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Medical Image Processing Using Transforms

Hongmei Zhu, Ph.DDepartment of Mathematics & Statistics

York Universityhmzhu@yorku.ca

MR

What do we want when we see a doctor?

patients doctorsus

Medical image processing

Increase the chance of making right decisions on diagnosis, treatment, prediction, prevention, …

Purpose of medical image processing

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UltraSound

MRI

X-ray CT

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General Questions Concerned

• Are the images good enough to make diagnosis?

• If not, how can we improve image quality? • What information can we draw from images?• Can the information aid disease diagnosis?

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Outlines

• Image Quality•Gray value transforms•Histogram processing•Transforms in image space • Transforms in Fourier space • Transforms in Time-frequency space

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Registration

Filtering

Correction

Segmentation

Analysis

Visualization

Validation

Standard pipeline for medical image processing

Filtering

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References

1. Gonzalez, R. C., Woods, R. E. and Eddins S. L. (2004). Digital Image processing. Pearson Prentice Hall; www.ImageProcessingPlace.com

2. www.sprawls.org2. http://docs.gimp.org/en/3. http://www.gnu.org/software/octave/doc/interpreter/4. http://www.mathworks.com/access/helpdesk/help/helpdesk.h

tml5. http://www.math.ufl.edu/help/matlab-tutorial/matlab-

tutorial.html

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1. Digital Image Quality

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Image Quality

www.sprawls.org

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Contrast Sensitivity

www.sprawls.org

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Contrast Sensitivity

www.sprawls.org

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Image Blurring

www.sprawls.org

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Image Blurring

www.sprawls.org

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Image Blurring

www.sprawls.org

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Image Blurring

www.sprawls.org

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Images with different noise levels

www.sprawls.org

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Effects of noise

www.sprawls.org

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Effects of Noise and Blur

www.sprawls.org

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2. Gray Value Transforms

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Histogram

Histogram shows the distribution of image intensity, often displayed as a bar graph.

( )

[ ]

The histogram of a digital image with gray levels in the range [0, L-1] is defined as

where : the th gray level, 0 1: the number of pixels having gray levels

k k

k k

k k

h g n

g k g , L-n g

=

∈

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Normalized Histogram

Normalized histogram estimates the probability distribution of occurrence of gray levels

( )

[ ]

Normalized histogram of a digital image with gray levels in the range [0, L-1] is defined as

where : the th gray level, 0 1: the number of pixels having gray levels : the total num

kk

k k

k k

np g n

g k g , L-n gn

=

∈

ber of pixels in the image

Histogram Processing

Dark: focuses on low values of thegray scales

Bright: biased towards the high sideof the gray scales

Low contrast: has a narrow histogramHigh contrast: covers a broad range

of the gray scales

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Gray Value Transformations

( )

A gray level transformation of a digital image with gray levels in the range [0, L-1] is defined as

where : the original/input gray levels: the transformed/output gray levels: gray level transfo

s T r

rsT

=

rmation

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Gray Value Transformations

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Image Negatives: s = L-1-r

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Gray Value Transformations

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Log Transformations: s = c log(1+r) (r ≥ 0)

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Power-Law Transformations

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Power-Law Transformations: s = c rγ

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Power-Law Transformations: s = c rγ

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Contrast stretching transforms

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Contrast stretching transforms

( )( )

11 Es T r

m r= =

+

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Slicing transforms

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Slicing transforms

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2. Histogram Processing

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Histogram Equalization

( ) ( ) ( )

( ) ( ) ( )

0

1 1

1

1

r

r

k kj

k k r jj j

s T r L p d

rs T r L p r

nn

ω ω

= =

= = −

= = − =

∫

∑ ∑

The probability density function of the output levels s is uniform. For digital images, the equalization transform becomes

where is the total number of the pixels in the image.

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Histogram Equalization

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Example: 3-bit image

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Example: 3-bit image

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Example: Pollen

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Example: Pollen

T(r)

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Example: Pollen

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Example: Mars_Moon

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Example: Mars_Moon

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Histogram Matching

( ) ( ) ( )

( ) ( ) ( )( )

0

0

1

1 ,

.

r

r

z

z

z

s T r L p d

z

H z L p d s

z p z

z

ω ω

ω ω

= = −

= − =

=

∫

∫

results in intensity levels s that is uniform distributed.Suppose we define a variable such that

where intensity level has the specific density Then we have

( ) ( )

( )( )

1 1 .

r

z

H s H T r

p r

p z

− −= ⎡ ⎤⎣ ⎦That is, we transform intensity levels r with density function to intensity levels z with specific density

function .

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Histogram Matching

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3. Application: MR Intensity Calibrations

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MR Image Intensity Calibration

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MR intensity variation

• MR signal intensity doesn’t have a fixed unit of measure

• Although the relative difference between tissue types will remain roughly constant from scan to scan, the absolute value of the scale is not fixed

• May pose a problem in image segmentation or quantification

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Overview

Methods• Scaling or windowing (quick, intra-patients)• Transforming to a “standard” histogram (inter-

patients, various setting)

Task• For the same tissues in all-same-settings, the

resulting image intensities should be same or close• For the same tissues in similar settings, the

resulting image intensities should be similar

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Method 1: Scaling or windowing

• Quick• Can achieve display uniformity

I1 I2 I3

• May not be fine enough for quantitative image analysis across different imaging protocols

( )( )CSFi

CSFii Imean

ImeanII

,

,1ˆ ⋅=

min,1min,max,

min,1max,1min, )(ˆ I

IIII

IIIii

iii +−−

⋅−=

or

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Method 2: transformation

ModelBimodal histogram

Range of intensity of interests

Method 2: transformationPiecewise linear mapping

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after scaling

original

aftertransformation

Example: histogram standardization

Eg: for different patients

Eg: For an non-brain region

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Eg3: From different scanners

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Remarks

• The transform chosen for standardization has to be 1-to-1 and monotonically increasing

• The intensity calibration for patients is better done in disease-removed images or in an non-disease homogenous region.