# Efficient visual search of local features. Cordelia Schmid

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1 Efficient visual search of local features Cordelia Schmid

2 Visual search change in viewing angle

3 Matches 22 correct matches

4 Image search system for large datasets Large image dataset (one million images or more) query ranked image list Image search system Issues for very large databases to reduce the query time to reduce the storage requirements

5 Two strategies 1. Efficient approximate nearest neighbour search on local feature descriptors. 2. Quantize descriptors into a visual vocabulary and use efficient techniques from text retrieval. (Bag-of-words representation)

6 Strategy 1: Efficient approximate NN search Local features invariant descriptor vectors Images invariant descriptor vectors 1. Compute local features in each image independently 2. Describe each feature by a descriptor vector 3. Find nearest neighbour vectors between query and database 4. Rank matched images by number of (tentatively) corresponding regions 5. Verify top ranked images based on spatial consistency

7 Finding nearest neighbour vectors Establish correspondences between query image and images in the database by nearest neighbour matching on SIFT vectors Model image 128D descriptor space Image database Solve following problem for all feature vectors,, in the query image: where,, are features from all the database images.

8 Quick look at the complexity of the NN-search N images M regions per image (~1000) D dimension of the descriptor (~128) Exhaustive linear search: O(M NMD) Example: Matching two images (N=1), each having 1000 SIFT descriptors Nearest neighbors search: 0.4 s (2 GHz CPU, implemenation in C) Memory footprint: 1000 * 128 = 128kB / image # of images CPU time Memory req. N = 1,000 ~7min (~100MB) N = 10,000 ~1h7min (~ 1GB) N = 10 7 ~115 days (~ 1TB) All images on Facebook: N = ~300 years (~ 1PB)

9 Nearest-neighbor matching Solve following problem for all feature vectors, x j, in the query image: where x i are features in database images. Nearest-neighbour matching is the major computational bottleneck Linear search performs dn operations for n features in the database and d dimensions No exact methods are faster than linear search for d>10 Approximate methods can be much faster, but at the cost of missing some correct matches. Failure rate gets worse for large datasets.

10 Approximate nearest neighbour search kd-trees (k dim. tree) Binary tree in which each node is a k-dimensional point Every split is associated with one dimension kd-tree decomposition kd-tree

11 K-d tree K-d tree is a binary tree data structure for organizing a set of points Each internal node is associated with an axis aligned hyper-plane splitting its associated points into two sub-trees. Dimensions with high variance are chosen first. Position of the splitting hyper-plane is chosen as the mean/median of the projected points balanced tree. l 1 4 l l 2 l 3 l l 4 l 5 l 7 l l l l 9 6 7

12 Simple 2D example K-d tree construction 4 l 1 l 9 l 5 l l 1 l 2 5 l 3 l 2 l 3 l l 10 l7 l 4 l 5 l 7 l 6 1 l l l 10 8 l

13 K-d tree query 4 6 l 5 l 1 l 9 l 6 7 l 1 l 2 q 5 8 l 3 l 2 l 3 l l l l 7 l 4 l 5 l 7 l l 4 11 l l 10 8 l

14 Large scale object/scene recognition Image dataset: > 1 million images query Image search system ranked image list Each image described by approximately 2000 descriptors 2 * 10 9 descriptors to index for one million images! Database representation in RAM: Size of descriptors : 1 TB, search+memory intractable

15 Bag-of-features [Sivic&Zisserman 03] Query image Set of SIFT descriptors centroids (visual words) sparse frequency vector Harris-Hessian-Laplace regions + SIFT descriptors Bag-of-features processing +tf-idf weighting visual words : 1 word (index) per local descriptor only images ids in inverted file => 8 GB fits! Re-ranked list Inverted file Geometric verification [Chum & al. 2007] querying ranked image short-list

16 Indexing text with inverted files Document collection: Inverted file: Term List of hits (occurrences in documents) People [d1:hit hit hit], [d4:hit hit] Common [d1:hit hit], [d3: hit], [d4: hit hit hit] Sculpture [d2:hit], [d3: hit hit hit] Need to map feature descriptors to visual words

17

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19 Example: each group of patches belongs to the same visual word Visual words K. Grauman, B. Leibe Figure from Sivic & Zisserman, ICCV

20 K-means clustering Minimizing sum of squared Euclidean distances between points x i and their nearest cluster centers Algorithm: Randomly initialize K cluster centers Iterate until convergence: Assign each data point to the nearest center Recompute each cluster center as the mean of all points assigned to it Local minimum, solution dependent on initialization Initialization important, run several times, select best

21 Inverted file index for images comprised of visual words Word number List of image numbers Score each image by the number of common visual words (tentative correspondences) Dot product between bag-of-features Fast for sparse vectors! Image credit: A. Zisserman K. Grauman, B. Leibe

22 Inverted file index for images comprised of visual words Weighting with tf-idf score: weight visual words based on their frequency Tf: normalized term (word) ti frequency in a document dj tf / ij = n ij n kj k Idf: inverse document frequency, total number of documents divided by number of documents containing the term ti idf i = log D { d : t d} i Tf-Idf: tf idf ij = tf Image credit: A. Zisserman ij idf i K. Grauman, B. Leibe

23 Visual words Map descriptors to words by quantizing the feature space Quantize via k-means clustering to obtain visual words Assign descriptor to closest visual word Bag-of-features as approximate nearest neighbor search Bag-of-features matching function where q(x) is a quantizer, i.e., assignment to visual word and δ a,b is the Kronecker operator (δ a,b =1 iff a=b)

24 Approximate nearest neighbor search evaluation ANN algorithms usually returns a short-list of nearest neighbors this short-list is supposed to contain the NN with high probability exact search may be performed to re-order this short-list Proposed quality evaluation of ANN search: trade-off between Accuracy: NN recall = probability that the NN is in this list against Ambiguity removal = proportion of vectors in the short-list - the lower this proportion, the more information we have about the vector - the lower this proportion, the lower the complexity if we perform exact search on the short-list ANN search algorithms usually have some parameters to handle this trade-off

25 ANN evaluation of bag-of-features NN re ecall k=100 ANN algorithms returns a list of potential neighbors Accuracy: NN recall = probability that the NN is in this list Ambiguity removal: = proportion of vectors in the short-list 0.1 BOW 0 1e-07 1e-06 1e rate of points retrieved In BOF, this trade-off is managed by the number of clusters k

26 Vocabulary size The intrinsic matching scheme performed by BOF is weak for a small visual dictionary: too many false matches for a large visual dictionary: complexity, true matches are missed No good trade-off between small and large! either the Voronoi cells are too big or these cells can t absorb the descriptor noise intrinsic approximate nearest neighbor search of BOF is not sufficient

27 20K visual word: false matches

28 200K visual word: good matches missed

29 Hamming Embedding [Jegou et al. ECCV 08] Representation of a descriptor x Vector-quantized to q(x) as in standard BOF + short binary vector b(x) for an additional localization in the Voronoi cell Two descriptors x and y match iif where h(a,b) Hamming distance

30 Hamming Embedding Nearest neighbors for Hamming distance those for Euclidean distance a metric in the embedded space reduces dimensionality curse effects Efficiency Hamming distance = very few operations Fewer random memory accesses: 3 x faster that BOF with same dictionary size!

31 Hamming Embedding Off-line (given a quantizer) draw an orthogonal projection matrix P of size d b d this defines d b random projection directions for each Voronoi cell and projection direction, compute the median value for a learning set On-line: compute the binary signature b(x) of a given descriptor project x onto the projection directions as z(x) = (z 1, z db ) b i (x) = 1 if z i (x) is above the learned median value, otherwise 0

32 ANN evaluation of Hamming Embedding 0.7 NN recall h t = k=100 compared to BOW: at least times less points in the short-list for the same level of accuracy Hamming Embedding provides a much better trade-off between recall and ambiguity removal 0.1 HE+BOW BOW 0 1e-08 1e-07 1e-06 1e rate of points retrieved

33 Matching points - 20k word vocabulary 201 matches 240 matches Many matches with the non-corresponding image!

34 Matching points - 200k word vocabulary 69 matches 35 matches Still many matches with the non-corresponding one

35 Matching points - 20k word vocabulary + HE 83 matches 8 matches 10x more matches with the corresponding image!

36 Bag-of-features [Sivic&Zisserman 03] Query image Set of SIFT descriptors centroids (visual words) sparse frequency vector Harris-Hessian-Laplace regions + SIFT descriptors Bag-of-features processing +tf-idf weighting visual words : 1 word (index) per local descriptor only images ids in inverted file => 8 GB fits! Re-ranked list Inverted file Geometric verification [Chum & al. 2007] querying ranked image short-list

37 Geometric verification Use the position and shape of the underlying features to improve retrieval quality Both images have many matches which is correct?

38 Geometric verification We can measure spatial consistency between the query and each result to improve retrieval quality Many spatially consistent matches correct result Few spatially consistent matches incorrect result

39 Geometric verification Gives localization of the object

40 Geometric verification Remove outliers, matches contain a high number of incorrect ones Estimate geometric transformation Robust strategies RANSAC Hough transform

41 Example: estimating 2D affine transformation Simple fitting procedure (linear least squares) Approximates viewpoint changes for roughly planar objects and roughly orthographic cameras Can be used to initialize fitting for more complex models Matches consistent with an affine transformation

42 Fitting an affine transformation Assume we know the correspondences, how do we get the transformation? ), ( i i y x ), ( i x i y + = t t y x m m m m y x i i i i = L L L L i i i i i i y x t t m m m m y x y x

43 Fitting an affine transformation L x i y i x i y i 0 1 L t 1 t 2 m 1 m 2 m 3 m 4 = L x i y i L Linear system with six unknowns Each match gives us two linearly independent equations: need at least three to solve for the transformation parameters

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