Definition:• Radon transform computes projections of an image matrix along specified

directions

),(

)sincos(),())}(,({

tC

dxdytyxyxftyxfR

where t = xcosθ + ysinθ is the line to the origin

Matlab calculation

• In Matlab, the Radon transform Rθ{f(x,y) is the line integral of function f(x,y) parallel to the y´-axis

Radon perspective of imagesRadon of 0 degree

Radon of 45 degree

Viewing the Radon Transform as an Image

The Radon transform for dinosaur head is computed at angles from 0° to 180°, in 1° increments

Why Radon?

• Higher accuracy rate: up to 70 ~ 80%

• Speed: 5 times faster than fft

• Simplicity: 1-D projection function

• Concentrate on the shape of object: take advantage of edges-detection

• Invariance of rotation, translation, and scaling movement (working progress)

Radon transform properties

• Rotation:

• Translation:

• Scale:

)())}(,({))}(,({ tRtyxfRtyxfR

)()]sincos()}[,({))}(,({ ttRtttyxfRttytxfR yxyx

)()())}(,({

tRtRt

yxfR

sincos: yx tttwhere

→Our GOAL: Make the Radon transform invariant of rotating, translating, scaling movements of the objects.

Rotation invariance

Original image Rotated 30

Auto_corrlation = 0.2036

Take radon transform of both images for 30 degree

Auto_corrlation = 0.8535

Radon transform for 60 degree Radon transform for 30 degree

Translation invariance

Original image I I_trans = circshift(I, [0, 80]);

Auto_corrlation = 0.0730Auto_corrlation = 1

R_sh = circshift(R1, [80, 0])

Scale invariance

Original image I Scaled image

Auto_corrlation = 0.1024

Scaled Radon transform

Auto_corrlation = 0.7169

)())}(,({))}(,({ tRtyxfRtyxfR

Original image Rotated by 30 degree

)()]sincos()}[,({))}(,({ ttRtttyxfRttytxfR yxyx

Original image Translation

)()())}(,({

tRtRt

yxfR

Original image Scale by half

Data Base (290)

Input Image

Edge Detection

Noise Removal:

(1) Median Filtering (medfilt2)

(2) Adaptive Filtering (wiener2)

Transform Auto-Correlation

Highest % = Best Match

Sort Result

Gray Scale

Median Filtering:

Output pixel is set to an "average" of the pixel values in the neighborhood of the corresponding input pixel.

The value of an output pixel is determined by the median of the neighborhood pixels rather than the mean. The median is much less sensitive than the mean to extreme values (outliers)

Median filtering is better able to remove these outliers without reducing the sharpness of the image.

Adaptive Filtering:

The adaptive filter tailor itself to the local image variance.

Where the variance is large, the filter performs little smoothing. Where the variance is small, the filter performs more smoothing.

The adaptive filter is selective and preserves edges and other high frequency parts of an image.

There are no design tasks; the filter handles all preliminary computations, and implements the filter for an input image.

Noises: A Nightmare for Recognition

Original Image and its Edge Detection:

Noise due to lack of focus, shakiness, material of the background, etc.

Solution: Filter them out

Original Image After Median Filter After Adaptive Filter

After Edge Detection

Example: Less Noise, Better Result

Matching %

=

0.7062

Matching Images Without Noise Removal

Example: Less Noise, Better Result

Matching %

=

0.7956

Matching Images With Median Filtering

Example: Less Noise, Better Result

Matching %

=

0.7715

Matching Images With Adaptive Filtering

Drawbacks: Noise Removal Removing a Little Too Much

Matching %

=

0.8884

Matching Images Without Noise Removal

Drawbacks: Noise Removal Removing a Little Too Much

Matching %

=

0.7475

Matching Images With Median Filtering

Drawbacks: Noise Removal Removing a Little Too Much

Matching %

=

0.7392

Matching Images With Adaptive Filtering