亚像素精度边缘检测：基于空间矩的方法

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"Lyvers等人提出了一种基于空间矩的亚像素边缘检测方法，该方法通过使用六个矩模板来获取图像中的灰度变化、背景灰度级别、像素中心到边缘的距离以及边缘的角度信息，从而实现亚像素级别的精确边缘定位。这种方法被详细阐述在一篇发表在1990年IEEE Transactions on Pattern Analysis and Machine Intelligence的文章中，由Lyvers、Owen Robert Mitchell、Mark L. Akey和Anthony P. Reeves等人共同撰写。文章受到广泛关注，至今已被引用257次，并在ResearchGate上可供查看和讨论作者的详细资料。" 本文的核心知识点如下： 1. **空间矩**：空间矩是图像处理中的一个基本概念，它通过对图像中每个像素的灰度值乘以其位置的矩坐标（即x和y坐标）的幂次后求和，形成一系列描述图像形状特征的数值。在这个边缘检测方法中，空间矩被用来捕捉图像边缘的几何特性。 2. **亚像素边缘检测**：传统的边缘检测算法通常只能确定边缘的大致位置，而亚像素边缘检测则可以提高边缘定位的精度，将边缘位置精确到小于一个像素的单位。这种方法对于需要高精度图像分析的领域如光学字符识别、医学图像分析等尤其重要。 3. **边缘检测算子**：本文提出的一种基于二维空间矩的边缘检测算子，能够实现亚像素级别的边缘定位。该算子通过计算不同方向上的矩模板响应，获取边缘的灰度变化、背景灰度和像素距离等信息，从而提高边缘检测的准确性和精细度。 4. **模板匹配**：在该方法中，使用了六个矩模板，这些模板可能对应于不同角度，以便从多个方向检测边缘，从而更全面地捕获边缘信息。 5. **论文影响力**：此研究在计算机视觉领域具有显著的影响力，被广泛引用，表明其提出的边缘检测技术对后续的研究和应用产生了深远影响。 6. **应用领域**：基于空间矩的亚像素边缘检测技术可以应用于各种领域，包括机器人视觉、遥感图像分析、半导体器件检测等，特别是在需要高精度定位和测量的场景中。 7. **实施窗口大小**：论文指出，该边缘检测算子可适应任意大小的窗口，这意味着它可以灵活地应用于不同分辨率和复杂度的图像。 Lyvers等人提出的这种基于空间矩的亚像素边缘检测方法通过精确的几何特性分析，显著提高了边缘检测的精度，为计算机视觉领域的精确测量提供了有力工具。

LYVERS

al.:

SUBPIXEL

MEASUREMENTS

USING

EDGE OPERATOR

1295

-1.5

-1.0

-.50

0.0

.50

1.0

1.5

Edge

Translation, pixels

Fig.

Edge translation error. The spatial moment operator is shown as a

solid line while the gray level moment operator is shown as a dashed

line.

h+k

-1

Fig.

Sampled ideal edge characterized

Ak,

l,,

and

12.

hx2

A,,

Akx'dr

J,,

kx2

-1

fAk(1;

I:)

fk(

1;).

(10)

Substitution of

(8),

(9),

and

(IO)

into the length solution,

(5),

gives

3M2

I,,.,

2Ml

Ak[ZI(l

1:)

12(1

l;)]

k12(1

1:)

Ak(1:

1:)

k( 1

1;)

- -

(11)

Equation

(11)

is the length result when the operator is

applied to sampled edges.

To determine the bias associated with the length given

(1 1)

and the ideal length, an image formation model

needed. Let the ideal sampled edge be generated by the

linear equation

which shows the incremental step gray value

is pro-

portional to the normalized distance that

is from the up-

per step

1,.

The sampling aperture is flat with a width of

one pixel. The constant

(

)

is the length of one pixel.

Solving (12) for

gives

(13)

11).

The bias error is simply the actual length

minus the mo-

ment calculated length

l,,.,,

B(ll,

12,

Ak, k)

1M.

(14)

Substituting

and

(12)

into the bias equation and sim-

plifying yields

where

(16)

p=-

orp51.

The bias error is zero for all roots of the numerator of

(13,

i.e.,

P(1,

11)2(Z,

12)(P

(17)

When

the length

equations agree and the bias error is zero. This condition

occurs when the edge coincides with a pixel boundary and

the three-level edge is reduced to a two-level edge. At

every pixel boundary, the bias error is zero.

The other situation where the length equations match is

when

-12.

This occurs when the length is near the

center of the window. It should be noted that the differ-

ence

remains constant and is equal to one pixel

width. For a window of

pixels, the lengths

and

equal

-0.2

and

0.2,

respectively, when the edge is lo-

cated within the center pixel of the window. All edge po-

sitions within this range produce no bias error from the

length equation, as can be seen in Fig.

The location error is well behaved and it was empiri-

cally verified that the calculated edge location versus true

edge location is a monotonic function. Therefore, to elim-

inate the bias error that is present when the edge location

is not within the center pixel and not centered on pixel

boundaries, a look-up table procedure can be used. In this

particular implementation, the bias table was created from

edge locations spaced at

0.05

pixel intervals. Linear in-

terpolation was used to determine the bias between the

intervals. Use of the bias table to correct the calculated

length resulted in a reduction of the maximum error over

the -1.5 to

+1.5

pixel range from

0.101

0.0016

pix-

els. Note that although this result is for a range of

1.5

pixels, the most critical range is

-0.5

0.5

pixels because for

0.5

pixels the edge is closer

to a window centered on an adjacent pixel. Note that for

0.5

pixels, the spatial moment operator has no er-

ror, and therefore a bias correction table is not necessary.

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