ece 484 digital image processing lec 04 - point operations ... · manipulating dynamic ranges of...

ECE 484 Digital Image Processing Lec 04 - Point Operations & Quantization

Zhu Li

Dept of CSEE, UMKC

Office: FH560E, Email: [email protected], Ph: x 2346.

http://l.web.umkc.edu/lizhu

office hour: Tu/Th 2:30-4pm@FH560E

Z. Li, ECE484 Digital Image Processing, 2019. p.1

Outline

Recap of Lec 03 Point Operations Image Quantization Summary


Camera Projection: Intrinsic + Extrinsic Parameters Camera Intrinsic + Extrinsic Parameters:

Intrinsic: camera center:[u0, v0], focus length/aspect ratio: Extrinsic: Rotation R=RxRyRz, translation t=[tx, ty, tz]


Ow

iw

kw

jwR,t

X

x

Homography In general, homography H maps 2d points

according to, x’=Hx

Up to a scale, as [x, y, w]=[sx, sy, sw], so H has 8 DoF

Affine Transform: 6 DoF: Contains a translation [t1, t2], and

invertable affine matrix A[2x2]

Similarity Transform, 4DoF: a rigid transform that preserves

distance if s=1:


Homography Estimation – SVD Pseudo Inv

In matrix form, we have, Ah = 0 If we have more than 4, then the system is over-determined, we

will find a solution by least squares, computationally via SVD

Pseudo Inverse: SVD: Pseudo inverse (minimizing least square error)

Details: https://www.ecse.rpi.edu/~qji/CV/svd_review.pdf


Outline



Types of Image Processing Opeartions

Geometrical Transform Rotation, translation Affine transform I(x,y) -> J(x, y): pixel values not

changed but moving around

Spatial Processing Directly operating on the pixel values

within the image Neighborhood based (3x3) Point operation: 1x1

Transform Domain Processing J(u,v) = T(I(x,y)), DCT, FFT, e.g.


Spatial Domain Processing

: Spatial operator defined on a neighborhood N of a given pixel

point processing mask/kernel processing

Application

Method

-8-Z. Li, ECE484 Digital Image Processing, 2019.

Intensity transformation / point operation Map a given gray or color level u to a

new level v

Memory-less, direction-less operation output at (x, y) only depend on the input

intensity at the same point Pixels of the same intensity gets the same

transformation Does not bring in new information,

may cause loss of information But can improve visual appearance or

make features easier to detect

input gray level u

outp

ut g

ray

leve

l

v


Simple Image Statistics

Pixel mean and variance im=imread('lenna.png'); m = mean(double(im(:))); v = var(double(im(:)));

Pixel histogram (pmf) [pixel_count, value]= hist(double(im(:)), [0 255]);


Binarization by pixel value thresholding %binarize mx = mean(double(im(:))); vx = var(double(im(:))); im2 = zeros(512,512); im2(find(im>mx))=255; imagesc(im2);


Point Operation - Histogram, Binarization, Negative

Z. Li, ECE484 Digital Image Processing, 2019.

Matlab % histogram figure(32); colormap('gray');im1=imread('data/lenna.png'); im = rgb2gray(im1); subplot(2,2,1); imagesc(im); title('lenna');[h, v]=hist(double(im(:)), [0:255]); subplot(2,2,2); bar(v, h); grid on; title('hist');

%binarizemx = mean(double(im(:))); vx = var(double(im(:)));im2 = zeros(512,512); im2(find(im>mx))=255;

% digital negativeim3 = 255*ones(512, 512) - double(im);

subplot(2,2,3); imagesc(im2); title('binarize'); subplot(2,2,4); imagesc(im3); title('negative');

p.12

image negatives

the appearance of photographic negatives

Enhance white or gray detail on dark regions, esp. when black areas are dominant in size


Basic intensity transform functions

monotonic, reversible compress or stretch certain range of gray-levels


Freq Domain Log Mapping

Freq domain operations


lena

FFT(lena) stretch:u 2 [0, .5] v 2 [0, .59]

compress:u 2 [.5, 1] v 2 [.59, 1]im = imread(‘lena.png’)

a = abs(fftshift(fft2(double(im))));c = log(1+double(im)); c = range_normalize(c);b = log(1+a); b=b/max(b(:));

Gamma Correction

Matching Display characteristics


power-law response functions in practiceCRT Intensity-to-voltage

function has ¼ 1.8~2.5Camera capturing distortion

with c = 1.0-1.7 Similar device curves in

scanners, printers, …

power-law transformations are also useful for general purpose contrast manipulation

Gamma Correction

Make image characteristis match display (voltage-brightness )


make linear input appear linear on displays method: calibration pattern + interactive adjustment

Effects of Gamma Correction

On images...


L02.2 L0

1/2.2L0

Histogram Equalization

Why histogram equalization ?


if pixel values are i.i.d random variables histogram is an estimate of the probability distribution of the r.v.

“unbalanced” histograms do not fully utilize the dynamic range Low contrast image: narrow

luminance range Under-exposed image:

concentrating on the dark side Over-exposed image:

concentrating on the bright side

“balanced” histogram gives more pleasant look and reveals rich details

Contrast Stretching

Map intensity to a larger range


0 L-1

L-1

Stretch the over-concentrated gray-levelsPiece-wise linear function, where the slope in the stretching region is greater than 1.

Objectives of Histogram Equalization Objectives


goal: map the each luminance level to a new value such that the output image has approximately uniform distribution of gray levels

two desired properties monotonic (non-decreasing) function: no value reversals [0,1][0.1] : the output range being the same as the input range

pdf

cdf

o

1

1o

1

1

Histogram Equalization

Algorithm

Matlab


make

show

o

1

1

im = imread(‘lena.png’)[im1,T]=histeq(rbg2gray(im));plot((0:255)/255,T);

T: cdf of lena

Equalization Algorithm

Alogrithm Sketch


Rounding or Uniform

quantization

u v v’

pu(xi)

compute histogram

equalize

round the output

or

Only depend on the input image histogram

Fast to implement For u in discrete prob.

distribution, the output v will be approximately uniform

Outline



Image Quantization

Image quantization transfer function


Uniform Quantizer The uniform quantizer’s design:

• Denote the input brightness range: • Let B – the number of bits of the quantizer => L=2B reconstruction levels• The expressions of the decision levels:

•quantization step size: q

E.g. B=2 => L=4


Uniform quantization error

Reconstruction

Reconstruction error in Mean Squred Error (MSE)

If P(x) is uniform: proof: homework


Uniform quantization and errors

Cameraman example B=1 => L=2

Non-quantized image Quantized image

Quantization error; MSE=36.2

The histogram


Uniform Quantization Example Finer quantization B=2 => L=4


Quantization error; MSE=15

histogram


Quantization: B=3, L=8

Finer quantization: B=3 => L=8; false contours present


Quantization error; MSE=7.33

histogram


Distribution Optimal Quantization

Lloyd-Max Quantization, optimize w.r.t to tk, rk

we have

which means, tk is the mid point between two reconstructions, while rk is the average.


LMQ error

The MSE from LMQ: assuming piece-wise linear PDF of X

The MSE is estimated as,


Lloyd－Max Quantization Example

LMQ example: B=1 => L=2Non-quantized image

Quantized image

The quantization error; MSE=19.5

The evolution of MSE in the optimization, startingfrom the uniform quantizer


LMQ example

LMQ: B=2, L=4

The quantization error; MSE=9.6




LMQ

LMQ: B=3, L=8

The quantization error; MSE=5




Summary

Point operations Operates on the imput intensity value, has no memory of the

neigbhouring pixels manipulating dynamic ranges of images, give it more resolution,

improves its quality Gamma correction (display adaptation) Histogram equalization Quantization - uniform: most widely used these days Lloyd-Max quantiztion: adaptive to distribution


ece 484 digital image processing lec 04 - point operations ... · manipulating dynamic ranges of...

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