基于多尺度图像特征的快速压缩跟踪算法

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快速压缩跟踪快速压缩跟踪是图像处理领域中的一种技术，主要应用于图像压缩感知、去噪和基于字典的描述图像。下面是关于快速压缩跟踪的详细知识点： 1. 压缩感知：快速压缩跟踪技术基于压缩感知理论，压缩感知是一种信号处理技术，它可以从少量的观测值中重构原始信号。压缩感知理论在图像处理领域中的应用可以实现图像压缩、去噪和重建。 2. 字典学习：快速压缩跟踪技术使用字典学习方法来学习图像特征，字典学习是一种机器学习方法，它可以学习图像特征并将其表示为稀疏表示。稀疏表示可以减少图像的维数，从而实现图像压缩和去噪。 3. 多尺度图像特征：快速压缩跟踪技术使用多尺度图像特征来描述图像，多尺度图像特征可以捕捉图像的多个方面信息，例如图像的texture、edge和corner信息。 4. 非自适应随机投影：快速压缩跟踪技术使用非自适应随机投影来实现图像压缩和去噪，非自适应随机投影可以将图像投影到低维空间中，从而实现图像压缩和去噪。 5. 图像跟踪：快速压缩跟踪技术可以应用于图像跟踪领域，图像跟踪是指在视频序列中跟踪目标对象的运动轨迹。快速压缩跟踪技术可以实现实时图像跟踪，具有高效和鲁棒性。 6. 在线学习：快速压缩跟踪技术可以在线学习目标对象的外观模型，在线学习可以实时更新目标对象的外观模型，以适应目标对象的变化。 7. 去噪：快速压缩跟踪技术可以实现图像去噪，去噪是指从图像中去除噪声和artifact，去噪可以提高图像的质量和可读性。 8. 基于字典的描述图像：快速压缩跟踪技术可以实现基于字典的描述图像，基于字典的描述图像可以将图像表示为稀疏表示，从而实现图像压缩和去噪。 9. 图像重建：快速压缩跟踪技术可以实现图像重建，图像重建是指从压缩图像中重建原始图像，图像重建可以用于图像压缩和去噪。 10. 机器学习：快速压缩跟踪技术基于机器学习理论，机器学习是一种人工智能技术，它可以使计算机自主学习和决策。

IEEE TRANSACTION ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 3

negative samples are selected via an online classiﬁer with

structural constraints. Wang et al. [30] present a discriminative

appearance model based on superpixels which is able to handle

heavy occlusions and recovery from drift. In [13], Hare et al.

use an online structured output support vector machine (SVM)

for robust tracking which can mitigate the effect of wrong

labeling samples. Recently, Henriques et al. [16] introduce a

fast tracking algorithm which exploits the circulant structure

of the kernel matrix in SVM classiﬁer that can be efﬁciently

computed by the fast Fourier transform algorithm.

3 PRELIMINARIES

We present some preliminaries of compressive sensing which

are used in the proposed tracking algorithm.

3.1 Random projection and compressive sensing

In random projection, a random matrix R ∈ R

n×m

whose

rows have unit length projects data from the high-dimensional

feature space x ∈ R

to a lower-dimensional space v ∈ R

v = Rx, (1)

where n  m. Each projection v is essentially equivalent

to a compressive measurement in the compressive sensing

encoding stage. The compressive sensing theory [19], [34]

states that if a signal is K-sparse (i.e., the signal is a linear

combination of only K basis [35]), it is possible to near

perfectly reconstruct the signal from a small number of random

measurements. The encoder in compressive sensing (using (1))

correlates signal with noise (using random matrix R) [19],

thereby it is a universal encoding which requires no prior

knowledge of the signal structure. In this paper, we adopt this

encoder to construct the appearance model for visual tracking.

Ideally, we expect R provides a stable embedding that

approximately preserves the salient information in any K-

sparse signal when projecting from x ∈ R

to v ∈ R

. A

necessary and sufﬁcient condition for this stable embedding is

that it approximately preserves distances between any pairs of

K-sparse signals that share the same K basis. That is, for any

two K-sparse vectors x

and x

sharing the same K basis,

(1−)kx

−x

≤ kRx

−Rx

≤ (1+)kx

−x

. (2)

The restricted isometry property [18], [19] in compressive

sensing shows the above results. This property is achieved with

high probability for some types of random matrix R whose

entries are identically and independently sampled from a

standard normal distribution, symmetric Bernoulli distribution

or Fourier matrix. Furthermore, the above result can be directly

obtained from the Johnson-Lindenstrauss (JL) lemma [20].

Lemma 1. (Johnson-Lindenstrauss lemma) [20]: Let Q be

a ﬁnite collection of d points in R

. Given 0 <  < 1 and

β > 0, let n be a positive integer such that

n ≥



4 + 2β



/2 − 



ln(d). (3)

Let R ∈ R

n×m

be a random matrix with R(i, j) = r

, where



+1 with probability

−1 with probability

(4)

√

3 ×







+1 with probability

0 with probability

−1 with probability

(5)

Then, with probability exceeding 1 − d

−β

, the following

statement holds: For every x

, x

∈ Q,

(1−)kx

−x

≤

√

kRx

−Rx

≤ (1+)kx

−x

(6)

Baraniuk et al. [36] prove that any random matrix satisfying

the Johnson-Lindenstrauss lemma also holds true for the

restricted isometry property in compressive sensing. Therefore,

if the random matrix R in (1) satisﬁes the JL lemma, x

can be reconstructed with minimum error from v with high

probability if x is K-sparse (e.g., audio or image signals).

This strong theoretical support motivates us to analyze the

high-dimensional signals via their low-dimensional random

projections. In the proposed algorithm, a very sparse matrix is

adopted that not only asymptotically satisﬁes the JL lemma,

but also can be efﬁciently computed for real-time tracking.

3.2 Very sparse random measurement matrix

A typical measurement matrix satisfying the restricted isome-

try property is the random Gaussian matrix R ∈ R

n×m

where

∼ N(0, 1) (i.e., zero mean and unit variance), as used in

recent work [11], [37], [38]. However, as the matrix is dense,

the memory and computational loads are very expensive when

m is large. In this paper, we adopt a very sparse random

measurement matrix with entries deﬁned as

√

ρ ×







1 with probability

2ρ

0 with probability 1 −

−1 with probability

2ρ

(7)

Achlioptas [20] proves that this type of matrix with ρ = 1

or 3 satisﬁes the Johnson-Lindenstrauss lemma (i.e., (4) and

(5)). This matrix is easy to compute which requires only a

uniform random generator. More importantly, when ρ = 3,

it is sparse where two thirds of the computation can be

avoided. In addition, Li et al. [39] show that for ρ = o(m)

(x ∈ R

), the random projections are almost as accurate as

the conventional random projections where r

∼ N(0, 1).

Therefore, the random matrix (7) with ρ = o(m) asymp-

totically satisﬁes the JL lemma. In this work, we set ρ =

o(m) = m/(a log

(m)) = m/(10a) ∼ m/(6a) with a

ﬁxed constant a because the dimensionality m is typically

in the order of 10

to 10

. For each row of R, only about

c = (

2ρ

) × m = a log

(m) ≤ 10a nonzero entries

need to be computed. We observe that good results can be

obtained by ﬁxing a = 0.4 in our experiments. Therefore, the

computational complexity is only o(cn) (n = 100 in this work)

which is very low. Furthermore, only the nonzero entries of R

need to be stored which makes the memory requirement also

very light.

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