[TOC]

# Overview

Types of Image Feature:

• Edges
• Corners (also known as interest points)
• Blobs (also known as regions of interest )

# Image Corners/Keypoints

Keypoints Structure (from OpenCV):

• pt: x & y coordinates of the keypoint
• size: keypoint diameter
• angle: keypoint orientation
• response: keypoint detector response on the keypoint (that is, strength of the keypoint)
• octave: pyramid octave in which the keypoint has been detected
• class_id: object id

feature detector + feature descriptor

## Harris

Define the auto-correlation surface or SSD surface or the weighted sum of squared differences:

\begin{aligned} E_{AC}(\Delta \mathbf{u}) &= \sum_{i} \omega(\mathbf{x}_i) [ \mathbf{I}_0 ( \mathbf{x}_i + \Delta \mathbf{u} ) - \mathbf{I}_0 (\mathbf{x}_i) ]^2 \\ &\approx \sum_{i} \omega(\mathbf{x}_i) [ \mathbf{I}_0(\mathbf{x}_i) + \nabla \mathbf{I}_0(\mathbf{x}_i) \cdot \Delta\mathbf{u} - \mathbf{I}_0(\mathbf{x}_i) ]^2 \\ &= \sum_{i} \omega(\mathbf{x}_i) [ \nabla \mathbf{I}_0(\mathbf{x}_i) \cdot \Delta\mathbf{u} ]^2 \\ &= \Delta\mathbf{u}^T \cdot \mathbf{M} \cdot \Delta\mathbf{u} \end{aligned}

where

$\nabla \mathbf{I}_0(\mathbf{x}_i) = ( \frac{\partial{\mathbf{I}_0}}{\partial{x}}, \frac{\partial{\mathbf{I}_0}}{\partial{y}} ) (\mathbf{x}_i)$

$\nabla \mathbf{I} = [\mathbf{I}_x, \mathbf{I}_y]$

and the auto-correlation matrix with the weighting kernel $\omega$

$\mathbf{M} = \omega * \begin{bmatrix} \mathbf{I}_x^2 & \mathbf{I}_x\mathbf{I}_y \\ \mathbf{I}_x\mathbf{I}_y & \mathbf{I}_y^2 \end{bmatrix}$

then create a score equation, which will determine if a window can contain a corner or not

$R = det(\mathbf{M}) - k (trace(\mathbf{M}))^2$

where

$det(\mathbf{M}) = \lambda_1 \lambda_2$ $trace(\mathbf{M}) = \lambda_1 + \lambda_2$

and, $\lambda_1$ and $\lambda_2$ are the eigen values of $\mathbf{M}$, we can compute it by

$det(\lambda E - M) = 0$

So the values of these eigen values decide whether a region is corner, edge or flat

• When $|R|$ is small, which happens when $\lambda_1$ and $\lambda_2$ are small, the region is flat.
• When $R<0$, which happens when $\lambda_1 » \lambda_2$ or vice versa, the region is edge.
• When $R$ is large, which happens when $\lambda_1$ and $\lambda_2$ are large and $\lambda_1 \sim \lambda_2$, the region is a corner.

## Shi-Tomas

• cv::goodFeaturesToTrack

The Shi-Tomasi corner detector is based entirely on the Harris corner detector. However, one slight variation in a “selection criteria” made this detector much better than the original. It works quite well where even the Harris corner detector fails.

Later in 1994, J. Shi and C. Tomasi made a small modification to it in their paper Good Features to Track which shows better results compared to Harris Corner Detector.

The scoring function in Harris Corner Detector was given by (Harris corner strength):

$\mathbf{R} = \lambda_1 \lambda_2 - k(\lambda_1 + \lambda_2)^2$

Instead of this, Shi-Tomasi proposed (get the minimum eigenvalue):

$R=min(\lambda_1,\lambda_2)$

If $R$ is greater than a certain predefined value, it can be marked as a corner

## FAST

FAST (Features from Accelerated Segment Test) algorithm was proposed by Edward Rosten and Tom Drummond in their paper “Machine learning for high-speed corner detection” in 2006 (Later revised it in 2010).

### Feature Detection

• 在图像中选取像素p，假设它的亮度为 $I_p$；
• 设置一个阈值 $Ｔ$；
• 以像素 $p$ 为中心，选取半径为3的 Bresenham圆 上的16个像素；
• 假设选取的圆上有连续的Ｎ个点的亮度大于 $I_p+T$ 或 $I_p-T$，则该点 $ｐ$ 可被认为是特征点（Ｎ通常取12，即为 FAST-12，其他常用的Ｎ取值有9和11，分别被成为 FAST-9FAST-11）；
• 循环以上四步；

### Non-maximal Suppression

FAST角点经常出现“扎堆”的情况，通过 非极大值抑制，在一定区域内仅保留响应极大值的角点，避免角点集中的问题。

## SIFT

in 2004, D.Lowe, University of British Columbia, came up with a new algorithm, Scale Invariant Feature Transform (SIFT) in his paper, Distinctive Image Features from Scale-Invariant Keypoints, which extract keypoints and compute its descriptors.

## SURF

In 2006, three people, Bay, H., Tuytelaars, T. and Van Gool, L, published another paper, “SURF: Speeded Up Robust Features” which introduced a new algorithm called SURF. As name suggests, it is a speeded-up version of SIFT.

2006年，Bay和Ess等人基于SIFT算法的思路，提出了加速鲁棒特征（SURF）,该算法主要针对于SIFT算法速度太慢，计算量大的缺点，使用了近似Harr小波方法来提取特征点，这种方法就是基于Hessian行列式（DoH）的斑点特征检测方法。通过在不同的尺度上利用积分图像可以有效地计算出近似Harr小波值，简化了二阶微分模板的构建，搞高了尺度空间的特征检测的效率。

SURF算法在积分图像上使用了盒子滤波器对二阶微分模板进行了简化，从而构建了Hessian矩阵元素值，进而缩短了特征提取的时间，提高了效率。

## BRIEF Descriptor

• BRIEF: Binary Robust Independent Elementary Features

## BRISK

BRISK算法在特征点检测部分没有选用FAST特征点检测，而是选用了稳定性更强的AGAST算法。在特征描述子的构建中，BRISK算法通过利用简单的像素灰度值比较，进而得到一个级联的二进制比特串来描述每个特征点，这一点上原理与BRIEF是一致的。BRISK算法里采用了邻域采样模式，即以特征点为圆心，构建多个不同半径的离散化Bresenham同心圆，然后再每一个同心圆上获得具有相同间距的N个采样点。

## ORB

As an OpenCV enthusiast, the most important thing about the ORB(Oriented FAST and Rotated BRIEF) is that it came from “OpenCV Labs”. This algorithm was brought up by Ethan Rublee, Vincent Rabaud, Kurt Konolige and Gary R. Bradski in their paper ORB: An efficient alternative to SIFT or SURF in 2011. As the title says, it is a good alternative to SIFT and SURF in computation cost, matching performance and mainly the patents. Yes, SIFT and SURF are patented and you are supposed to pay them for its use. But ORB is not !!!

ORB is basically a fusion of FAST keypoint detector and BRIEF descriptor with many modifications to enhance the performance.

### oriented FAST

• use FAST to find keypoints, then apply Harris corner measure to find top N points among them

• multiscale
• use pyramid to produce multiscale-features
• rotation invariance (Orientation): It computes the intensity weighted centroid of the patch with located corner at center. The direction of the vector from this corner point to centroid gives the orientation. To improve the rotation invariance, moments are computed with x and y which should be in a circular region of radius r, where r is the size of the patch. 旋转部分计算如下：

• 在一个小的图像块 B 中,定义 图像块的一阶矩 为: $M_{pq} = \sum_{x,y \in B} x^p y^q I(x,y), \quad p,q \in \{ 0,1\}$

• 通过矩找到图像块的质心: $C = \bigg( \frac{M_{10}}{M_{00}}, \frac{M_{01}}{M_{00}} \bigg)$

• 几何中心 $O$ 与 质心 $C$ 连接得到 方向向量$\vec{OC}$，于是特征点的方向定义为： $\theta = arctan( \frac{M_{01}}{M_{10}} )$

### rotated BRIEF

• Binary Robust Independent Elementary Features

## FREAK

• Fast Retina KeyPoint

## SubPixel Corners

• Subpixel Corners: Increasing accuracy
• Use the OpenCV function cornerSubPix to find more exact corner positions (more exact than integer pixels)

$<\nabla \mathbf{I}(\mathbf{p}_i), \mathbf{q} - \mathbf{p}_i> = 0$

## 图像金字塔

use pyramid to produce multiscale-features

• 均值金字塔：2*2邻域均值滤波
• 高斯金字塔：向下降采样图像(4层)，高斯核5*5
• 拉普拉斯金字塔

• 正交表示
• 普吕克坐标表示

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