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Affine shape adaptation is a methodology for iteratively adapting the shape of the smoothing kernels in an affine group of smoothing kernels to the local image structure in neighbourhood region of a specific image point. Equivalently, affine shape adaptation can be accomplished by iteratively warping a local image patch with affine transformations while applying a rotationally symmetric filter to the warped image patches. Provided that this iterative process converges, the resulting fixed point will be affine invariant. In the area of computer vision, this idea has been used for defining affine invariant interest point operators as well as affine invariant texture analysis methods.

Affine-adapted interest point operators

The interest points obtained from the scale-adapted Laplacian blob detector or the multi-scale Harris corner detector with automatic scale selection are invariant to translations, rotations and uniform rescalings in the spatial domain. The images that constitute the input to a computer vision system are, however, also subject to perspective distortions. To obtain interest points that are more robust to perspective transformations, a natural approach is to devise a feature detector that is invariant to affine transformations.

Interestingly, affine invariance can be accomplished from measurements of the same multi-scale windowed second moment matrix $\mu$ as is used in the multi-scale Harris operator provided that we extend the regular scale space concept obtained by convolution with rotationally symmetric Gaussian kernels to an affine Gaussian scale-space obtained by shape-adapted Gaussian kernels (Lindeberg 1994 section 15.3; Lindeberg and Garding 1997). For a two-dimensional image $I$, let $\overline{x}=(x,y{)}^T$ and let ${\mathrm{\Sigma }}_t$ be a positive definite 2×2 matrix. Then, a non-uniform Gaussian kernel can be defined as

$g(\overline{x};\mathrm{\Sigma })=\frac{1}{2\pi \sqrt{\det {\mathrm{\Sigma }}_t}}{e}^{-\overline{x}{\mathrm{\Sigma }}_t^{-1}\overline{x}/2}$

and given any input image $I_L$ the affine Gaussian scale-space is the three-parameter scale-space defined as

$L(\overline{x};{\mathrm{\Sigma }}_t)=\underset{\overline{xi}}{\int }{I}_L(x-\xi )g(\overline{\xi };{\mathrm{\Sigma }}_t)d\overline{\xi }.$

Next, introduce an affine transformation $\eta =B\xi$ where $B$ is a 2×2-matrix, and define a transformed image $I_R$ as

$I_L(\overline{\xi })=I_R(\overline{\eta })$.

Then, the affine scale-space representations $L$ and $R$ of $I_L$ and $I_R$, respectively, are related according to

$L(\overline{\xi },{\mathrm{\Sigma }}_L)=R(\overline{\eta },{\mathrm{\Sigma }}_R)$

provided that the affine shape matrices ${\mathrm{\Sigma }}_L$ and ${\mathrm{\Sigma }}_R$ are related according to

${\mathrm{\Sigma }}_R=B{\mathrm{\Sigma }}_L{B}^T$.

Disregarding mathematical details, which unfortunately become somewhat technical if one aims at a precise description of what is going on, the important message is that the affine Gaussian scale-space is closed under affine transformations.

If we, given the notation $\mathrm{\nabla }L=(L_x,L_y{)}^T$ as well as local shape matrix ${\mathrm{\Sigma }}_t$ and an integration shape matrix ${\mathrm{\Sigma }}_s$, introduce an affine-adapted multi-scale second-moment matrix according to

${\mu }_L(\overline{x};{\mathrm{\Sigma }}_t,{\mathrm{\Sigma }}_s)=g(\overline{x}-\overline{\xi };{\mathrm{\Sigma }}_s)({\mathrm{\nabla }}_L(\overline{\xi };{\mathrm{\Sigma }}_t)\underset{L}{\overset{T}{\mathrm{\nabla }}}(\overline{\xi };{\mathrm{\Sigma }}_t))$

it can be shown that under any affine transformation $\overline{q}=B\overline{p}$ the affine-adapted multi-scale second-moment matrix transforms according to

${\mu }_L(\overline{p};{\mathrm{\Sigma }}_t,{\mathrm{\Sigma }}_s)=B^T{\mu }_R(q;B{\mathrm{\Sigma }}_t{B}^T,B{\mathrm{\Sigma }}_s{B}^T)B$.

Again, disregarding somewhat messy technical details, the important message here is that given a correspondence between the image points $\overline{p}$ and $\overline{q}$, the affine transformation $B$ can be estimated from measurements of the multi-scale second-moment matrices ${\mu }_L$ and ${\mu }_R$ in the two domains.

An important consequence of this study is that if we can find an affine transformation $B$ such that ${\mu }_R$ is a constant times the unit matrix, then we obtain a fixed-point that is invariant to affine transformations (Lindeberg 1994 section 15.4; Lindeberg and Garding 1997). For the purpose of practical implementation, this property can often be reached by in either of two main ways. The first approach is based on transformations of the smoothing filters and consists of:

• estimating the second-moment matrix $\mu$ in the image domain,
• determining a new adapted smoothing kernel with covariance matrix proportional to ${\mu }^{-1}$,
• smoothing the original image by the shape-adapted smoothing kernel, and
• repeating this operation until the difference between two successive second-moment matrices is sufficiently small.

The second approach is based on warpings in the image domain and implies:

• estimating $\mu$ in the image domain,
• estimating a local affine transformation proportional to $\hat{B}={\mu }^{1/2}$ where ${\mu }^{1/2}$ denotes the square root matrix of $\mu$,
• warping the input image by the affine transformation ${\hat{B}}^{-1}$ and
• repeating this operation until $\mu$ is sufficiently close to a constant times the unit matrix.

This overall process is referred to as affine shape adaptation (Lindeberg and Garding 1997; Baumberg 2000; Mikolajczyk and Schmid 2004; Tuytelaars and van Gool 2004; Lindeberg 2008). In the ideal continuous case, the two approaches are mathematically equivalent. In practical implementations, however, the first filter-based approach is usually more accurate in the presence of noise while the second warping-based approach is usually faster.

In practice, the affine shape adaptation process described here is often combined with interest point detection automatic scale selection as described in the articles on blob detection and corner detection, to obtain interest points that are invariant to the full affine group, including scale changes. Besides the commonly used multi-scale Harris operator, this affine shape adaptation can also be applied to other types of interest point operators such as the Laplacian/Difference of Gaussian blob operator and the determinant of the Hessian (Lindeberg 2008). Affine shape adaptation can also be used for affine invariant texture recognition and affine invariant texture segmentation.

See also

• Corner detection
• Blob detection
• Harris affine region detector
• Hessian affine region detector
• Scale space
• Gaussian function

References

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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
  
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  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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• 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