Scale-invariant feature transform: differenze tra le versioni

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{{S|informaticaprogrammazione}}

Nell'ambito della [[visione artificiale]], lo '''''scale-invariant feature transform''''' (o '''SIFT''') è un [[algoritmo]] che permette di rilevare e descrivere [[Caratteristica (apprendimento automatico)|caratteristiche]] locali in immagini. L'algoritmo è stato pubblicato da [[David G. Lowe]] nel [[1999]].<ref name="lowe99">{{cita conferenza

<!--{{FeatureDetectionCompVisNavbox}}-->

|autore = Lowe, David G.

|anno = 1999

'''Scale-invariant feature transform''' (o '''SIFT''') è un [[algoritmo]] utilizzato in [[computer vision]] che permette di rilevare e descrivere caratteristiche, o feature, locali in immagini. L'algoritmo è stato pubblicato da [[:en:David G. Lowe|David G. Lowe]] nel [[1999]].<ref name=lowe99>{{cita conferenza

|titolo = Object recognition from local scale-invariant features

| autore = Lowe, David G.

|conferenza = Proceedings of the International Conference on Computer Vision

| anno = 1999

|volume = 2

| titolo = Object recognition from local scale-invariant features

|pagine = 1150–1157

| conferenza = Proceedings of the International Conference on Computer Vision

|doi = 10.1109/ICCV.1999.790410

| volume = 2

|url = http://doi.ieeecs.org/10.1109/ICCV.1999.790410

| pagine = 1150–1157

|urlmorto = sì

| doi = 10.1109/ICCV.1999.790410

| url = http://doi.ieeecs.org/10.1109/ICCV.1999.790410

}}</ref>

Applicazioni includono: [[objectriconoscimento recognitiondi oggetti]], [[robotic mapping]] e navigation, [[image stitching]], [[modellazione 3D]], [[gesture recognition|riconoscimento dei gesti]], [[video tracking]], e [[match moving]].

L'algoritmo è brevettato negli Stati Uniti; il proprietario è la [[University of British Columbia]].<ref>Brevetto US 6,711,293</ref>

== Scale-invariant feature transform ==

Per ogni oggetto in un'immagine, alcuni punti interessanti possono essere estratti per fornire una descrizione delle caratteristiche dell'oggetto. Questa descrizione, ricavata da un'immagine selezionata per l'apprendimento, può essere poi usata per identificare l'oggetto quando si prova ad individuarlo in una immagine di test che contiene molti altri oggetti. Per un riconoscimento affidabile, è importante che le caratteristiche estratte dall'immagine di apprendimento siano rilevabili anche con cambiamenti di scala, rumore e illuminazione. Questi punti di solito sono posizionati in regioni ad alto contrasto dell'immagine, come contorni di oggetti.

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Per ogni oggetto in un'immagine, punti interessanti possono essere estratti per fornire una descrizione delle caratteristiche dell'oggetto. Questa descrizione, ricavata da un'immagine selezionata per l'apprendimento, può essere poi usata per identificare l'oggetto quando si prova ad individuarlo in una immagine di test che contiene molti altri oggetti. Per un riconoscimento affidabile, è importante che le caratteristiche estratte dall'immagine di apprendimento siano rilevabili anche con cambiamenti di scala, rumore e illuminazione. Tali punti di solito sono posizionati in regioni ad alto contrasto dell'immagine, come contorni di oggetti.

Un'altra importante caratteristica di queste feature è che le loro posizioni relative nella scena originale non dovrebbero cambiarecambiano tra due immagini affini. Per esempio, se come featurecaratteristiche vengono presi gli angoli di una porta, esse dovrebbero funzionare indipendentemente dalla posizione della porta; invece se però vengonovenissero presi in considerazione anche punti del contorno, il riconoscimento fallirebbe seo meno a seconda che la porta fossesia chiusa o aperta. In modo simile, le feature situate in oggetti articolati o flessibili tipicamente non funzionano sequando avvenisseavviene un qualsiasi cambiamento della loro geometria interna tra due immagini nell'insieme elaborato. In praticaComunque, comunqueSIFT, SIFTper rilevaridurre eil contributo degli errori causati da variazioni locali, usa un numero molto più grande di feature dalle immagini, ilin modo che riducevariazioni illocali contributosulla degligeometria erroriinterna causatiabbiano daun questepeso variazioni localipiccolo sull'errore medio di tutte le feature.

Il metodo di Lowe<ref>Brevetto US 6,711,293, "Method and apparatus for identifying scale invariant features in an image and use of same for locating an object in an image", brevetto del metodo di Lowe per l'algoritmo SIFT</ref> può identificare in modo robusto oggetti anche nella confusione o parzialmente nascosti, perché il descrittore di feature SIFT è invariante per scala, orientazione e distorsione affine e parzialmente invariante a cambi di illuminazione<ref name="lowe">Lowe, D. G., “Object recognition from local scale-invariant features”, International Conference on Computer Vision, Corfu, Greece, September 1999.</ref>. Questa sezione riassume il metodo di object recognition di Lowe e introduce alcune tecniche alternative disponibili per il riconoscimenti di oggetti in condizioni di confusione o parziale occlusione.

=== Il metodo di Lowe ===

Il metodo di Lowe<ref>Brevetto US 6,711,293, "Method and apparatus for identifying scale invariant features in an image and use of same for locating an object in an image", brevetto del metodo di Lowe per l'algoritmo SIFT</ref> può identificare, in modo robusto, oggetti anche nella confusione o parzialmente nascosti, perché il descrittore di feature SIFT è invariante per scala, orientamento, distorsione affine e parzialmente invariante a cambi di illuminazione<ref name="lowe">Lowe, D. G., “Object recognition from local scale-invariant features”, International Conference on Computer Vision, Corfu, Greece, September 1999.</ref>. Questa sezione riassume il metodo di object recognition di Lowe e introduce alcune tecniche alternative disponibili per il riconoscimenti di oggetti in condizioni di confusione o parziale occlusione.

I punti chiave SIFT di oggetti vengono prima estratti da un insieme di immagini di riferimento e immagazzinati in un database. Un oggetto viene riconosciuto in una nuova immagine confrontando individualmente ogni feature della nuova immagine con il database precedentemente ottenuto e cercando feature basandosi sulla [[distanza euclidea]] dei loro vettori di feature. Dall'insieme completo di corrispondenze, nella nuova immagine vengono identificati sottoinsiemi di punti chiave che concordano con l'oggetto e la sua posizione, scala, orientazione per filtrare le corrispondenze migliori. La determinazione di cluster consistenti viene effettuata in modo rapido usando un'implementazione con [[hash table]] efficiente della [[trasformata di Hough]] generalizzata. Ogni cluster di 3 o più feature concordanti su un oggetto e la sua posizione è soggetto ad un'ulteriore verifica dettagliata del modello e, di conseguenza, gli outliers vengono scartati. Infine viene calcolata la probabilità che un particolare insieme di features indichi la presenza di un oggetto, data l'accuratezza di fit e un numero di possibili false corrispondenze. Quelle che passano tutti questi test vengono identificate come corrette con alta confidenza<ref name="lowe04">Lowe, D. G., “Distinctive Image Features from Scale-Invariant Keypoints”, International Journal of Computer Vision, 60, 2, pp. 91-110, 2004.</ref>.

I punti chiave SIFT degli oggetti vengono prima estratti da un insieme di immagini di riferimento e immagazzinati in un database. Un oggetto viene riconosciuto in una nuova immagine confrontando individualmente ogni caratteristica della nuova immagine con quelle nel database precedentemente ottenuto basandosi sulla [[distanza euclidea]] dei loro vettori di caratteristiche. Dall'insieme completo di corrispondenze, nella nuova immagine vengono identificati sottoinsiemi di punti chiave che concordano con l'oggetto la sua posizione, la sua scala e il suo orientamento per filtrare le corrispondenze migliori. La determinazione di cluster consistenti viene effettuata in modo rapido usando un'implementazione con [[hash table]] efficiente della [[trasformata di Hough]] generalizzata. Ogni cluster di 3 o più feature concordanti su un oggetto e la sua posizione è soggetto a un'ulteriore verifica dettagliata del modello e, di conseguenza, gli outlier vengono scartati. Infine, viene calcolata la probabilità che un particolare insieme di caratteristiche indichi la presenza di un oggetto, data l'accuratezza di fit e un numero di possibili false corrispondenze. Quelle che passano tutti questi test vengono identificate come corrette con alta confidenza<ref name="lowe04">Lowe, D. G., “Distinctive Image Features from Scale-Invariant Keypoints”, International Journal of Computer Vision, 60, 2, pp. 91-110, 2004.</ref>.

<!--

{| class="wikitable"

|-

! Problem

! Technique

! Advantage

|-

| key localization / scale / rotation

| DoG / scale - space pyramid / orientation assignment

| accuracy, stability, scale & rotational invariance

|-

| geometric distortion

| blurring / resampling of local image orientation planes

| affine invariance

|-

| indexing and matching

| nearest neighbor / Best Bin First search

| Efficiency / speed

|-

| Cluster identification

| Hough Transform voting

| reliable pose models

|-

| Model verification / outlier detection

| Linear least squares

| better error tolerance with fewer matches

|-

| Hypothesis acceptance

| Bayesian Probability analysis

| reliability

|}

===Key stages===

====Scale-invariant feature detection====

Lowe's method for image feature generation transforms an image into a large collection of feature vectors, each of which is invariant to image translation, scaling, and rotation, partially invariant to illumination changes and robust to local geometric distortion. These features share similar properties with neurons in [[inferior temporal cortex]] that are used for object recognition in primate vision<ref>Serre, T., Kouh, M., Cadieu, C., Knoblich, U., Kreiman, G., Poggio, T., “[http://web.mit.edu/serre/www/publications/serre_etal-AIM-2005-036.pdf A Theory of Object Recognition: Computations and Circuits in the Feedforward Path of the Ventral Stream in Primate Visual Cortex]”, Computer Science and Artificial Intelligence Laboratory Technical Report, December 19, 2005 MIT-CSAIL-TR-2005-082.</ref>. Key locations are defined as maxima and minima of the result of [[difference of Gaussians]] function applied in [[scale-space]] to a series of smoothed and resampled images. Low contrast candidate points and edge response points along an edge are discarded. Dominant orientations are assigned to localized keypoints. These steps ensure that the keypoints are more stable for matching and recognition. SIFT descriptors robust to local affine distortion are then obtained by considering pixels around a radius of the key ___location, blurring and resampling of local image orientation planes.

====Feature matching and indexing====

Indexing consists of storing SIFT keys and identifying matching keys from the new image. Lowe used a modification of the [[k-d tree]] algorithm called the '''[[Best Bin First|Best-bin-first]] search''' method <ref>Beis, J., and Lowe, D.G “Shape indexing using approximate nearest-neighbour search in high-dimensional spaces”, Conference on Computer Vision and Pattern Recognition, Puerto Rico, 1997, pp. 1000–1006.</ref> that can identify the [[nearest neighbor]]s with high probability using only a limited amount of computation. The BBF algorithm uses a modified search ordering for the [[k-d tree]] algorithm so that bins in feature space are searched in the order of their closest distance from the query ___location. This search order requires the use of a [[heap (data structure)|heap]]-based [[priority queue]] for efficient determination of the search order. The best candidate match for each keypoint is found by identifying its nearest neighbor in the database of keypoints from training images. The nearest neighbors are defined as the keypoints with minimum [[Euclidean distance]] from the given descriptor vector. The probability that a match is correct can be determined by taking the ratio of distance from the closest neighbor to the distance of the second closest.

Lowe<ref name="lowe04" /> rejected all matches in which the distance ratio is greater than 0.8, which eliminates 90% of the false matches while discarding less than 5% of the correct matches. To further improve the efficiency of the best-bin-first algorithm search was cut off after checking the first 200 [[nearest neighbor]] candidates. For a database of 100,000 keypoints, this provides a speedup over exact [[nearest neighbor]] search by about 2 orders of magnitude, yet results in less than a 5% loss in the number of correct matches.

====Cluster identification by Hough transform voting====

[[Hough Transform]] is used to cluster reliable model hypotheses to search for keys that agree upon a particular model [[Pose (computer vision)|pose]]. Hough transform identifies clusters of features with a consistent interpretation by using each feature to vote for all object poses that are consistent with the feature. When clusters of features are found to vote for the same pose of an object, the probability of the interpretation being correct is much higher than for any single feature. An entry in a [[hash table]] is created predicting the model ___location, orientation, and scale from the match hypothesis.The [[hash table]] is searched to identify all clusters of at least 3 entries in a bin, and the bins are sorted into decreasing order of size.

Each of the SIFT keypoints specifies 2D ___location, scale, and orientation, and each matched keypoint in the database has a record of its parameters relative to the training image in which it was found. The similarity transform implied by these 4 parameters is only an approximation to the full 6 degree-of-freedom pose space for a 3D object and also does not account for any non-rigid deformations. Therefore, Lowe<ref name="lowe04" /> used broad bin sizes of 30 degrees for orientation, a factor of 2 for scale, and 0.25 times the maximum projected training image dimension (using the predicted scale) for ___location. The SIFT key samples generated at the larger scale are given twice the weight of those at the smaller scale. This means that the larger scale is in effect able to filter the most likely neighbours for checking at the smaller scale. This also improves recognition performance by giving more weight to the least-noisy scale. To avoid the problem of boundary effects in bin assignment, each keypoint match votes for the 2 closest bins in each dimension, giving a total of 16 entries for each hypothesis and further broadening the pose range.

====Model verification by linear least squares====

Each identified cluster is then subject to a verification procedure in which a [[linear least squares]] solution is performed for the parameters of the [[affine transformation]] relating the model to the image. The [[affine transformation]] of a model point [x y]^T to an image point [u v]^T can be written as below

:<math>

\begin{bmatrix} u \\ v \end{bmatrix} = \begin{bmatrix} m1 & m2 \\ m3 & m4 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} + \begin{bmatrix} tx \\ ty \end{bmatrix}

</math>

where the model translation is [tx ty]^T and the affine rotation, scale, and stretch are represented by the parameters m1, m2, m3 and m4. To solve for the transformation parameters the equation above can be rewritten to gather the unknowns into a column vector.

:<math>

\begin{bmatrix} x & y & 0 & 0 & 1 & 0 \\ 0 & 0 & x & y & 0 & 1 \\ ....\\ ....\end{bmatrix} \begin{bmatrix}m1 \\ m2 \\ m3 \\ m4 \\ tx \\ ty \end{bmatrix} = \begin{bmatrix} u \\ v \\ . \\ . \end{bmatrix}

</math>

This equation shows a single match, but any number of further matches can be added, with each match contributing two more rows to the first and last matrix. At least 3 matches are needed to provide a solution.

We can write this linear system as

:<math>A\hat{\mathbf{x}} \approx \mathbf{b},</math>

where ''A'' is a known ''m''-by-''n'' [[Matrix (mathematics)|matrix]] (usually with ''m'' > ''n''), '''x''' is an unknown ''n''-dimensional parameter [[vector space|vector]], and '''b''' is a known ''m''-dimensional measurement vector.

Therefore the minimizing vector <math>\hat{\mathbf{x}}</math> is a solution of the '''normal equation'''

:<math> A^T \! A \hat{\mathbf{x}} = A^T \mathbf{b}. </math>

The solution of the system of linear equations is given in terms of the matrix <math>(A^TA)^{-1}A^T</math> , called the [[Moore-Penrose pseudoinverse|pseudoinverse]] of ''A'', by

:<math> \hat{\mathbf{x}} = (A^T\!A)^{-1} A^T \mathbf{b}. </math>

which minimizes the sum of the squares of the distances from the projected model locations to the corresponding image locations.

====Outlier detection====

[[Outlier]]s can now be removed by checking for agreement between each image feature and the model, given the parameter solution. Given the [[linear least squares]] solution, each match is required to agree within half the error range that was used for the parameters in the [[Hough transform]] bins. As outliers are discarded, the [[linear least squares]] solution is re-solved with the remaining points, and the process iterated. If fewer than 3 points remain after discarding [[outlier]]s, then the match is rejected. In addition, a top-down matching phase is used to add any further matches that agree with the projected model position, which may have been missed from the [[Hough transform]] bin due to the similarity transform approximation or other errors.

The final decision to accept or reject a model hypothesis is based on a detailed probabilistic model<ref>Lowe, D.G., Local feature view clustering for 3D object recognition. IEEE Conference on Computer Vision and Pattern Recognition,Kauai, Hawaii, 2001, pp. 682-688.</ref>. This method first computes the expected number of false matches to the model pose, given the projected size of the model, the number of features within the region, and the accuracy of the fit. A [[Bayesian probability]] analysis then gives the probability that the object is present based on the actual number of matching features found. A model is accepted if the final probability for a correct interpretation is greater than 0.98. Lowe's SIFT based object recognition gives excellent results except under wide illumination variations and under non-rigid transformations.

===Competing methods for scale invariant object recognition under clutter / partial occlusion===

RIFT <ref>Lazebnik, S., Schmid, C., and Ponce, J., "[http://www-cvr.ai.uiuc.edu/ponce_grp/publication/paper/bmvc04.pdf Semi-Local Affine Parts for Object Recognition]", Proceedings of the British Machine Vision Conference, 2004.</ref> is a rotation-invariant generalization of SIFT. The RIFT descriptor is constructed using circular normalized patches divided into concentric rings of equal width and within each ring a gradient orientation histogram is computed. To maintain rotation invariance, the orientation is measured at each point relative to the direction pointing outward from the center.

G-RIF<ref>Sungho Kim, Kuk-Jin Yoon, In So Kweon, "Object Recognition Using a Generalized Robust Invariant Feature and Gestalt’s Law of Proximity and Similarity", Conference on Computer Vision and Pattern Recognition Workshop (CVPRW'06), 2006</ref> : Generalized Robust Invariant Feature is a general context descriptor which encodes edge orientation, edge density and hue information in a unified form combining perceptual information with spatial encoding. The object recognition scheme uses neighbouring context based voting to estimate object models.

"[[SURF]]<ref>Bay, H., Tuytelaars, T., Gool, L.V., "[http://www.vision.ee.ethz.ch/~surf/eccv06.pdf SURF: Speeded Up Robust Features]", Proceedings of the ninth European Conference on Computer Vision, May 2006.</ref> : Speeded Up Robust Features" is a high-performance scale and rotation-invariant interest point detector / descriptor claimed to approximate or even outperform previously proposed schemes with respect to repeatability, distinctiveness, and robustness. [[SURF]] relies on integral images for image convolutions to reduce computation time, builds on the strengths of the leading existing detectors and descriptors (using a fast [[Hessian matrix]]-based measure for the detector and a distribution-based descriptor). It describes a distribution of [[Haar wavelet]] responses within the interest point neighbourhood. Integral images are used for speed and only 64 dimensions are used reducing the time for feature computation and matching. The indexing step is based on the sign of the [[Laplacian]],which increases the matching speed and the robustness of the descriptor.

PCA-SIFT <ref>Ke, Y., and Sukthankar, R., "[http://www.cs.cmu.edu/~rahuls/pub/cvpr2004-keypoint-rahuls.pdf PCA-SIFT: A More Distinctive Representation for Local Image Descriptors]", Computer Vision and Pattern Recognition, 2004.</ref> and [[GLOH]] <ref>Mikolajczyk, K., and Schmid, C., "A performance evaluation of local descriptors", IEEE Transactions on Pattern Analysis and Machine Intelligence, 10, 27, pp 1615--1630, 2005.</ref> are variants of SIFT. PCA-SIFT descriptor is a vector of image gradients in x and y direction computed within the support region. The gradient region is sampled at 39x39 locations, therefore the vector is of dimension 3042. The dimension is reduced

to 36 with [[PCA]]. Gradient ___location-orientation histogram ([[GLOH]]) is an extension of the SIFT descriptor designed to increase its robustness and distinctiveness. The SIFT descriptor is computed for a log-polar ___location grid with three bins in radial direction (the radius set to 6, 11, and 15) and 8 in angular direction, which results in 17 ___location bins. The central bin is not divided in angular directions. The gradient orientations are quantized in 16 bins resulting in 272 bin histogram. The size of this descriptor is reduced with [[PCA]]. The [[covariance matrix]] for [[PCA]] is estimated on image patches collected from various images. The 128 largest [[eigenvector]]s are used for description.

Wagner et al. developed two object recognition algorithms especially designed with the limitations of current mobile phones in mind <ref>D. Wagner, G. Reitmayr, A. Mulloni, T. Drummond, and D. Schmalstieg, "[http://mi.eng.cam.ac.uk/~gr281/docs/WagnerIsmar08NFT.pdf Pose tracking from natural features on mobile phones]" Proceedings of the International Symposium on Mixed and Augmented Reality, 2008.</ref>. In contrast to the classic SIFT approach Wagner et al. use the FAST [[Corner detection|corner detector]] for feature detection. The algorithm also distinguishes between the off-line preparation phase where features are created at different scale levels and the on-line phase where features are only created at the current fixed scale level of the phone's camera image. In addition, features are created from a fixed patch size of 15x15 pixels and form a SIFT descriptor with only 36 dimensions. The approach has been further extended by integrating a [[Scalable Vocabulary Tree]] in the recognition pipeline <ref>N. Henze, T. Schinke, and S. Boll, "What is That? Object Recognition from Natural Features on a Mobile Phone" Proceedings of the Workshop on Mobile Interaction with the Real World, 2009.</ref>. This allows the efficient recognition of a larger number of objects on mobile phones. The approach is mainly restricted by the amount of available [[Random-access memory|RAM]].

== Features ==

The detection and description of local image features can help in object recognition. The SIFT features are local and based on the appearance of the object at particular interest points, and are invariant to image scale and rotation. They are also robust to changes in illumination, noise, and minor changes in viewpoint. In addition to these properties, they are highly distinctive, relatively easy to extract, allow for correct object identification with low probability of mismatch and are easy to match against a (large) database of local features. Object description by set of SIFT features is also robust to partial occlusion; as few as 3 SIFT features from an object are enough to compute its ___location and pose. Recognition can be performed in close-to-real time, at least for small databases and on modern computer hardware.{{Citation needed|date=August 2008}}

== Algorithm ==

=== Scale-space extrema detection ===

This is the stage where the interest points, which are called keypoints in the SIFT framework, are detected. For this, the image is convolved with Gaussian filters at different scales, and then the difference of successive Gaussian-blurred images are taken. Keypoints are then taken as maxima/minima of the [[Difference of Gaussians]] (DoG) that occur at multiple scales. Specifically, a DoG image <math>D \left( x, y, \sigma \right)</math> is given by

:<math>D \left( x, y, \sigma \right) = L \left( x, y, k_i\sigma \right) - L \left( x, y, k_j\sigma \right)</math>,

:where <math>L \left( x, y, k\sigma \right)</math> is the convolution of the original image <math>I \left( x, y \right)</math> with the [[Gaussian blur]] <math>G \left( x, y, k\sigma \right)</math> at scale <math>k\sigma</math>, i.e.,

:<math>L \left( x, y, k\sigma \right) = G \left( x, y, k\sigma \right) * I \left( x, y \right)</math>

Hence a DoG image between scales <math>k_i\sigma</math> and <math>k_j\sigma</math> is just the difference of the Gaussian-blurred images at scales <math>k_i\sigma</math> and <math>k_j\sigma</math>. For [[scale-space]] extrema detection in the SIFT algorithm, the image is first convolved with Gaussian-blurs at different scales. The convolved images are grouped by octave (an octave corresponds to doubling the value of <math>\sigma</math>), and the value of <math>k_i</math> is selected so that we obtain a fixed number of convolved images per octave. Then the Difference-of-Gaussian images are taken from adjacent Gaussian-blurred images per octave.

Once DoG images have been obtained, keypoints are identified as local minima/maxima of the DoG images across scales. This is done by comparing each pixel in the DoG images to its eight neighbors at the same scale and nine corresponding neighboring pixels in each of the neighboring scales. If the pixel value is the maximum or minimum among all compared pixels, it is selected as a candidate keypoint.

This keypoint detection step is a variation of one of the [[blob detection]] methods developed

by Lindeberg by detecting scale-space extrema of the scale normalized Laplacian,<ref name=Lindeberg1998>{{cite journal

| author = Lindeberg, Tony

| year = 1998

| title = Feature detection with automatic scale selection

| journal = International Journal of Computer Vision

| volume = 30

| issue = 2

| pages = 79–116

| doi = 10.1023/A:1008045108935

| url = http://www.nada.kth.se/cvap/abstracts/cvap198.html

}}</ref> that is detecting points that are local extrema with respect to both space and scale, in the discrete case by comparisons with the nearest 26 neighbours in a discretized scale-space volume. The difference of Gaussians operator can be seen as an approximation to the Laplacian, here expressed in a [[pyramid (image processing)|pyramid]] setting.

=== Keypoint localization ===

[[Image:Sift keypoints filtering.jpg|thumb|After scale space extrema are detected (their ___location being shown in the uppermost image) the SIFT algorithm discards low contrast keypoints (remaining points are shown in the middle image) and then filters out those located on edges. Resulting set of keypoints is shown on last image.]] Scale-space extrema detection produces too many keypoint candidates, some of which are unstable. The next step in the algorithm is to perform a detailed fit to the nearby data for accurate ___location, scale, and ratio of [[principal curvatures]]. This information allows points to be rejected that have low contrast (and are therefore sensitive to noise) or are poorly localized along an edge.

==== Interpolation of nearby data for accurate position ====

First, for each candidate keypoint, interpolation of nearby data is used to accurately determine its position. The initial approach was to just locate each keypoint at the ___location and scale of the candidate keypoint.<ref name=lowe99/> The new approach calculates the interpolated ___location of the extremum, which substantially improves matching and stability.<ref name=lowe04>{{cite journal

| author = Lowe, David G.

| year = 2004

| title = Distinctive Image Features from Scale-Invariant Keypoints

| journal = International Journal of Computer Vision

| volume = 60

| issue = 2

| pages = 91–110

| doi = 10.1023/B:VISI.0000029664.99615.94

| url = http://citeseer.ist.psu.edu/lowe04distinctive.html

}}</ref> The interpolation is done using the quadratic [[Taylor expansion]] of the Difference-of-Gaussian scale-space function, <math>D \left( x, y, \sigma \right)</math> with the candidate keypoint as the origin. This Taylor expansion is given by:

:<math>D(\textbf{x}) = D + \frac{\partial D^T}{\partial \textbf{x}}\textbf{x} + \frac{1}{2}\textbf{x}^T \frac{\partial^2 D}{\partial \textbf{x}^2} \textbf{x}</math>

where D and its derivatives are evaluated at the candidate keypoint and <math>\textbf{x} = \left( x, y, \sigma \right)</math> is the offset

from this point. The ___location of the extremum, <math>\hat{\textbf{x}}</math>, is determined by taking the derivative of this function with respect to <math>\textbf{x}</math> and setting it to zero. If the offset <math>\hat{\textbf{x}}</math> is larger than <math>0.5</math> in any dimension, then that's an indication that the extremum lies closer to another candidate keypoint. In this case, the candidate keypoint is changed and

the interpolation performed instead about that point. Otherwise the offset is added to its candidate keypoint to get the interpolated estimate for the ___location of the extremum.

A similar subpixel determation of the locations of scale-space extrema is performed

in the real-time implementation based on hybrid pyramids

developed by Lindeberg and his co-workers<ref>

{{cite journal

| author = Lindeberg, Tony and Bretzner, Lars

| year = 2003

| title = Real-time scale selection in hybrid multi-scale representations

| journal = Proc. Scale-Space'03, Springer Lecture Notes in Computer Science

| volume = 2695

| pages = 148–163

| doi = 10.1007/3-540-44935-3_11

| isbn = 978-3-540-40368-5

| url = http://www.nada.kth.se/cvap/abstracts/cvap279.html

}}</ref>

==== Discarding low-contrast keypoints ====

To discard the keypoints with low contrast, the value of the second-order Taylor expansion <math>D(\textbf{x})</math> is computed at the offset <math>\hat{\textbf{x}}</math>. If this value is less than <math>0.03</math>, the candidate keypoint is discarded. Otherwise it is kept, with final ___location <math>\textbf{y} + \hat{\textbf{x}}</math> and scale <math>\sigma</math>, where <math>\textbf{y}</math> is the original ___location of the keypoint at scale <math>\sigma</math>.

==== Eliminating edge responses ====

The DoG function will have strong responses along edges, even if the candidate keypoint is not robust to small amounts of noise. Therefore, in order to increase stability, we need to eliminate the keypoints that have poorly determined locations but have high edge responses.

For poorly defined peaks in the DoG function, the [[principal curvature]] across the edge would be much larger than the principal curvature along it. Finding these principal curvatures amounts to solving for the [[eigenvalues]] of the second-order [[Hessian matrix]], '''H''':

:<math> \textbf{H} = \begin{bmatrix}

D_{xx} & D_{xy} \\

D_{xy} & D_{yy}

\end{bmatrix} </math>

The eigenvalues of '''H''' are proportional to the principal curvatures of D. It turns out that the ratio of the two eigenvalues, say <math>\alpha</math> is the larger one, and <math>\beta</math> the smaller one, with ratio <math>r = \alpha/\beta</math>, is sufficient for SIFT's purposes. The trace of '''H''', i.e., <math>D_{xx} + D_{yy}</math>, gives us the sum of the two eigenvalues, while its determinant, i.e., <math>D_{xx} D_{yy} - D_{xy}^2</math>, yields the product. The ratio <math> \text{R} = \operatorname{Tr} \left( \textbf{H} \right)^2/\operatorname{Det} \left( \textbf{H} \right)</math> can be shown to be equal to <math>\left( r+1 \right)^2/r</math>, which depends only on the ratio of the eigenvalues rather than their individual values. R is minimum when the eigenvalues are equal to each other. Therefore the higher the [[absolute difference]] between the two eigenvalues, which is equivalent to a higher absolute difference between the two principal curvatures of D, the higher the value of R. It follows that, for some threshold eigenvalue ratio <math>r_{\text{th}}</math>, if R for a candidate keypoint is larger than <math>\left( r_{\text{th}} + 1 \right)^2/r_{\text{th}}</math>, that keypoint is poorly localized and hence rejected. The new approach uses <math>r_{\text{th}} = 10</math>.<ref name=lowe04/>

This processing step for suppressing responses at edges is a transfer of a corresponding approach in the Harris operator for [[corner detection]]. The difference is that the measure for thresholding is computed from the Hessian matrix instead of a second-moment matrix (see [[structure tensor]]).

=== Orientation assignment ===

In this step, each keypoint is assigned one or more orientations based on local image gradient directions. This is the key step in achieving [[rotational invariance|invariance to rotation]] as the keypoint descriptor can be represented relative to this orientation and therefore achieve invariance to image rotation.

First, the Gaussian-smoothed image <math>L \left( x, y, \sigma \right)</math> at the keypoint's scale <math>\sigma</math> is taken so that all computations are performed in a scale-invariant manner. For an image sample <math>L \left( x, y \right)</math> at scale <math>\sigma</math>, the gradient magnitude, <math>m \left( x, y \right)</math>, and orientation, <math>\theta \left( x, y \right)</math>, are precomputed using pixel

differences:

:<math>m \left( x, y \right) = \sqrt{\left( L \left( x+1, y \right) - L \left( x-1, y \right) \right)^2 + \left( L \left( x, y+1 \right) - L \left( x, y-1 \right) \right)^2}</math>

:<math>\theta \left( x, y \right) = \tan^{-1}\left(\frac{L \left( x, y+1 \right) - L \left( x, y-1 \right)}{L \left( x+1, y \right) - L \left( x-1, y \right)} \right)</math>

The magnitude and direction calculations for the gradient are done for every pixel in a neighboring region around the keypoint in the Gaussian-blurred image L. An orientation histogram with 36 bins is formed, with each bin covering 10 degrees. Each sample in the neighboring window added to a histogram bin is weighted by its gradient magnitude and by a Gaussian-weighted circular window with a <math>\sigma</math> that is 1.5 times that of the scale of the keypoint. The peaks in this histogram correspond to dominant orientations. Once the histogram is filled, the orientations corresponding to the highest peak and local peaks that are within 80% of the highest peaks are assigned to the keypoint. In the case of multiple orientations being assigned, an additional keypoint is created having the same ___location and scale as the original keypoint for each additional orientation.

=== Keypoint descriptor ===

Previous steps found keypoint locations at particular scales and assigned orientations to them. This ensured invariance to image ___location, scale and rotation. Now we want to compute a descriptor vector for each keypoint such that the descriptor is highly distinctive and partially invariant to the remaining variations such as illumination, 3D viewpoint, etc. This step is performed on the image closest in scale to the keypoint's scale.

First a set of orientation histograms are created on 4x4 pixel neighborhoods with 8 bins each. These histograms are computed from magnitude and orientation values of samples in a 16 x 16 region around the keypoint such that each histogram contains samples from a 4 x 4 subregion of the original neighborhood region. The magnitudes are further weighted by a Gaussian function with <math>\sigma</math> equal to one half the width of the descriptor window. The descriptor then becomes a vector of all the values of these histograms. Since there are 4 x 4 = 16 histograms each with 8 bins the vector has 128 elements. This vector is then normalized to unit length in order to enhance invariance to affine changes in illumination. To reduce the effects of non-linear illumination a threshold of 0.2 is applied and the vector is again normalized.

Although the dimension of the descriptor, i.e. 128, seems high descriptors with lower dimension than this don't perform as well across the range of matching tasks<ref name=lowe04/> and the computational cost remains low due to the approximate BBF (see below) method used for finding the nearest-neighbor. Longer descriptors continue to do better but not by much and there is an additional danger of increased sensitivity to distortion and occlusion. It is also shown that feature matching accuracy is above 50% for viewpoint changes of up to 50 degrees. Therefore SIFT descriptors are invariant to minor affine changes. To test the distinctiveness of the SIFT descriptors, matching accuracy is also measured against varying number of keypoints in the testing database, and it is shown that matching accuracy decreases only very slightly for very large database sizes, thus indicating that SIFT features are highly distinctive.

==Comparison of SIFT features with other local features==

There has been an extensive study done on the performance evaluation of different local descriptors, including SIFT, using a range of detectors.<ref name=Mikolajczyk2005>{{cite journal

| author = Mikolajczyk, K.; Schmid, C.

| year = 2005

| title = A performance evaluation of local descriptors

| journal = IEEE Transactions on Pattern Analysis and Machine Intelligence

| volume = 27

| pages = 1615–1630

| doi = 10.1109/TPAMI.2005.188

| url = http://research.microsoft.com/users/manik/projects/trade-off/papers/MikolajczykPAMI05.pdf

| pmid = 16237996

| issue = 10

}}</ref> The main results are summarized below:

* SIFT and SIFT-like [[GLOH]] features exhibit the highest matching accuracies (recall rates) for an affine transformation of 50 degrees. After this transformation limit, results start to become unreliable.

* Distinctiveness of descriptors is measured by summing the eigenvalues of the descriptors, obtained by the [[Principal components analysis]] of the descriptors normalized by their variance. This corresponds to the amount of variance captured by different descriptors, therefore, to their distinctiveness. PCA-SIFT (Principal Components Analysis applied to SIFT descriptors), GLOH and SIFT features give the highest values.

* SIFT-based descriptors outperform other local descriptors on both textured and structured scenes, with the difference in performance larger on the textured scene.

* For scale changes in the range 2-2.5 and image rotations in the range 30 to 45 degrees, SIFT and SIFT-based descriptors again outperform other local descriptors with both textured and structured scene content.

* Performance for all local descriptors degraded on images introduced with a significant amount of blur, with the descriptors that are based on edges, like [[shape context]], performing increasingly poorly with increasing amount blur. This is because edges disappear in the case of a strong blur. But GLOH, PCA-SIFT and SIFT still performed better than the others. This is also true for evaluation in the case of illumination changes.

The evaluations carried out suggests strongly that SIFT-based descriptors, which are region-based, are the most robust and distinctive, and are therefore best suited for feature matching. However, most recent feature descriptors such as [[SURF]] have not been evaluated in this study.

[[SURF]] has later been shown to have similar performance to [[SIFT]], while at the same time being much faster.<ref>http://www.tu-chemnitz.de/etit/proaut/rsrc/iav07-surf.pdf</ref>

Recently, a slight variation of the descriptor employing an irregular histogram grid has been proposed that significantly improves its performance<ref name=irrgrid>{{cite conference

| first = Y.

| last = Cui

| authorlink =

| coauthors = Hasler, N.; Thormaehlen, T.; Seidel, H.-P.

| title = Scale Invariant Feature Transform with Irregular Orientation Histogram Binning

| booktitle = Proceedings of the International Conference on Image Analysis and Recognition (ICIAR 2009)

| pages =

| publisher = Springer

| date = July 2009

| ___location = Halifax, Canada

| url = http://www.mpi-inf.mpg.de/~hasler/download/CuiHasThoSei09igSIFT.pdf

| doi =

| id =

| accessdate = }}</ref>. Instead of using a 4x4 grid of histogram bins, all bins extend to the center of the feature. This improves the descriptor's robustness to scale changes.

The SIFT-Rank technique was shown to improve the performance of the standard SIFT descriptor for affine feature matching<ref name=ToewsCVPR:10>{{cite conference

| author = Matthew Toews, William M. Wells III

| year = 2009

| title = SIFT-Rank: Ordinal Descriptors for Invariant Feature Correspondence

| booktitle = IEEE International Conference on Computer Vision and Pattern Recognition

| pages = 172–177

| doi = 10.1109/CVPR.2009.5206849

| url = http://www.cim.mcgill.ca/~mtoews/papers/cvpr09-matt.final.pdf

}}</ref>. A SIFT-Rank descriptor is generated from a standard SIFT descriptor, by setting each histogram bin to its rank in a sorted array of bins. The Euclidean distance between SIFT-Rank descriptors is invariant to arbitrary monotonic changes in histogram bin values, and is related to [[Spearman's rank correlation coefficient]].

==Applications==

===Object recognition using SIFT features===

Given SIFT's ability to find distinctive keypoints that are invariant to ___location, scale and rotation, and robust to [[affine transformations]] (changes in [[scale]], [[rotation]], [[Shear mapping|shear]], and position) and changes in illumination, they are usable for object recognition. The steps are given below.

* First, SIFT features are obtained from the input image using the algorithm described above.

* These features are matched to the SIFT feature database obtained from the training images. This feature matching is done through a Euclidean-distance based nearest neighbor approach. To increase robustness, matches are rejected for those keypoints for which the ratio of the nearest neighbor distance to the second nearest neighbor distance is greater than 0.8. This discards many of the false matches arising from background clutter. Finally, to avoid the expensive search required for finding the Euclidean-distance-based nearest neighbor, an approximate algorithm called the best-bin-first algorithm is used.<ref name=Beis1997>{{cite conference

| author = Beis, J.

| coauthors=Lowe, David G.

| year = 1997

| title = Shape indexing using approximate nearest-neighbour search in high-dimensional spaces

| booktitle = Conference on Computer Vision and Pattern Recognition, Puerto Rico: sn

| pages = 1000–1006

| doi = 10.1109/CVPR.1997.609451

| url = http://www.cs.ubc.ca/~lowe/papers/cvpr97.pdf

}}</ref> This is a fast method for returning the nearest neighbor with high probability, and can give speedup by factor of 1000 while finding nearest neighbor (of interest) 95% of the time.

* Although the distance ratio test described above discards many of the false matches arising from background clutter, we still have matches that belong to different objects. Therefore to increase robustness to object identification, we want to cluster those features that belong to the same object and reject the matches that are left out in the clustering process. This is done using the [[Hough transform]]. This will identify clusters of features that vote for the same object pose. When clusters of features are found to vote for the same pose of an object, the probability of the interpretation being correct is much higher than for any single feature. Each keypoint votes for the set of object poses that are consistent with the keypoint's ___location, scale, and orientation. ''Bins'' that accumulate at least 3 votes are identified as candidate object/pose matches.

* For each candidate cluster, a least-squares solution for the best estimated affine projection parameters relating the training image to the input image is obtained. If the projection of a keypoint through these parameters lies within half the error range that was used for the parameters in the Hough transform bins, the keypoint match is kept. If fewer than 3 points remain after discarding outliers for a bin, then the object match is rejected. The least-squares fitting is repeated until no more rejections take place. This works better for planar surface recognition than 3D object recognition since the affine model is no longer accurate for 3D objects.

SIFT features can essentially be applied to any task that requires identification of matching locations between images. Work has been done on applications such as recognition of particular object categories in 2D images, 3D reconstruction,

motion tracking and segmentation, robot localization, image panorama stitching and epipolar calibration. Some of these are discussed in more detail below.

===Robot localization and mapping===

In this application,<ref name=Se2001>{{cite conference

| author = Se, S.; Lowe, David G.; Little, J.

| year = 2001

| booktitle = Proceedings of the IEEE International Conference on Robotics and Automation (ICRA)

| title = Vision-based mobile robot localization and mapping using scale-invariant features

| volume = 2

| pages = 2051

| doi = 10.1109/ROBOT.2001.932909

| url = http://citeseer.ist.psu.edu/425735.html

}}</ref> a trinocular stereo system is used to determine 3D estimates for keypoint locations. Keypoints are used only when they appear in all 3 images with consistent disparities, resulting in very few outliers. As the robot moves, it localizes itself using feature matches to the existing 3D map, and then incrementally adds features to the map while updating their 3D positions using a Kalman filter. This provides a robust and accurate solution to the problem of robot localization in unknown environments.

===Panorama stitching===

SIFT feature matching can be used in [[image stitching]] for fully automated [[panorama]] reconstruction from non-panoramic images. The SIFT features extracted from the input images are matched against each other to find ''k'' nearest-neighbors for each feature. These correspondences are then used to find ''m'' candidate matching images for each image. [[homography|Homographies]] between pairs of images are then computed using [[RANSAC]] and a probabilistic model is used for verification. Because there is no restriction on the input images, graph search is applied to find connected components of image matches such that each connected component will correspond to a panorama. Finally for each connected component [[Bundle adjustment]] is performed to solve for joint camera parameters, and the panorama is rendered using [[multi-band blending]]. Because of the SIFT-inspired object recognition approach to panorama stitching, the resulting system is insensitive to the ordering, orientation, scale and illumination of the images. The input images can contain multiple panoramas and noise images (some of which may not even be part of the composite image), and panoramic sequences are recognized and rendered as output.<ref name=Brown2003>{{cite conference

| author = Brown, M.; Lowe, David G.

| year = 2003

| title = Recognising Panoramas

| booktitle = Proceedings of the ninth IEEE International Conference on Computer Vision

| pages = 1218–1225

| volume = 2

| doi = 10.1109/ICCV.2003.1238630

| url = http://graphics.cs.cmu.edu/courses/15-463/2005_fall/www/Papers/BrownLowe.pdf

}}</ref>

===3D scene modeling, recognition and tracking===

This application uses SIFT features for [[3D object recognition]] and [[3D modeling]] in context of [[augmented reality]], in which synthetic objects with accurate pose are superimposed on real images. SIFT matching is done for a number of 2D images of a scene or object taken from different angles. This is used with [[bundle adjustment]] to build a sparse 3D model of the viewed scene and to simultaneously recover camera poses and calibration parameters. Then the position, orientation and size of the virtual object are defined relative to the coordinate frame of the recovered model. For online [[match moving]], SIFT features again are extracted from the current video frame and matched to the features already computed for the world mode, resulting in a set of 2D-to-3D correspondences. These correspondences are then used to compute the current camera pose for the virtual projection and final rendering. A regularization technique is used to reduce the jitter in the virtual projection.<ref>Iryna Gordon and David G. Lowe, "[http://www.cs.ubc.ca/labs/lci/papers/docs2006/lowe_gordon.pdf What and where: 3D object recognition with accurate pose]," in Toward Category-Level Object Recognition, (Springer-Verlag, 2006), pp. 67-82</ref>

===3D SIFT descriptors for human action recognition===

Extensions of the SIFT descriptor to 2+1-dimensional spatio-temporal data in context of human action recognition in video sequences have been studied by

<ref name=LapLin04>{{cite conference

| author = Laptev, Ivan and Lindeberg, Tony

| year = 2004

| title = Local descriptors for spatio-temporal recognition

| booktitle = ECCV'04 Workshop on Spatial Coherence for Visual Motion Analysis, Springer Lecture Notes in Computer Science, Volume 3667

| pages = 91–103

| doi = 10.1007/11676959_8

| url = ftp://ftp.nada.kth.se/CVAP/reports/LapLin04-SCVMA.pdf

}}</ref>

<ref name=Lap>{{cite journal

| author = Ivan Laptev, Barbara Caputo, Christian Schuldt and Tony Lindeberg

| year = 2007

| title = Local velocity-adapted motion events for spatio-temporal recognition

| journal = Computer Vision and Image Understanding

| volume = 108

| pages = 207–229

| doi = 10.1016/j.cviu.2006.11.023

| url = http://www.csc.kth.se/cvap/abstracts/LapCapSchLin07-CVIU.html

}}</ref><ref name=Scovanner2007>{{cite conference

| author = Scovanner, Paul

| coauthors = Ali, S; Shah, M

| year = 2007

| title = A 3-dimensional sift descriptor and its application to action recognition

| booktitle = Proceedings of the 15th International Conference on Multimedia

| pages = 357–360

| doi = 10.1145/1291233.1291311

}}</ref>. The computation of local position-dependent histograms in the 2D SIFT algorithm are extended from two to three dimensions to describe SIFT features in a spatio-temporal ___domain. For application to human action recognition in a video sequence, sampling of the training videos is carried out either at spatio-temporal interest points or at randomly determined locations, times and scales. The spatio-temporal regions around these interest points are then described using the 3D SIFT descriptor. These descriptors are then clustered to form a spatio-temporal [[Bag of words model]]. 3D SIFT descriptors extracted from the test videos are then matched against these ''words'' for human action classification.

The authors report much better results with their 3D SIFT descriptor approach than with other approaches like simple 2D SIFT descriptors and Gradient Magnitude.<ref>{{cite conference|author=Niebles, J. C. Wang, H. and Li, Fei-Fei|url=http://vision.cs.princeton.edu/niebles/humanactions.htm |title=Unsupervised Learning of Human Action Categories Using Spatial-Temporal Words |booktitle=Proceedings of the British Machine Vision Conference (BMVC)|___location=Edinburgh|year=2006|accessdate=2008-08-20}}</ref>

===Analyzing the Human Brain in 3D Magnetic Resonance Images===

The Feature-based Morphometry (FBM) technique<ref name=ToewsNI:10>{{cite journal

| author = Matthew Toews, William M. Wells III, D. Louis Collins, Tal Arbel

| year = 2010

| title = Feature-based Morphometry: Discovering Group-related Anatomical Patterns

| journal = NeuroImage

| volume = 49

| pages = 2318–2327

| doi = 10.1016/j.neuroimage.2009.10.032

| url = http://www.cim.mcgill.ca/~mtoews/papers/matt_neuroimage09.pdf

| pmid = 19853047

| issue = 3

}}</ref> uses extrema in a difference of Gaussian scale-space to analyze and classify 3D magnetic resonance images (MRIs) of the human brain. FBM models the image probabilistically as a collage of independent features, conditional on image geometry and group labels, e.g. healthy subjects and subjects with Alzheimer's disease (AD). Features are first extracted in individual images from a 4D difference of Gaussian scale-space, then modeled in terms of their appearance, geometry and group co-occurrence statistics across a set of images. FBM was validated in the analysis of AD using a set of ~200 volumetric MRIs of the human brain, automatically identifying established indicators of AD in the brain and classifying mild AD in new images with a rate of 80%<ref name=ToewsNI:10 />.

== See also ==

*[[3D single object recognition]]

*[[autostitch]]

*[[Feature detection (computer vision)]]

*[[SURF]] (Speeded Up Robust Features)

*[[scale-space]]

*[[scale-space implementation]]-->

== Note ==

<references/>

<!--

==External links==

* [http://web.engr.oregonstate.edu/~hess/#%5B%5BSIFT%20Feature%20Detector%5D%5D Rob Hess's implementation of SIFT] accessed 20 Mar 2010

* [http://www.jprr.org/index.php/jprr/article/view/26 The Invariant Relations of 3D to 2D Projection of Point Sets, Journal of Pattern Recognition Research][http://www.jprr.org (JPRR)], Vol. 3, No 1, 2008.

* [http://citeseer.ist.psu.edu/lowe04distinctive.html Lowe, D. G., “Distinctive Image Features from Scale-Invariant Keypoints”, International Journal of Computer Vision, 60, 2, pp. 91-110, 2004.]

* [http://lear.inrialpes.fr/pubs/2005/MS05/ Mikolajczyk, K., and Schmid, C., "A performance evaluation of local descriptors", IEEE Transactions on Pattern Analysis and Machine Intelligence, 10, 27, pp 1615--1630, 2005.]

* [http://www.cs.cmu.edu/~yke/pcasift/ PCA-SIFT: A More Distinctive Representation for Local Image Descriptors]

* [http://www-cvr.ai.uiuc.edu/ponce_grp/publication/paper/bmvc04.pdf Lazebnik, S., Schmid, C., and Ponce, J., Semi-Local Affine Parts for Object Recognition, BMVC, 2004. ]

* [http://www.ipol.im/pub/algo/my_affine_sift/ ASIFT (Affine SIFT)]: large viewpoint matching with SIFT, with source code and online demonstration

* [http://www.vlfeat.org/ VLFeat], an open source computer vision library in C (with a MEX interface to MATLAB), including an implementation of SIFT

* [http://www.cs.cityu.edu.hk/~wzhao2/lip-vireo.htm LIP-VIREO], A toolkit for keypoint feature extraction (binaries for Windows, Linux and SunOS), including an implementation of SIFT

* [https://sites.google.com/site/btabibian/projects/3d-reconstruction/code (Parallel) SIFT in C#],SIFT algorithm in C# using Emgu CV and also a modified parallel version of the algorithm.

-->

== Voci correlate ==

* [[Speeded_Up_Robust_FeatureSpeeded Up Robust Feature|SURF]]

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