From 4df39bd9f5c3426d976896c37f5c91a5e199d30c Mon Sep 17 00:00:00 2001 From: Ankit Date: Thu, 1 Oct 2020 08:38:08 -0700 Subject: [PATCH 01/10] set url --- _config.yml | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/_config.yml b/_config.yml index 8fada278..aaf781da 100644 --- a/_config.yml +++ b/_config.yml @@ -25,7 +25,7 @@ baseurl: /cs131_notes_dev # the subpath of your site, e.g. /blog ########### ONLY CHANGE THESE FIELDS # replace with your own email for build notifications or leave it blank -url: "https://anarcomey.github.io" # the base hostname & protocol for your site, e.g. http://example.com +url: "https://gt3.github.io" # the base hostname & protocol for your site, e.g. http://example.com ########### # Theme-specific From e7027325c7ea1b550c71dd5ef9565e33daf53bde Mon Sep 17 00:00:00 2001 From: cheekyiscool Date: Sat, 3 Oct 2020 17:42:23 -0700 Subject: [PATCH 02/10] started intro --- _chapters/pixels/features_and_fitting.md | 39 +++++++++++++++++++----- 1 file changed, 32 insertions(+), 7 deletions(-) diff --git a/_chapters/pixels/features_and_fitting.md b/_chapters/pixels/features_and_fitting.md index 63e89603..d61fea2f 100644 --- a/_chapters/pixels/features_and_fitting.md +++ b/_chapters/pixels/features_and_fitting.md @@ -1,12 +1,37 @@ --- title: Features and fitting -keywords: (insert comma-separated keywords here) +keywords: RANSAC, order: 5 # Lecture number for 2020 --- -**Lorem ipsum** dolor sit amet, consectetur adipiscing elit, sed do eiusmod -tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, -quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo -consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum -dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, -sunt in culpa qui officia deserunt mollit anim id est laborum. +In this lecture, we will introduce model fitting for line detection using the RANSAC (Random Sample Consensus) algorithm. + + + +## Line Fitting + +Straight lines characterize many objects around us. Here are few examples where line fitting is used: + +**Example 1: Shape analysis** +<> + +**Example 2: Navigation** +<> + + + +_Note:_ Edge detection is a technique that is used to identify boundaries. It does not provide information on the orientation of pixels. + +**Challenges** + +As with any fitting problem, we ned to take into account missing information, noise, and outliers. Our solution should be able to able to detect the best fitting line from the given edge points despite any clutter or noise. It should also be able to detect lines that bridge missing evidence. + +<> + +## Model Fitting + +We would like to form a higher-level succinct representation of the features in the image by grouping multiple features based on a simple model. This section focuses on lines described as one such simple model that relies on edge points as features. + +Least squares regression is a common technique to find a line of best fit for a dataset. However, least squares is prone to the negative effect of outliers on fitting. In the case of images, it is often the case that the number of outliers is large. A better approach is to find a set of inliers to initiate model fitting. + + From e93f6c1358eced5d074c954be3b67df107214d60 Mon Sep 17 00:00:00 2001 From: cheekyiscool Date: Sat, 3 Oct 2020 21:20:45 -0700 Subject: [PATCH 03/10] rearrange intro --- _chapters/pixels/features_and_fitting.md | 17 ++++++++++------- 1 file changed, 10 insertions(+), 7 deletions(-) diff --git a/_chapters/pixels/features_and_fitting.md b/_chapters/pixels/features_and_fitting.md index d61fea2f..1d61d56b 100644 --- a/_chapters/pixels/features_and_fitting.md +++ b/_chapters/pixels/features_and_fitting.md @@ -8,9 +8,9 @@ In this lecture, we will introduce model fitting for line detection using the RA -## Line Fitting +## Lines -Straight lines characterize many objects around us. Here are few examples where line fitting is used: +Straight lines characterize many objects around us. Here are few examples where line detection is useful: **Example 1: Shape analysis** <> @@ -20,18 +20,21 @@ Straight lines characterize many objects around us. Here are few examples where -_Note:_ Edge detection is a technique that is used to identify boundaries. It does not provide information on the orientation of pixels. +## Model Fitting + +We would like to form a higher-level succinct representation of the features in the image by grouping multiple features based on a simple model. This section focuses on lines described as one such model (i.e. line fitting) with edge points as features. + +_Note:_ Edge detection is a technique that is used to identify boundaries. It does not provide information on the orientation of pixels (shape). **Challenges** -As with any fitting problem, we ned to take into account missing information, noise, and outliers. Our solution should be able to able to detect the best fitting line from the given edge points despite any clutter or noise. It should also be able to detect lines that bridge missing evidence. +As with any fitting problem, we ned to take into account missing information, noise, and outliers. Our solution should be able to able to detect the best fitting line from edge points that could have clutter and noise. It should also be able to detect lines that bridge missing evidence. <> -## Model Fitting +Least squares regression is a common technique to find a line of best fit for a dataset. However, least squares is prone to the negative effect of outliers. When dealing with images it is often the case that the number of outliers is large. A better approach is to find a set of inliers to initiate model fitting. + -We would like to form a higher-level succinct representation of the features in the image by grouping multiple features based on a simple model. This section focuses on lines described as one such simple model that relies on edge points as features. -Least squares regression is a common technique to find a line of best fit for a dataset. However, least squares is prone to the negative effect of outliers on fitting. In the case of images, it is often the case that the number of outliers is large. A better approach is to find a set of inliers to initiate model fitting. From 986dbe0c1c9914f68d3ee93fe8da993595c633e1 Mon Sep 17 00:00:00 2001 From: cheekyiscool Date: Sat, 3 Oct 2020 22:53:33 -0700 Subject: [PATCH 04/10] voting scheme, added placeholders for all sections --- _chapters/pixels/features_and_fitting.md | 25 +++++++++++++++++++++++- 1 file changed, 24 insertions(+), 1 deletion(-) diff --git a/_chapters/pixels/features_and_fitting.md b/_chapters/pixels/features_and_fitting.md index 1d61d56b..c2bf8bd9 100644 --- a/_chapters/pixels/features_and_fitting.md +++ b/_chapters/pixels/features_and_fitting.md @@ -28,13 +28,36 @@ _Note:_ Edge detection is a technique that is used to identify boundaries. It do **Challenges** -As with any fitting problem, we ned to take into account missing information, noise, and outliers. Our solution should be able to able to detect the best fitting line from edge points that could have clutter and noise. It should also be able to detect lines that bridge missing evidence. +As with any fitting problem, we ned to take into account missing information, noise, and outliers. Our solution should be able to detect the best fitting line from edge points that could have clutter and noise. It should also be able to detect lines that bridge missing evidence. <> Least squares regression is a common technique to find a line of best fit for a dataset. However, least squares is prone to the negative effect of outliers. When dealing with images it is often the case that the number of outliers is large. A better approach is to find a set of inliers to initiate model fitting. +## Voting Scheme +We could try to fit the model by exhausitively checking all combination of features. However, this approach is inefficient with O(N2) time complexity. +Voting is a decent alternative. The idea is to have features vote for compatible models and to capture model parameters that form a majority. It turns out that this set of winning model parameters are not affected by votes from clutter and noisy features as their voting will be inconsistent with the consensus. The problem of missing observations can also be overlooked as the fitted model can span multiple fragments. In general, voting is considered to be robust to outliers and missing data. +[Hough Transform](#edge_detection) uses this voting strategy for detecting lines. Check out this video for a quick review of Hough Transform: +https://www.youtube.com/watch?v=4zHbI-fFIlI&feature=youtu.be&t=3m35s + +<> + +## RANSAC + +**Algorithm** + +**Walk-through** + +**Model parameters** + +**Estimate number of iterations** + +**Refined RANSAC** + +**Analysis** + +**Summary** From 2b4eac912eb0b29da9659a011a49926b1c4699c5 Mon Sep 17 00:00:00 2001 From: cheekyiscool Date: Sun, 4 Oct 2020 10:08:19 -0700 Subject: [PATCH 05/10] added RANSAC algorithm --- _chapters/pixels/features_and_fitting.md | 45 +++++++++++++++++++----- 1 file changed, 37 insertions(+), 8 deletions(-) diff --git a/_chapters/pixels/features_and_fitting.md b/_chapters/pixels/features_and_fitting.md index c2bf8bd9..f9ced16e 100644 --- a/_chapters/pixels/features_and_fitting.md +++ b/_chapters/pixels/features_and_fitting.md @@ -20,42 +20,71 @@ Straight lines characterize many objects around us. Here are few examples where +Looking at these examples and intuitively thinking of line detection, few questions arise. +- Once we have points that belong to a line, what is the line? +- How many lines exist? +- Which points belong to which line? + +_Note:_ Edge detection is a technique that is used to identify boundaries. It does not provide information on the orientation of pixels. For example, whether these pixels form a line or corner. + ## Model Fitting We would like to form a higher-level succinct representation of the features in the image by grouping multiple features based on a simple model. This section focuses on lines described as one such model (i.e. line fitting) with edge points as features. -_Note:_ Edge detection is a technique that is used to identify boundaries. It does not provide information on the orientation of pixels (shape). - **Challenges** As with any fitting problem, we ned to take into account missing information, noise, and outliers. Our solution should be able to detect the best fitting line from edge points that could have clutter and noise. It should also be able to detect lines that bridge missing evidence. <> -Least squares regression is a common technique to find a line of best fit for a dataset. However, least squares is prone to the negative effect of outliers. When dealing with images it is often the case that the number of outliers is large. A better approach is to find a set of inliers to initiate model fitting. +Least squares regression is a common technique to find a line of best fit for a dataset. However, least squares is prone to the negative effect of outliers. When dealing with images it is often the case that the number of outliers is large. A better approach is to find a set of inliers to initiate model fitting. In the next section, we'll explore such a technique that helps us find consistent matches. + +<> -## Voting Scheme +## Voting-based Fitting We could try to fit the model by exhausitively checking all combination of features. However, this approach is inefficient with O(N2) time complexity. Voting is a decent alternative. The idea is to have features vote for compatible models and to capture model parameters that form a majority. It turns out that this set of winning model parameters are not affected by votes from clutter and noisy features as their voting will be inconsistent with the consensus. The problem of missing observations can also be overlooked as the fitted model can span multiple fragments. In general, voting is considered to be robust to outliers and missing data. -[Hough Transform](#edge_detection) uses this voting strategy for detecting lines. Check out this video for a quick review of Hough Transform: +[Hough Transform](#edge_detection) uses this voting strategy for detecting lines. Check out this video for a quick review of Hough transform: https://www.youtube.com/watch?v=4zHbI-fFIlI&feature=youtu.be&t=3m35s <> -## RANSAC +## Random Sample Consensus (RANSAC) + +Both Hough transform and RANSAC rely on voting to arrive at the optimum model. The part where they differ is in how the model is chosen. As the name suggests, RANSAC introduces randomness in the model selection process. A number of models are proposed until one is found that is supported by a consensus of features (voters). Let's try to understand the algorithm. **Algorithm** -**Walk-through** +RANSAC algorithm was developed by Fischler and Bolles in 1981. Here's the gist: + +1. Randomly sample a group of points required to fit the model +2. Find model parameters using this sample +3. Calculate the votes from the fraction of inliers within a predetermined threshold of the model +4. Repeat steps 1-3 until model is found with high confidence + +** Algorithm Pseudocode ** -**Model parameters** +<> + +**Model Parameters** + +The alogrithm can be tuned with the following parameters: + +- *n* &ndash minimum number of points required to estimate model parameters +- *k* &ndash maximum number of iterations +- *t* &ndash distance threshold to determine points that fit well to model +- *d* &ndash number of close points required to assert a model fits well + +The parameter that needs most attention is *k* - number of iterations. Next, we'll see how to determine its value. **Estimate number of iterations** + + **Refined RANSAC** **Analysis** From ca8fda347f2f9266fba32e8603fdb6ba365e800c Mon Sep 17 00:00:00 2001 From: cheekyiscool Date: Sun, 4 Oct 2020 11:03:19 -0700 Subject: [PATCH 06/10] add images --- _chapters/pixels/features_and_fitting.md | 43 +++++++++++----------- assets/pixels/line-fitting-challenges.png | Bin 0 -> 504824 bytes assets/pixels/line-fitting-outliers.png | Bin 0 -> 155511 bytes assets/pixels/lines-intro.png | Bin 0 -> 1296950 bytes assets/pixels/ransac-pseudocode.png | Bin 0 -> 359287 bytes 5 files changed, 22 insertions(+), 21 deletions(-) create mode 100644 assets/pixels/line-fitting-challenges.png create mode 100644 assets/pixels/line-fitting-outliers.png create mode 100644 assets/pixels/lines-intro.png create mode 100644 assets/pixels/ransac-pseudocode.png diff --git a/_chapters/pixels/features_and_fitting.md b/_chapters/pixels/features_and_fitting.md index f9ced16e..dc215c1f 100644 --- a/_chapters/pixels/features_and_fitting.md +++ b/_chapters/pixels/features_and_fitting.md @@ -12,13 +12,9 @@ In this lecture, we will introduce model fitting for line detection using the RA Straight lines characterize many objects around us. Here are few examples where line detection is useful: -**Example 1: Shape analysis** -<> - -**Example 2: Navigation** -<> - - +
+ +
Looking at these examples and intuitively thinking of line detection, few questions arise. - Once we have points that belong to a line, what is the line? @@ -35,23 +31,27 @@ We would like to form a higher-level succinct representation of the features in As with any fitting problem, we ned to take into account missing information, noise, and outliers. Our solution should be able to detect the best fitting line from edge points that could have clutter and noise. It should also be able to detect lines that bridge missing evidence. -<> +
+ +
Least squares regression is a common technique to find a line of best fit for a dataset. However, least squares is prone to the negative effect of outliers. When dealing with images it is often the case that the number of outliers is large. A better approach is to find a set of inliers to initiate model fitting. In the next section, we'll explore such a technique that helps us find consistent matches. -<> +
+ +
## Voting-based Fitting -We could try to fit the model by exhausitively checking all combination of features. However, this approach is inefficient with O(N2) time complexity. +We could try to fit the model by exhausitively checking all combination of features. However, this approach is inefficient and bears a O(N2) time complexity. Voting is a decent alternative. The idea is to have features vote for compatible models and to capture model parameters that form a majority. It turns out that this set of winning model parameters are not affected by votes from clutter and noisy features as their voting will be inconsistent with the consensus. The problem of missing observations can also be overlooked as the fitted model can span multiple fragments. In general, voting is considered to be robust to outliers and missing data. -[Hough Transform](#edge_detection) uses this voting strategy for detecting lines. Check out this video for a quick review of Hough transform: +[Hough Transform](../edge_detection) uses this voting strategy for detecting lines. Check out this video for a quick review of Hough transform: -https://www.youtube.com/watch?v=4zHbI-fFIlI&feature=youtu.be&t=3m35s + -<> + ## Random Sample Consensus (RANSAC) @@ -66,25 +66,26 @@ RANSAC algorithm was developed by Fischler and Bolles in 1981. Here's the gist: 3. Calculate the votes from the fraction of inliers within a predetermined threshold of the model 4. Repeat steps 1-3 until model is found with high confidence -** Algorithm Pseudocode ** +**Algorithm Pseudocode** -<> +
+ +
**Model Parameters** The alogrithm can be tuned with the following parameters: -- *n* &ndash minimum number of points required to estimate model parameters -- *k* &ndash maximum number of iterations -- *t* &ndash distance threshold to determine points that fit well to model -- *d* &ndash number of close points required to assert a model fits well +- *n* – minimum number of points required to estimate model parameters +- *k* – maximum number of iterations +- *t* – distance threshold to determine points that fit well to model +- *d* – number of close points required to assert a model fits well -The parameter that needs most attention is *k* - number of iterations. Next, we'll see how to determine its value. +The parameter that needs most attention is *k* – number of iterations. Next, we'll see how to determine its value. **Estimate number of iterations** - **Refined RANSAC** **Analysis** diff --git a/assets/pixels/line-fitting-challenges.png b/assets/pixels/line-fitting-challenges.png new file mode 100644 index 0000000000000000000000000000000000000000..f97ca46c3ccf1da9b9b2ea8ef1144b4995b39a90 GIT binary patch literal 504824 zcmeFY_gB+f^ew71sY>r1LJ1uK0RbV@P^2Y<5~NG-iqZwCp$CHWCN)6lU__*b-g^@e z5tJe#O`2TJx$oX{&Kuvq;EngwXC!+hJ6U6`J=dCZ?pTAzniQnWq_=L}q5x~D8Qr>t z54&}XfRmW$<{eT{>784*IB$W~R3I;(?6#e?xI04uvM*j_Eq%Dia*83MO}<5-hDXkv zVqm~5A;?ymRT56XU5Y3LnVEem4=vEhe0ulx?d%j3s{L!ZE)9ztvAS9spHmUPf%vPJ z6Ph13Wwv%Pz3?52z|g?eL-`=@=RFquy$OM<&L<0jKlYbfH}iZ)P>t?OU7m@rc-Dwz z@&5a`79gPJhg}O?UgG|{@b@cXyvLXzuK)LEZkD{LD7}21Qz;+m{SS+48-sKI(_Rd> zZ}Y<#ZmX?E{0~3mW+RIK;k*9-jsA|+{})3Sh7W)}I^Dl!X@0>_zXV41p9&8dw1{Mt zsxR^WVjJ&UXUQL{NxwH#t&VK$ucrkM?lozp!t?PxYDn~5Q%abSyz0GMw=!s%sJ$%C zm7E9nKRf=UFvp7Now*fYScF-u&1Qm{FI$%sm^qy%qS(Hi{&4gVytFuNQ_p@{ulFN= 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zgk%)t=I$`2;7MT3n^v|R2w&}a$+9FxBdtwWM_2h^GsBsXK# zbC2bcD)F9_++Z`j~2yO2Zkn_Y{ccCt3!-O*`c zavx7eB#KlH2YLl$z4T1Ih*{cEGFzNBtn>D6Gkt%?Zxos?8M?Oo|GKfA8TE=R^R;mhM~ouWBP}SA{H7Tmv}*V zGoE%hue4*}b4-}OzX_TG!cT8KTmMp>y(635vom{P7xV7@49x%KfM?gEMfDz#n^Njo zolI8{T~)}Vr@WEPCr-~hcE2V4*7wjHmV|<9nqvC&*R06T9!S3==rysa4bcA9*Lz48 zeim%gT;*>1^ta9z-<>e9vuCcE!bSfM=;!(U>H6Op`X98RZARyoeWUy3DG&F6e;R0A KRDrT(;Qs-1+S7*s literal 0 HcmV?d00001 From 185f910bdd41a196f22af878532e15b767c2fba6 Mon Sep 17 00:00:00 2001 From: cheekyiscool Date: Sun, 4 Oct 2020 11:18:55 -0700 Subject: [PATCH 07/10] spelling --- _chapters/pixels/features_and_fitting.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/_chapters/pixels/features_and_fitting.md b/_chapters/pixels/features_and_fitting.md index dc215c1f..4d65e678 100644 --- a/_chapters/pixels/features_and_fitting.md +++ b/_chapters/pixels/features_and_fitting.md @@ -29,7 +29,7 @@ We would like to form a higher-level succinct representation of the features in **Challenges** -As with any fitting problem, we ned to take into account missing information, noise, and outliers. Our solution should be able to detect the best fitting line from edge points that could have clutter and noise. It should also be able to detect lines that bridge missing evidence. +As with any fitting problem, we need to take into account missing information, noise, and outliers. Our solution should be able to detect the best fitting line from edge points that could have clutter and noise. It should also be able to detect lines that bridge missing evidence.

@@ -43,7 +43,7 @@ Least squares regression is a common technique to find a line of best fit for a ## Voting-based Fitting -We could try to fit the model by exhausitively checking all combination of features. However, this approach is inefficient and bears a O(N2) time complexity. +We could try to fit the model by exhaustively checking all combination of features. However, this approach is inefficient and bears a O(N2) time complexity. Voting is a decent alternative. The idea is to have features vote for compatible models and to capture model parameters that form a majority. It turns out that this set of winning model parameters are not affected by votes from clutter and noisy features as their voting will be inconsistent with the consensus. The problem of missing observations can also be overlooked as the fitted model can span multiple fragments. In general, voting is considered to be robust to outliers and missing data. @@ -74,7 +74,7 @@ RANSAC algorithm was developed by Fischler and Bolles in 1981. Here's the gist: **Model Parameters** -The alogrithm can be tuned with the following parameters: +The algorithm can be tuned with the following parameters: - *n* – minimum number of points required to estimate model parameters - *k* – maximum number of iterations From 06878c15c37700e8f84660ebdf31b62fb2d91805 Mon Sep 17 00:00:00 2001 From: cheekyiscool Date: Sun, 4 Oct 2020 14:15:40 -0700 Subject: [PATCH 08/10] calculation of number of iterations --- _chapters/pixels/features_and_fitting.md | 22 ++++++++++++++++------ 1 file changed, 16 insertions(+), 6 deletions(-) diff --git a/_chapters/pixels/features_and_fitting.md b/_chapters/pixels/features_and_fitting.md index 4d65e678..b2d07905 100644 --- a/_chapters/pixels/features_and_fitting.md +++ b/_chapters/pixels/features_and_fitting.md @@ -62,7 +62,7 @@ Both Hough transform and RANSAC rely on voting to arrive at the optimum model. T RANSAC algorithm was developed by Fischler and Bolles in 1981. Here's the gist: 1. Randomly sample a group of points required to fit the model -2. Find model parameters using this sample +2. Find model parameters from this sample 3. Calculate the votes from the fraction of inliers within a predetermined threshold of the model 4. Repeat steps 1-3 until model is found with high confidence @@ -76,15 +76,25 @@ RANSAC algorithm was developed by Fischler and Bolles in 1981. Here's the gist: The algorithm can be tuned with the following parameters: -- *n* – minimum number of points required to estimate model parameters -- *k* – maximum number of iterations -- *t* – distance threshold to determine points that fit well to model -- *d* – number of close points required to assert a model fits well +- _n_ – minimum number of points required to estimate model parameters +- _k_ – maximum number of iterations +- _t_ – distance threshold to determine points that fit well to model +- _d_ – number of close points required to assert a model fits well -The parameter that needs most attention is *k* – number of iterations. Next, we'll see how to determine its value. +The parameter that needs most attention is _k_ – number of iterations. Next, we'll see how to determine its value. **Estimate number of iterations** +Random sampling requires a minimum number of samples to be drawn to provide a high confidence estimate of parameters. The number of samples largely depends on outliers and sample size. In other words, the number of iterations is directly proportional to noise in the data and complexity of the model. This can be described by the following probabilities: + +$$ \begin{equation} P_f = (1 - W^n)^k = 1 - p \end{equation} $$ + +where _k_ is the number of samples, _n_ is the minimum number of points required to estimate model parameters (2 parameters for line), _W_ is the fraction of inliers, and $P_f$ represents probability of k samples failing. + +Solving for _k_ gives us the minimum number of samples: + +$$ \begin{equation} k = \frac{\log(1 - p)}{\log(1 - W^n)} \end{equation} $$ + **Refined RANSAC** From cce4d14166b893117a075c5571196b9bb879429f Mon Sep 17 00:00:00 2001 From: cheekyiscool Date: Sun, 4 Oct 2020 14:52:39 -0700 Subject: [PATCH 09/10] number of iterations - table --- _chapters/pixels/features_and_fitting.md | 17 +++++++++++------ assets/pixels/RANSAC-number-of-iterations.png | Bin 0 -> 248959 bytes 2 files changed, 11 insertions(+), 6 deletions(-) create mode 100644 assets/pixels/RANSAC-number-of-iterations.png diff --git a/_chapters/pixels/features_and_fitting.md b/_chapters/pixels/features_and_fitting.md index b2d07905..8b507250 100644 --- a/_chapters/pixels/features_and_fitting.md +++ b/_chapters/pixels/features_and_fitting.md @@ -76,10 +76,10 @@ RANSAC algorithm was developed by Fischler and Bolles in 1981. Here's the gist: The algorithm can be tuned with the following parameters: -- _n_ – minimum number of points required to estimate model parameters -- _k_ – maximum number of iterations -- _t_ – distance threshold to determine points that fit well to model -- _d_ – number of close points required to assert a model fits well +- $n$ – minimum number of points required to estimate model parameters +- $k$ – maximum number of iterations +- $t$ – distance threshold to determine points that fit well to model +- $d$ – number of close points required to assert a model fits well The parameter that needs most attention is _k_ – number of iterations. Next, we'll see how to determine its value. @@ -89,12 +89,17 @@ Random sampling requires a minimum number of samples to be drawn to provide a hi $$ \begin{equation} P_f = (1 - W^n)^k = 1 - p \end{equation} $$ -where _k_ is the number of samples, _n_ is the minimum number of points required to estimate model parameters (2 parameters for line), _W_ is the fraction of inliers, and $P_f$ represents probability of k samples failing. +where $k$ is the number of samples, $n$ is the minimum number of points required to estimate model parameters (e.g. line has 2 parameters), $W$ is the fraction of inliers, and $P_f$ represents probability of _k_ samples failing. -Solving for _k_ gives us the minimum number of samples: +Solving for $k$ gives us the minimum number of samples required to keep the failure rate low. $$ \begin{equation} k = \frac{\log(1 - p)}{\log(1 - W^n)} \end{equation} $$ +The table below shows the number of samples required for different choices of noise and model size. It's evident that $k$ increases sharply as $n$ increases. + +
+ +
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+1,35 @@ --- title: Features and fitting -keywords: RANSAC, +keywords: RANSAC, Harris order: 5 # Lecture number for 2020 --- -In this lecture, we will introduce model fitting for line detection using the RANSAC (Random Sample Consensus) algorithm. +[//]: # (TODO!!! ** overall intro **) + + +Table of content + +- [RANSAC](#ransac) + - [Lines](#lines) + - [Model Fitting](#model-fitting) + - [Voting Based Fitting](#voting-based-fitting) + - [Random Sample Consensus](#random-sample-consensus) +- [Local Invariant Features](#local-invariant-features) +- [Harris Corner Detector](#harris-corner-detector) + +[//]: # (This is how you can make a comment that won't appear in the web page! It might be visible on some machines/browsers so use this only for development.) + +[//]: # (Notice in the table of contents that [First Big Topic] matches #first-big-topic, except for all lowercase and spaces are replaced with dashes. This is important so that the table of contents links properly to the sections) - +[//]: # (Leave this line here, but you can replace the name field with anything! It's used in the HTML structure of the page but isn't visible to users) + + +#RANSAC + +In this lecture, we will introduce model fitting for line detection using the RANSAC (Random Sample Consensus) algorithm. + + ## Lines Straight lines characterize many objects around us. Here are few examples where line detection is useful: @@ -23,6 +45,7 @@ Looking at these examples and intuitively thinking of line detection, few questi _Note:_ Edge detection is a technique that is used to identify boundaries. It does not provide information on the orientation of pixels. For example, whether these pixels form a line or corner. + ## Model Fitting We would like to form a higher-level succinct representation of the features in the image by grouping multiple features based on a simple model. This section focuses on lines described as one such model (i.e. line fitting) with edge points as features. @@ -41,6 +64,7 @@ Least squares regression is a common technique to find a line of best fit for a

+ ## Voting-based Fitting We could try to fit the model by exhaustively checking all combination of features. However, this approach is inefficient and bears a O(N2) time complexity. @@ -53,6 +77,7 @@ Voting is a decent alternative. The idea is to have features vote for compatible + ## Random Sample Consensus (RANSAC) Both Hough transform and RANSAC rely on voting to arrive at the optimum model. The part where they differ is in how the model is chosen. As the name suggests, RANSAC introduces randomness in the model selection process. A number of models are proposed until one is found that is supported by a consensus of features (voters). Let's try to understand the algorithm. @@ -106,3 +131,9 @@ The table below shows the number of samples required for different choices of no **Analysis** **Summary** + + +# Local Invariant Features + + +# Harris Corner Detector

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