Choi–Williams distribution function: Difference between revisions

Revision as of 21:08, 20 April 2013

Shape context is a feature descriptor used in object recognition. Serge Belongie and Jitendra Malik proposed the term in their paper "Matching with Shape Contexts" in 2000.^[1]

Theory

The shape context is intended to be a way of describing shapes that allows for measuring shape similarity and the recovering of point correspondences.^[1] The basic idea is to pick n points on the contours of a shape. For each point p_i on the shape, consider the n − 1 vectors obtained by connecting p_i to all other points. The set of all these vectors is a rich description of the shape localized at that point but is far too detailed. The key idea is that the distribution over relative positions is a robust, compact, and highly discriminative descriptor. So, for the point p_i, the coarse histogram of the relative coordinates of the remaining n − 1 points,

h_{i} (k) = # {q \neq p_{i} : (q - p_{i}) \in bin (k)}

is defined to be the shape context of $p_{i}$ . The bins are normally taken to be uniform in log-polar space. The fact that the shape context is a rich and discriminative descriptor can be seen in the figure below, in which the shape contexts of two different versions of the letter "A" are shown.

(a) and (b) are the sampled edge points of the two shapes. (c) is the diagram of the log-polar bins used to compute the shape context. (d) is the shape context for the circle, (e) is that for the diamond, and (f) is that for the triangle. As can be seen, since (d) and (e) are the shape contexts for two closely related points, they are quite similar, while the shape context in (f) is very different.

Now in order for a feature descriptor to be useful, it needs to have certain invariances. In particular it needs to be invariant to translation, scale, small perturbations, and depending on application rotation. Translational invariance come naturally to shape context. Scale invariance is obtained by normalizing all radial distances by the mean distance $α$ between all the point pairs in the shape ^[2]^[3] although the median distance can also be used.^[1]^[4] Shape contexts are empirically demonstrated to be robust to deformations, noise, and outliers^[4] using synthetic point set matching experiments.^[5]

One can provide complete rotation invariance in shape contexts. One way is to measure angles at each point relative to the direction of the tangent at that point (since the points are chosen on edges). This results in a completely rotationally invariant descriptor. But of course this is not always desired since some local features lose their discriminative power if not measured relative to the same frame. Many applications in fact forbid rotation invariance e.g. distinguishing a "6" from a "9".

Use in shape matching

A complete system that uses shape contexts for shape matching consists of the following steps (which will be covered in more detail in the Details of Implementation section):

Randomly select a set of points that lie on the edges of a known shape and another set of points on an unknown shape.
Compute the shape context of each point found in step 1.
Match each point from the known shape to a point on an unknown shape. To minimize the cost of matching, first choose a transformation (e.g. affine, thin plate spline, etc.) that warps the edges of the known shape to the unknown (essentially aligning the two shapes). Then select the point on the unknown shape that most closely corresponds to each warped point on the known shape.
Calculate the "shape distance" between each pair of points on the two shapes. Use a weighted sum of the shape context distance, the image appearance distance, and the bending energy (a measure of how much transformation is required to bring the two shapes into alignment).
To identify the unknown shape, use a nearest-neighbor classifier to compare its shape distance to shape distances of known objects.

Details of implementation

Step 1: Finding a list of points on shape edges

The approach assumes that the shape of an object is essentially captured by a finite subset of the points on the internal or external contours on the object. These can be simply obtained using the Canny edge detector and picking a random set of points from the edges. Note that these points need not and in general do not correspond to key-points such as maxima of curvature or inflection points. It is preferable to sample the shape with roughly uniform spacing, though it is not critical.^[2]

Step 2: Computing the shape context

This step is described in detail in the Theory section.

Step 3: Computing the cost matrix

Consider two points p and q that have normalized K-bin histograms (i.e. shape contexts) g(k) and h(k). As shape contexts are distributions represented as histograms, it is natural to use the χ² test statistic as the "shape context cost" of matching the two points:

C_{S} = \frac{1}{2} \sum_{k = 1}^{K} \frac{[g (k) - h (k)]^{2}}{g (k) + h (k)}

The values of this range from 0 to 1.^[1] In addition to the shape context cost, an extra cost based on the appearance can be added. For instance, it could be a measure of tangent angle dissimilarity (particularly useful in digit recognition):

C_{A} = \frac{1}{2} ‖ \begin{matrix} (\binom{\cos (θ_{1})}{\sin (θ_{1})}) - (\binom{\cos (θ_{2})}{\sin (θ_{2})}) \end{matrix} ‖

This is half the length of the chord in unit circle between the unit vectors with angles $θ_{1}$ and $θ_{2}$ . Its values also range from 0 to 1. Now the total cost of matching the two points could be a weighted-sum of the two costs:

C = (1 - β) C_{S} + β C_{A}

Now for each point p_i on the first shape and a point q_j on the second shape, calculate the cost as described and call it C_i,j. This is the cost matrix.

Step 4: Finding the matching that minimizes total cost

Now, a one-to-one matching p_i that matches each point p_i on shape 1 and q_j on shape 2 that minimizes the total cost of matching,

H (π) = \sum_{i} C (p_{i}, q_{π (i)})

is needed. This can be done in $O (N^{3})$ time using the Hungarian method, although there are more efficient algorithms.^[6] To have robust handling of outliers, one can add "dummy" nodes that have a constant but reasonably large cost of matching to the cost matrix. This would cause the matching algorithm to match outliers to a "dummy" if there is no real match.

Step 5: Modeling transformation

Given the set of correspondences between a finite set of points on the two shapes, a transformation $T : ℝ^{2} \to ℝ^{2}$ can be estimated to map any point from one shape to the other. There are several choices for this transformation, described below.

Affine

The affine model is a standard choice: $T (p) = A p + o$ . The least squares solution for the matrix $A$ and the translational offset vector o is obtained by:

o = \frac{1}{n} \sum_{i = 1}^{n} (p_{i} - q_{π (i)}), A = (Q^{+} P)^{t}

Where $P = (\begin{matrix} 1 & p_{11} & p_{12} \\ ⋮ & ⋮ & ⋮ \\ 1 & p_{n 1} & p_{n 2} \end{matrix})$ with a similar expression for $Q$ . $Q^{+}$ is the pseudoinverse of $Q$ .

Thin plate spline

The thin plate spline (TPS) model is the most widely used model for transformations when working with shape contexts. A 2D transformation can be separated into two TPS function to model a coordinate transform:

T (x, y) = (f_{x} (x, y), f_{y} (x, y))

where each of the ƒ_x and ƒ_y have the form:

f (x, y) = a_{1} + a_{x} x + a_{y} y + \sum_{i = 1}^{n} ω_{i} U (‖ \begin{matrix} (x_{i}, y_{i}) - (x, y) \end{matrix} ‖),

and the kernel function $U (r)$ is defined by $U (r) = r^{2} \log r^{2}$ . The exact details of how to solve for the parameters can be found elsewhere^[7]^[8] but it essentially involves solving a linear system of equations. The bending energy (a measure of how much transformation is needed to align the points) will also be easily obtained.

Regularized TPS

The TPS formulation above has exact matching requirement for the pairs of points on the two shapes. For noisy data, it is best to relax this exact requirement. If we let $v_{i}$ denote the target function values at corresponding locations $p_{i} = (x_{i}, y_{i})$ (Note that for $f_{x}$ , $v_{i}$ would $x^{'}$ the x-coordinate of the point corresponding to $p_{i}$ and for $f_{y}$ it would be the y-coordinate, $y^{'}$ ), relaxing the requirement amounts to minimizing

H [f] = \sum_{i = 1}^{n} (v_{i} - f (x_{i}, y_{i}))^{2} + λ I_{f}

where $I_{f}$ is the bending energy and $λ$ is called the regularization parameter. This ƒ that minimizes H[ƒ] can be found in a fairly straightforward way.^[9] If one uses normalize coordinates for $(x_{i}, y_{i}) and (x'_{i}, y'_{i})$ , then scale invariance is kept. However, if one uses the original non-normalized coordinates, then the regularization parameter needs to be normalized.

Note that in many cases, regardless of the transformation used, the initial estimate of the correspondences contains some errors which could reduce the quality of the transformation. If we iterate the steps of finding correspondences and estimating transformations (i.e. repeating steps 2–5 with the newly transformed shape) we can overcome this problem. Typically, three iterations are all that is needed to obtain reasonable results.

Step 6: Computing the shape distance

Now, a shape distance between two shapes $P$ and $Q$ . This distance is going to be a weighted sum of three potential terms:

Shape context distance: this is the symmetric sum of shape context matching costs over best matching points:

D_{s c} (P, Q) = \frac{1}{n} \sum_{p \in P} \arg min_{q \in Q} C (p, T (q)) + \frac{1}{m} \sum_{q \in Q} \arg min_{p \in P} C (p, T (q))

where T(·) is the estimated TPS transform that maps the points in Q to those in P.

Appearance cost: After establishing image correspondences and properly warping one image to match the other, one can define an appearance cost as the sum of squared brightness differences in Gaussian windows around corresponding image points:

D_{a c} (P, Q) = \frac{1}{n} \sum_{i = 1}^{n} \sum_{Δ \in Z^{2}} G (Δ) {[I_{P} (p_{i} + Δ) - I_{Q} (T (q_{π (i)}) + Δ)]}^{2}

where $I_{P}$ and $I_{Q}$ are the gray-level images ( $I_{Q}$ is the image after warping) and $G$ is a Gaussian windowing function.

Transformation cost: The final cost $D_{b e} (P, Q)$ measures how much transformation is necessary to bring the two images into alignment. In the case of TPS, it is assigned to be the bending energy.

Now that we have a way of calculating the distance between two shapes, we can use a nearest neighbor classifier (k-NN) with distance defined as the shape distance calculated here. The results of applying this to different situations is given in the following section.

Results

Digit recognition

Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church. The authors Serge Belongie and Jitendra Malik tested their approach on the MNIST database. Currently, more than 50 algorithms have been tested on the database. The database has a training set of 60,000 examples, and a test set of 10,000 examples. The error rate for this approach was 0.63% using 20,000 training examples and 3-NN. At the time of publication, this error rate was the lowest. Currently, the lowest error rate is 0.23%.

Silhouette similarity-based retrieval

The authors experimented with the MPEG-7 shape silhouette database, performing Core Experiment CE-Shape-1 part B, which measures performance of similarity-based retrieval.^[10] The database has 70 shape categories and 20 images per shape category. Performance of a retrieval scheme is tested by using each image as a query and counting the number of correct images in the top 40 matches. For this experiment, the authors increased the amount of points sampled from each shape. Also, since the shapes in the database sometimes were rotated or flipped, the authors took defined the distance between a reference shape and query shape to be minimum shape distance between the query shape and either the unchanged reference, the vertically flipped, or the reference horizontally flipped.^[1]^[2]^[3]^[4] With these changes, they obtained a retrieval rate of 76.45%, which by 2002 was the best.

3D object recognition

The next experiment performed on shape contexts involved the 20 common household objects in the Columbia Object Image Library (COIL-20). Each object has 72 views in the database. In the experiment, the method was trained on a number of equally spaced views for each object and the remaining views were used for testing. A 1-NN classifier was used. The authors also developed an editing algorithm based on shape context similarity and k-medoid clustering that improved on their performance.^[4]

Trademark retrieval

Shape contexts were used to retrieve the closest matching trademarks from a database to a query trademark (useful in detecting trademark infringement). No visually similar trademark was missed by the algorithm (verified manually by the authors). Potter or Ceramic Artist Truman Bedell from Rexton, has interests which include ceramics, best property developers in singapore developers in singapore and scrabble. Was especially enthused after visiting Alejandro de Humboldt National Park.

External links

References

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The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
↑ 55 years old Systems Administrator Antony from Clarence Creek, really loves learning, PC Software and aerobics. Likes to travel and was inspired after making a journey to Historic Ensemble of the Potala Palace.

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↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
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In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[shape_1-1] 1.0 ^1.1 ^1.2 ^1.3 ^1.4 55 years old Systems Administrator Antony from Clarence Creek, really loves learning, PC Software and aerobics. Likes to travel and was inspired after making a journey to Historic Ensemble of the Potala Palace.

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In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

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[6] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[7] 55 years old Systems Administrator Antony from Clarence Creek, really loves learning, PC Software and aerobics. Likes to travel and was inspired after making a journey to Historic Ensemble of the Potala Palace.

You can view that web-site... ccleaner free download

[8] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

[9] 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.

My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

[10] One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting

In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang

Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules

Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.

A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running

The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more

There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang

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en>Yobot

Revision as of 21:08, 20 April 2013

Contents

Theory

Use in shape matching

Details of implementation

Step 1: Finding a list of points on shape edges

Step 2: Computing the shape context

Step 3: Computing the cost matrix

Step 4: Finding the matching that minimizes total cost

Step 5: Modeling transformation

Affine

Thin plate spline

Regularized TPS

Step 6: Computing the shape distance

Results

Digit recognition

Silhouette similarity-based retrieval

3D object recognition

Trademark retrieval

External links

References

Navigation menu

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+'''Shape context''' is a feature descriptor used in [[object recognition]]. [[Serge Belongie]] and [[Jitendra Malik]] proposed the term in their paper "Matching with Shape Contexts" in 2000.<ref name="shape_1">
+{{cite conference
+ | author = S. Belongie and J. Malik
+ | title = Matching with Shape Contexts
+ | url = http://ieeexplore.ieee.org/iel5/6885/18538/00853834.pdf
+ | booktitle = IEEE Workshop on Contentbased Access of Image and Video Libraries (CBAIVL-2000)
+ | year = 2000}}</ref>
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+==Theory==
+The shape context is intended to be a way of describing shapes that allows for measuring shape similarity and the recovering of point correspondences.<ref name="shape_1"/> The basic idea is to pick ''n'' points on the contours of a shape. For each point ''p''<sub>''i''</sub> on the shape, consider the ''n''&nbsp;&minus;&nbsp;1 vectors obtained by connecting ''p''<sub>''i''</sub> to all other points. The set of all these vectors is a rich description of the shape localized at that point but is far too detailed. The key idea is that the distribution over relative positions is a robust, compact, and highly discriminative descriptor. So, for the point ''p''<sub>''i''</sub>, the coarse histogram of the relative coordinates of the remaining ''n''&nbsp;&minus;&nbsp;1 points,
+: <math>h_i(k) = \#\{q \ne p_i  :  (q - p_i) \in \mbox{bin}(k)\}</math>
+is defined to be the shape context of <math>p_i</math>. The bins are normally taken to be uniform in log-polar space. The fact that the shape context is a rich and discriminative descriptor can be seen in the figure below, in which the shape contexts of two different versions of the letter "A" are shown.
+[[Image:shapecontext.jpg|400px]]
+(a) and (b) are the sampled edge points of the two shapes. (c) is the diagram of the log-polar bins used to compute the shape context. (d) is the shape context for the circle, (e) is that for the diamond, and (f) is that for the triangle. As can be seen, since (d) and (e) are the shape contexts for two closely related points, they are quite similar, while the shape context in (f) is very different.
+Now in order for a feature descriptor to be useful, it needs to have certain invariances. In particular it needs to be invariant to translation, scale, small perturbations, and depending on application rotation. Translational invariance come naturally to shape context. Scale invariance is obtained by normalizing all radial distances by the mean distance <math>\alpha</math> between all the point pairs in the shape <ref name="shape_2">
+{{cite journal
+ | author = S. Belongie, J. Malik, and J. Puzicha
+ | title = Shape Matching and Object Recognition Using Shape Contexts
+ | url = http://www.eecs.berkeley.edu/Research/Projects/CS/vision/shape/belongie-pami02.pdf
+ | journal = IEEE Transactions on Pattern Analysis and Machine Intelligence
+ | volume = 24
+ | issue = 24
+ | date = April 2002
+ | pages = 509–521
+ | doi = 10.1109/34.993558}}</ref><ref name="shape_3">
+{{cite conference
+ | author = S. Belongie, J. Malik, and J. Puzicha
+ | title = Matching Shapes
+ | url = http://www.eecs.berkeley.edu/Research/Projects/CS/vision/shape/belongie-iccv01.pdf
+ | booktitle = Eighth IEEE International Conference on Computer Vision (July 2001)
+ |date=July 2001}}</ref> although the median distance can also be used.<ref name="shape_1"/><ref name="shape_4">
+{{cite conference
+ | author = S. Belongie, J. Malik, and J. Puzicha
+ | title = Shape Context: A new descriptor for shape matching and object recognition
+ | booktitle = NIPS 2000
+ | year = 2000}}</ref> Shape contexts are empirically demonstrated to be robust to deformations, noise, and outliers<ref name="shape_4"/> using synthetic point set matching experiments.<ref>
+{{cite conference
+ | author = H. Chui and A. Rangarajan
+ | title = A new algorithm for non-rigid point matching
+ | booktitle = CVPR
+ | volume = 2
+ | pages = 44–51
+ |date=June 2000}}</ref>
+One can provide complete rotation invariance in shape contexts. One way is to measure angles at each point relative to the direction of the tangent at that point (since the points are chosen on edges). This results in a completely rotationally invariant descriptor. But of course this is not always desired since some local features lose their discriminative power if not measured relative to the same frame. Many applications in fact forbid rotation invariance e.g. distinguishing a "6" from a "9".
+==Use in shape matching==
+A complete system that uses shape contexts for shape matching consists of the following steps (which will be covered in more detail in the [[#Details of implementation|Details of Implementation]] section):
+# Randomly select a set of points that lie on the edges of a known shape and another set of points on an unknown shape.
+# Compute the shape context of each point found in step 1.
+# Match each point from the known shape to a point on an unknown shape.  To minimize the cost of matching, first choose a transformation (e.g. [[affine transformation|affine]], [[thin plate spline]], etc.) that warps the edges of the known shape to the unknown (essentially aligning the two shapes).  Then select the point on the unknown shape that most closely corresponds to each warped point on the known shape.
+# Calculate the "shape distance" between each pair of points on the two shapes.  Use a weighted sum of the shape context distance, the image appearance distance, and the bending energy (a measure of how much transformation is required to bring the two shapes into alignment).
+# To identify the unknown shape, use a [[Nearest neighbor search|nearest-neighbor classifier]] to compare its shape distance to shape distances of known objects.
+==Details of implementation==
+===Step 1: Finding a list of points on shape edges===
+The approach assumes that the shape of an object is essentially captured by a finite subset of the points on the internal or external contours on the object. These can be simply obtained using the [[Canny edge detector]] and picking a random set of points from the edges. Note that these points need not and in general do not correspond to key-points such as maxima of curvature or [[inflection point]]s. It is preferable to sample the shape with roughly uniform spacing, though it is not critical.<ref name="shape_2"/>
+===Step 2: Computing the shape context===
+This step is described in detail in the [[#Theory|Theory section]].
+===Step 3: Computing the cost matrix===
+Consider two points ''p'' and ''q'' that have normalized ''K''-bin histograms (i.e. shape contexts) ''g''(''k'') and ''h''(''k''). As shape contexts are distributions represented as histograms, it is natural to use the [[chi-squared statistic|''&chi;''<sup>2</sup> test statistic]] as the "shape context cost" of matching the two points:
+: <math>C_S = \frac{1}{2}\sum_{k=1}^K \frac{[g(k) - h(k)]^2}{g(k) + h(k)}</math>
+The values of this range from 0 to 1.<ref name="shape_1"/>
+In addition to the shape context cost, an extra cost based on the appearance can be added. For instance, it could be a measure of tangent angle dissimilarity (particularly useful in digit recognition):
+: <math>C_A = \frac{1}{2}\begin{Vmatrix}
+ \dbinom{\cos(\theta_1)}{\sin(\theta_1)} - \dbinom{\cos(\theta_2)}{\sin(\theta_2)}
+\end{Vmatrix}</math>
+This is half the length of the chord in unit circle between the unit vectors with angles <math>\theta_1</math> and <math>\theta_2</math>. Its values also range from 0 to 1. Now the total cost of matching the two points could be a weighted-sum of the two costs:
+: <math>C = (1 - \beta)C_S + \beta C_A\!\,</math>
+Now for each point ''p''<sub>''i''</sub> on the first shape and a point ''q''<sub>''j''</sub> on the second shape, calculate the cost as described and call it ''C''<sub>''i'',''j''</sub>. This is the cost matrix.
+<!--  Commented out because image was deleted: [[Image:Matching.png|thumb|200px|right|An example of the matching found between two handwritten "A"'s with costs defined only as the "shape context cost" between the two histograms]] -->
+===Step 4: Finding the matching that minimizes total cost===
+[[File:Matching example using the Shape Context descriptor.jpg|thumb|100px|right|Results of matching]]
+Now, a one-to-one matching ''p''<sub>''i''</sub> that matches each point ''p''<sub>''i''</sub> on shape 1 and ''q''<sub>''j''</sub> on shape 2 that minimizes the total cost of matching,
+: <math>H(\pi) = \sum_i C\left (p_i,q_{\pi (i)} \right )</math>
+is needed. This can be done in <math>O(N^3)</math> time using the [[Hungarian method]], although there are more efficient algorithms.<ref>
+{{cite journal
+ |author = R. Jonker and A. Volgenant
+ |title = A Shortest Augmenting Path Algorithm for Dense and Sparse Linear Assignment Problems
+ |journal = Computing
+ |volume = 38
+ |pages = 325–340
+ |year = 1987
+ |doi = 10.1007/BF02278710
+ |issue = 4}}</ref>
+To have robust handling of outliers, one can add "dummy" nodes that have a constant but reasonably large cost of matching to the cost matrix. This would cause the matching algorithm to match outliers to a "dummy" if there is no real match.
+===Step 5: Modeling transformation===
+Given the set of correspondences between a finite set of points on the two shapes, a transformation <math>T : \Bbb{R}^2 \to \Bbb{R}^2</math> can be estimated to map any point from one shape to the other. There are several choices for this transformation, described below.
+====Affine====
+The [[Affine transformation|affine model]] is a standard choice: <math>T(p) = Ap + o\!</math>. The [[least squares]] solution for the matrix <math>A</math> and the translational offset vector ''o'' is obtained by:
+: <math>o = \frac{1}{n}\sum_{i=1}^n \left (p_i - q_{\pi(i)} \right ),
+      A = (Q^+ P)^t</math>
+Where <math>P = \begin{pmatrix}
+& p_{11} & p_{12} \\
+             \vdots & \vdots & \vdots \\
+& p_{n1} & p_{n2}
+\end{pmatrix} </math> with a similar expression for <math>Q\!</math>. <math>Q^+\!</math> is the [[Moore-Penrose pseudoinverse|pseudoinverse]] of <math>Q\!</math>.
+====Thin plate spline====
+The [[Thin plate spline|thin plate spline (TPS)]] model is the most widely used model for transformations when working with shape contexts. A 2D transformation can be separated into two TPS function to model a coordinate transform:
+: <math> T(x,y) = \left (f_x(x,y),f_y(x,y)\right )</math>
+where each of the ''ƒ''<sub>''x''</sub> and ''ƒ''<sub>''y''</sub> have  the form:
+: <math> f(x,y) = a_1 + a_xx + a_yy + \sum_{i=1}^n\omega_iU\left (\begin{Vmatrix}
+(x_i,y_i) - (x,y) \end{Vmatrix} \right ),</math>
+and the kernel function <math>U(r)\!</math> is defined by <math>U(r) = r^2\log r^2\!</math>. The exact details of how to solve for the parameters can be found elsewhere<ref>
+{{cite conference
+ |author = M.J.D. Powell
+ |title = A Thin Plate Spline Method for Mapping Curves into Curves in Two Dimensions
+ |booktitle = Computational Techniques and Applications (CTAC '95)
+ |year = 1995}}</ref><ref>
+{{cite journal
+ |author = J. Duchon
+ |title = Splines Minimizing Rotation-Invariant Semi-Norms in Sobolev Spaces
+ |journal = Constructive Theory of Functions of Several Variables
+ |pages = 85–100}}</ref> but it essentially involves solving a [[System of linear equations|linear system of equations]]. The bending energy (a measure of how much transformation is needed to align the points) will also be easily obtained.
+====Regularized TPS====
+The TPS formulation above has exact matching requirement for the pairs of points on the two shapes. For noisy data, it is best to relax this exact requirement. If we let <math>v_i</math> denote the target function values at corresponding locations <math>p_i = (x_i,y_i)</math> (Note that for <math>f_x</math>, <math>v_i</math> would <math>x'</math> the x-coordinate of the point corresponding to <math>p_i</math> and for <math>f_y</math> it would be the y-coordinate, <math>y'</math>), relaxing the requirement amounts to minimizing
+: <math> H[f] = \sum_{i=1}^n(v_i - f(x_i,y_i))^2 + \lambda I_f</math>
+where <math>I_f\!</math> is the bending energy and <math>\lambda\!</math> is called the regularization parameter. This ''ƒ'' that minimizes ''H''[''ƒ''] can be found in a fairly straightforward way.<ref>
+{{cite book
+ |author = G. Wahba
+ |title = Spline Models for Observational Data
+ |publisher = Soc. Industrial and Applied Math
+ |year = 1990}}</ref> If one uses normalize coordinates for <math>(x_i,y_i)\mbox{ and } (x'_i,y'_i)</math>, then scale invariance is kept. However, if one uses the original non-normalized coordinates, then the regularization parameter needs to be normalized.
+Note that in many cases, regardless of the transformation used, the initial estimate of the correspondences contains some errors which could reduce the quality of the transformation. If we iterate the steps of finding correspondences and estimating transformations (i.e. repeating steps 2&ndash;5 with the newly transformed shape) we can overcome this problem. Typically, three iterations are all that is needed to obtain reasonable results.
+===Step 6: Computing the shape distance===
+Now, a shape distance between two shapes <math>P\!</math> and <math>Q\!</math>. This distance is going to be a weighted sum of three potential terms:
+'''Shape context distance''': this is the symmetric sum of shape context matching costs over best matching points:
+: <math>D_{sc}(P,Q) = \frac{1}{n}\sum_{p \in P} \arg \underset{q \in Q}{\min} C(p,T(q)) + \frac{1}{m}\sum_{q \in Q} \arg \underset{p \in P}{\min} C(p,T(q))</math>
+where ''T''(·) is the estimated TPS transform that maps the points in ''Q'' to those in ''P''.
+'''Appearance cost''': After establishing image correspondences and properly warping one image to match the other, one can define an appearance cost as the sum of squared brightness differences in [[Gaussian filter|Gaussian windows]] around corresponding image points:
+: <math>D_{ac}(P,Q) = \frac{1}{n}\sum_{i=1}^n\sum_{\Delta \in Z^2} G(\Delta)\left [I_P(p_i + \Delta) - I_Q(T(q_{\pi(i)}) + \Delta)\right ]^2</math>
+where <math>I_P\!</math> and <math>I_Q\!</math> are the gray-level images (<math>I_Q\!</math> is the image after warping) and <math>G\!</math> is a Gaussian windowing function.
+'''Transformation cost''': The final cost <math>D_{be}(P,Q)\!\,</math> measures how much transformation is necessary to bring the two images into alignment. In the case of TPS, it is assigned to be the bending energy.
+Now that we have a way of calculating the distance between two shapes, we can use a [[k-nearest neighbor algorithm|nearest neighbor]] [[Classifier (mathematics)|classifier]] (k-NN) with distance defined as the shape distance calculated here. The results of applying this to different situations is given in the following section.
+==Results==
+===Digit recognition===
+{{Main|MNIST database}}
+The authors [[Serge Belongie]] and [[Jitendra Malik]] tested their approach on the [[MNIST database]]. Currently, more than 50 algorithms have been tested on the database. The database has a training set of 60,000 examples, and a test set of 10,000 examples. The error rate for this approach was 0.63% using 20,000 training examples and 3-NN. At the time of publication, this error rate was the lowest. Currently, the lowest error rate is 0.23%.
+===Silhouette similarity-based retrieval===
+The authors experimented with the MPEG-7 shape silhouette database, performing Core Experiment CE-Shape-1 part B, which measures performance of similarity-based retrieval.<ref>
+{{cite journal
+ |author = S. Jeannin and M. Bober
+ |title = Description of core experiments for MPEG-7 motion/shape. Technical Report ISO/IEC JTC 1/SC 29/WG 11 MPEG99/N2690, MPEG-7, Seoul
+ |date=March 1999}}</ref> The database has 70 shape categories and 20 images per shape category. Performance of a retrieval scheme is tested by using each image as a query and counting the number of correct images in the top 40 matches. For this experiment, the authors increased the amount of points sampled from each shape. Also, since the shapes in the database sometimes were rotated or flipped, the authors took defined the distance between a reference shape and query shape to be minimum shape distance between the query shape and either the unchanged reference, the vertically flipped, or the reference horizontally flipped.<ref name="shape_1"/><ref name="shape_2"/><ref name="shape_3"/><ref name="shape_4"/> With these changes, they obtained a retrieval rate of 76.45%, which by 2002 was the best.
+<!--  Commented out because image was deleted: [[Image:Coil_20.PNG|thumb|200px|right|Results of the experiment on COIL-20]] -->
+===3D object recognition===
+The next experiment performed on shape contexts involved the 20 common household objects in the [http://www1.cs.columbia.edu/CAVE/software/softlib/coil-20.php Columbia Object Image Library (COIL-20)]. Each object has 72 views in the database. In the experiment, the method was trained on a number of equally spaced views for each object and the remaining views were used for testing. A 1-NN classifier was used. The authors also developed an ''editing'' algorithm based on shape context similarity and [[k-medoid]] clustering that improved on their performance.<ref name=shape_4/>
+===Trademark retrieval===
+Shape contexts were used to retrieve the closest matching trademarks from a database to a query trademark (useful in detecting trademark infringement). No visually similar trademark was missed by the algorithm (verified manually by the authors). {{Citation needed|date=November 2009}}
+==External links==
+* [http://www.eecs.berkeley.edu/Research/Projects/CS/vision/shape/sc_digits.html Matching with Shape Contexts]
+* [http://yann.lecun.com/exdb/mnist/ MNIST database of handwritten digits]
+* [http://www1.cs.columbia.edu/CAVE/software/softlib/coil-20.php Columbia Object Image Library (COIL-20)]
+* [http://www.vision.caltech.edu/Image_Datasets/Caltech101/ Caltech101 Database]
+==References==
+<references/>
+[[Category:Computer vision]]

Choi–Williams distribution function: Difference between revisions

en>Yobot

Revision as of 21:08, 20 April 2013

Theory

Use in shape matching

Details of implementation

Step 1: Finding a list of points on shape edges

Step 2: Computing the shape context

Step 3: Computing the cost matrix

Step 4: Finding the matching that minimizes total cost

Step 5: Modeling transformation

Affine

Thin plate spline

Regularized TPS

Step 6: Computing the shape distance

Results

Digit recognition

Silhouette similarity-based retrieval

3D object recognition

Trademark retrieval

External links

References

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