Chapter 4 Optical Date Process

a simple lens → an optical data processor

a 2D array of data → another 2D array of data

(the object) ( the image)

Electrical data processing, 1D

Optical data processing, 2D without scanning

Hieroglyphics, letters, maps and paintings.

4.1 Abbe’s Theory of Image formation

More than a century ago, he studied the image formation in the microscope

1. Abbe’s theory:

an image is the result of two successive Fourier transformations.

1, Light source; 2, condenser; 3, object;

4, objective lens; 5, image;

A, B, Fourier transformations.

2. Frauhofer type diffraction pattern: 0th order→overall illumination

1th, 2th order → The transform information

more orders → more detail information

4.1 Abbe’s Theory of Image formation

3. Experimental illustration A grid (horizontal 50 lines/mm), illuminated by collimated white light.

a +10-diopter lens as the objective, a 10× Huygens eyepiece.

Place a sheet of ground glass at back focal plane of the objective.

(the Fourier transform plane)

zeroth-rder maximum: bright, center,white

higher-order maxima: less intense, colored

Block out all maxima except the 0th → no image is seen

the 0th maximum: provide the overall illumination

Admitting two 1th maxima → The grid is visible

Admitting all maxima → even better image

Higher-order maxima are needed for seeing more detail?

4.2 2D-transform

1. 4f configuration Object plane:

At left focal point of the first lens

a cross-grating object

Transform plane

At right focal point of the first lens

left focal point of the second lens

Fourier transform pattern

Image plane

At right focal point of the second lens. The image of the cross-grating

4.2 2D-transform

Spatial filters at transform plane

4.2 2D-transform

2. Theta modulation Different color of objects is modulated by different angles of grating White light is used for color reconstruction Filter is used on transform plane to reconstruct the color

Diffraction equation: House—redLawn—greenSky—blue

original image Theta modulated patterns on transform plane

smSin /

4.2 2D-transform

3. Input and output (a) 4f system: Object plane: input object f(x,y) Transform plane: spatial frequency spectrum F(,,) Image plane: output image f(x’, y’)The image is the result of two successive Fourier transformations:

The spatial frequency spectrum :

The output image:

“—”sign: upside down image(compare to the object)

dxexfF xi 2)(),(

ddeFyxf yxi2),(),(

4.2 2D-transform

(b) Fliters Low-pass filter(a): eliminates the high spatial frequencies,

make a laser beam more uniform, eliminating its granularity High-pass filter(b): eliminates the low spatial frequencies,

enhances the details of an image and emphasizes its edges. Band-pass filter(c): a ring-shaped, annular aperture

enhances details of a certain, predetermined size.

Example: A cytological specimen containing: epithelial cells (large) leukocytes (medium-sized) particles of undetermined origin (small)

4.2 2D-transform

(c) Phase filterThe object is an amplitude grating

The object may be a phase grating! (living tissue specimen without stain) An amplitude grating: diffraction maxima in the same phase A phase grating: more intense 0th maximum (longer vector!), but differ in phase by /2. Zernike, a Dutch physicist phase-strip method—look like an amplitude object for observing phase objects in good contrast Living tissue without staining – phase object

4.2 2D-transform

4.Correlation and Convolution

Convolution—two patterns interact with each other.

Multiple pinhole camera or projector

some applications of image Convolution

去光照影响 光照对图像的对比度有重大影响，光照少的话对比度也低，一般希望图像中的物体不受

光照影响，那怎么做呢？

图 a 是原始图像，图 b 是光照，将两者逐点相乘，得到图 c 的图像，图 c 就是受到光照影响的图像，所以目的就是尽量使图 c 恢复到图 a 的样子。

some applications of image Convolution

去光照影响

图 d 是将图 c 进行平滑滤波得到的图像，然后将图 c- 图 d 得到图 e ，可以看到效果有点担不是很明显，图 f 是用图 c/ 图 d 的结果，效果十分明显。

图 f 效果好的原因是，光照是通过和原始图像相乘来影响它的，所示是非线性运算，线性滤波器不能分离由非线性运算得到的图像。

some applications of image Convolution

通过卷积获得低分辨率图像

some applications of image Convolution

通过卷积获得低分辨率图像

注意掩膜图像的反变换，它在 4 个角上有值，并且被圆环围绕。

some applications of image Convolution

通过卷积获得低分辨率图像从卷积的输入角度，来看原始图像与 PSF左上角卷积的结果 :

通过将 4 个角卷积的结果加起来就是所得到的低分辨率图像。

some applications of image Convolution

去掉鸟笼子

some applications of image Convolution

The intensity distribution after M:

the convolution of the two:

Either M(x,y) or N(x,y) moved in the x-y plane while the pattern does not change!!

),(),(),( 0 yxNyxMIyxP

Suppose: light of intensity I0 the transmittance function: M(x,y), N(x,y)

),(),( 0 yxMIyxI

Internal coordinate for the output pattern: P(, )

The output P: (cross-correlation)

Autocorrelation: a pattern is convolved with an identical pattern:

the center output pattern reaches a maximum, forming a bright spot of light

dydxyxNyxMP ),(),(),(

dydxyxMdydxyxMyxMP ),(),(),(),( 2

The auto-correlation describes the relations between neighboring pixels.

Spatial Autocorrelation Function (ACF)

Image i(x,y)

} lag variable pixel shift in y

}

lag variable pixel shift in x

Spatial ACFr11(,)

Correlation FunctionMathematical CorrelationOf Image with Itself

CorrelateImageWithItself

Fluorescence Correlation Spectroscopy (FCS)FCS Instrumentation

LaserM1

M2BE

Sample

Dichroic

Pinhole

Mirror

Filters

APD AMP

SignalAutocorrelator

PhotonDetector

Computer

TemporalACF

4.3 Optical & Electronic Data Processing

Parallel data processing Optical: 2D→ Fourier transform→ 2D Electronic: in form of bits High speed, less interference Optical pulses transmit in the speed of light, an efficient way! Optical pulses do not interact with one another as electrons do. Perform difficult taskFourier transformation —perform by a simple lens, Speedily correlate any two-dimensional arrays of data—optical convolution,Can be served as a fan-out device that spreads data with good synchronization.

Optical computers are likely to make significant contributions to the science and art of high-speed computation.

Homework:

1, 2, 3, 4

Self study: Phase contrast