Point Operations – Chapter 5

Definition

• Some of the things we do to an image involve performing the same operation on each and every pixel (point)– We apply some function f to all pixels

for all pixels

• If the function is independent of (u,v), the pixel coordinates, it is called homogeneous

)),((),(' vuIfvuI

Usage

• Image enhancement– Contrast, brightness

• Applying a look-up table (LUT)

• Changing the color space

• others

The alternative

• If the function involves not only the current pixel value but also it’s location it is a nonhomogeneous operation

• These are good for local, adaptive processes

),),,((),(' vuvuIgvuI

Histogram

• All of these operations will modify the histogram!

Contrast enhancement

• This is basically a stretch (or compress – but not typically) of the histogram

• It’s nothing more than a multiplication

• but…remember you have to clamp to the intensity range

• a > 1 stretches the histogram,a < 1 compresses the histogram

),(),(' vuIavuI

Contrast enhancement

Contrast enhancement

Contrast enhancement

• Note the introduction of gaps (multiply) and spikes (divide) in the histogram– Bins are either vacated or combined

• Process is reversible except when pixels become saturated (dark or light)

Brightness enhancement

• This is basically a shift (left or right) of the histogram

• It’s nothing more than an addition

• but…remember you have to clamp to the intensity range

• a > 0 shifts the histogram right, a < 0 shifts the histogram left

),(),(' vuIavuI

Brightness enhancement

Brightness enhancement

Brightness enhancement

• Histogram will retain it’s original shape (no gaps or spikes) except when pixels become saturated (dark or light)

• Process is reversible except when pixels become saturated (dark or light)

• Operation is linear until clamping comes into play

Auto-contrast enhance

• The operations are typically combined into a single operation called “auto-contrast enhancement”

• Recall the definition of contrast is the “effective used of histogram bins”

• To utilize bins to the left of the original distribution a subtraction is performed

• To utilized bins to the right of the original distribution a multiplication is performed

IIIIIIlowhigh

lowvuIvuI

minmaxmin

)),((),('

Auto-contrast enhance

• Imin is typically 0

• Imax is the maximum range (e.g. 255 for an 8-bit image)

• Ilow is the lowest value of the actual distribution

• Ihigh is the largest value of the actual distribution

IIIIIIlowhigh

lowvuIvuI

minmaxmin

)),((),('

Contrast/Brightness enhance

Auto-contrast enhance

• In a typical image (e.g. the snake) there may be only a few pixels at Ilow and Ihigh so the enhancement is not “dramatic” (or even noticeable)

IIIIIIlowhigh

lowvuIvuI

minmaxmin

)),((),('

Auto-contrast enhance

• The solution is to not use the actual min and max values but to specify them instead

• This is typically done by specifying some percentage of pixels to saturate at the dark (low) and light (high) ends of the histogram

Contrast/Brightness enhance

Contrast and brightness• Note that in all the previous point

operations histogram bins may be combined but a single bin is never split

ImageJ

• Process->Math->Add or Subtract – Brightness• Process->Math->Multiply or Divide – Contrast• Process->Enhance Contrast

– Set the % of saturated pixels – the system will try to evenly distribute the percentage to the dark and light ends of the histogram

– Check Normalize to actually modify the pixel values (otherwise only the display is modified)

• If you don’t check Normalize the histogram will not change

Histogram Equalization

• Recall the smooth gradient image that I showed previously

Histogram Equalization

• Aside from the good contrast, dynamic range, exposure, etc. another interesting point is the shape of the histogram and cumulative histogram

• The histogram is uniform (intensities form a uniform distribution)

• The cumulative histogram is a linear ramp

Histogram Equalization

• One theory is that converting the histogram of any image to an approximately uniform histogram will create a good picture

• “approximately” because we can’t always get “perfectly” uniform

• Why?– Because we can only merge bins, we cannot

split them

Histogram Equalization

• Without going through the derivation (which is about 3 pages in another classic image processing book), the point operation to achieve the desired distribution is

• └ ┘is the floor operation (integer truncation)• MxN is the size of the image• K-1 is the largest possible intensity value

NM

KvuIHvuI

1)),((),('

Histogram Equalization

• Unfortunately, theory is not always correct

Histogram Equalization

• Note that the gaps in the histogram are not uniformly spaced• Note that the cumulative histogram is nearly perfectly uniform• Note that the resultant image looked really bad

Histogram specification• An alternative approach…

• Rather than go for the [arbitrarily chosen] uniform histogram, try to match the histogram to an given histogram

• But first, some definitions…

Normalized histogram

• Also known as the probability density function (probability distribution)

NM

ihip

)()(

Normalized cumulative histogram• Also known as the discrete distribution

function (cumulative distribution function)

i

j

jpNM

iHiP

0

)()(

)(

Histogram specification• The goal is to apply a point operation to

the image causing it’s cumulative distribution function (PA) to match a given cdf PR

• Again, without going through a long derivation

),(),('1

vuIvuI PP AR

Histogram specification

),(),('1

vuIvuI PP AR

PA(i)PR(i)

I(u,v)I’(u,v)

Histogram specification

Histogram specification

Histogram specification• The resulting cdf’s are of comparable shape

Histogram specification• The resulting images have a similar “look”

Histogram specification• And yet again, the resultant image looks

awful

• That’s because I matched two arbitrarily selected images– Normally you would use this on sets of

images of which you want to give the impression that the exposures were similar or were taken in similar lighting conditions

Histogram specification• You have to watch the implementation

in the event that the reference histogram has bins with no pixels in them

• The book gives those details if you’re interested