model - model parameters whichbest fit the data (or nil if no good model is found)

best_consensus_set - data point fromwhich this model has been estimated

best_error - the error of this modelrelative to the data

iterations:= 0

best_model:= nil

best_consensus_set:= nil

best_error:= infinity

whileiterations < k //进行K次迭代

maybe_inliers:= n randomly selected values from data

maybe_model:= model parameters fitted to maybe_inliers

consensus_set:= maybe_inliers

forevery point in data not in maybe_inliers

ifpoint fits maybe_model with an error smaller than t //错误小于阈值t

addpoint to consensus_set //成为同盟，加入consensus set

if thenumber of elements in consensus_set is > d //同盟军已经大于d个人，够了

(thisimplies that we may have found a good model,

nowtest how good it is)

better_model:= model parameters fitted to all points in consensus_set

this_error:= a measure of how well better_model fits these points

ifthis_error < best_error

(wehave found a model which is better than any of the previous ones,

keepit until a better one is found)

best_model:= better_model

best_consensus_set:= consensus_set

best_error:= this_error

incrementiterations

returnbest_model, best_consensus_set, best_error

2.RANSAC去除错配：

H= ransac_xform( feat1, n1, FEATURE_FWD_MATCH, lsq_homog, 4,0.01,homog_xfer_err, 3.0, NULL, NULL );

nm = get_matched_features( features, n, mtype, &matched );

/* initialize random number generator */

rng = gsl_rng_alloc( gsl_rng_mt19937 );

gsl_rng_set( rng, time(NULL) );

in_min = calc_min_inliers( nm, m, RANSAC_PROB_BAD_SUPP, p_badxform ); //符合这一要求的内点至少得有多少个

p = pow( 1.0 - pow( in_frac, m ), k );

i = 0;

while( p > p_badxform )//p>0.01

{

sample = draw_ransac_sample( matched, nm, m, rng );

extract_corresp_pts( sample, m, mtype, &pts, &mpts );

M = xform_fn( pts, mpts, m );

if( ! M )

goto iteration_end;

in = find_consensus( matched, nm, mtype, M, err_fn, err_tol, &consensus);

if( in > in_max ) {

if( consensus_max )

free( consensus_max );

consensus_max = consensus;

in_max = in;

in_frac = (double)in_max / nm;

}

else

free( consensus );

cvReleaseMat( &M );

iteration_end:

release_mem( pts, mpts, sample );

p = pow( 1.0 - pow( in_frac, m ), ++k );

}

/* calculate final transform based on best consensus set */

if( in_max >= in_min )

{

extract_corresp_pts( consensus_max, in_max, mtype, &pts, &mpts );

M = xform_fn( pts, mpts, in_max );

in = find_consensus( matched, nm, mtype, M, err_fn, err_tol, &consensus);

cvReleaseMat( &M );

release_mem( pts, mpts, consensus_max );

extract_corresp_pts( consensus, in, mtype, &pts, &mpts );

M = xform_fn( pts, mpts, in );

思考中的一些问题：

features间的对应关系，记录在features->fwd_match里（matching feature from forward

imge）。

1.数据是nm个特征点间的对应关系，由它们产生一个3*3变换矩阵（xform_fn= hsq_homog函数，此要>=4对的对应才可能计算出来咯~），此乃模型model。

2.然后开始找同盟军（find_consensus函数），判断除了sample的其它对应关系是否满足这个模型（err_fn= homog_xfer_err函数，<=err_tol就OK~），满足则留下。

3.一旦大于当前的in_max，那么该模型就升级为目前最牛的模型。（最最原始的RANSAC是按错误率最小走的，我们这会儿已经保证了错误率在err_tol范围内，按符合要求的对应数最大走，尽量多的特征能匹配地上）

4.重复以上3步，直到(1-w^m)^k <=p_badxform (即0.01)，模型就算找定~

5.最后再把模型和同盟军定一下，齐活儿~

声明：以上代码参考Rob Hess的SIFT实现。

其它参考文献：

1、http://www.cnblogs.com/slysky/archive/2011/11/08/2241247.html

2、http://en.wikipedia.org/wiki/Kd_tree

3、http://www.cnblogs.com/tjulxh/archive/2011/12/31/2308921.html

4、http://grunt1223.iteye.com/blog/961063