2. Tutorial¶

Hint

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Load modules:

import numpy as np
from SimpleMorphoMath.SimpleMorphoMath1D import *

Create a new function f(x) and plot its umbra:

function = Function((5,7,6,9,9,4,5,10,8,9,8,6,3,5,7,9,2,6))
print function.plot(plot_type='umbra')

Result:

┌──┬──────────────────┐
│11│                  │
│10│       ◼          │
│ 9│   ◼◼  ◼ ◼     ◼  │
│ 8│   ◼◼  ◼◼◼◼    ◼  │
│ 7│ ◼ ◼◼  ◼◼◼◼   ◼◼  │
│ 6│ ◼◼◼◼  ◼◼◼◼◼  ◼◼ ◼│
│ 5│◼◼◼◼◼ ◼◼◼◼◼◼ ◼◼◼ ◼│
│ 4│◼◼◼◼◼◼◼◼◼◼◼◼ ◼◼◼ ◼│
│ 3│◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼ ◼│
│ 2│◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼│
│ 1│◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼│
└──┴──────────────────┘

The umbra of a function f(x) is the set of all values (x, v) such that value v is less than or equal to f(x): \({(x,v) | v \leq f(x)}\).

Translate on the left the function f(x): \(f_{-1}(x) = f(x-1)\):

function_translated = function.clone().translate(-1)
print function_translated.plot(plot_type='umbra')

Result:

┌──┬──────────────────┐
│11│                  │
│10│      ◼           │
│ 9│  ◼◼  ◼ ◼     ◼   │
│ 8│  ◼◼  ◼◼◼◼    ◼   │
│ 7│◼ ◼◼  ◼◼◼◼   ◼◼   │
│ 6│◼◼◼◼  ◼◼◼◼◼  ◼◼ ◼ │
│ 5│◼◼◼◼ ◼◼◼◼◼◼ ◼◼◼ ◼ │
│ 4│◼◼◼◼◼◼◼◼◼◼◼ ◼◼◼ ◼ │
│ 3│◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼ ◼ │
│ 2│◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼ │
│ 1│◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼ │
└──┴──────────────────┘

The function is padded on the right with the infinimum of the function domain, thus zero.

Translate on the right the function f(x): \(f_{+1}(x) = f(x+1)\):

function_translated = function.clone().translate(1)
print function_translated.plot(plot_type='umbra')

Result:

┌──┬──────────────────┐
│11│                  │
│10│        ◼         │
│ 9│    ◼◼  ◼ ◼     ◼ │
│ 8│    ◼◼  ◼◼◼◼    ◼ │
│ 7│  ◼ ◼◼  ◼◼◼◼   ◼◼ │
│ 6│  ◼◼◼◼  ◼◼◼◼◼  ◼◼ │
│ 5│ ◼◼◼◼◼ ◼◼◼◼◼◼ ◼◼◼ │
│ 4│ ◼◼◼◼◼◼◼◼◼◼◼◼ ◼◼◼ │
│ 3│ ◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼ │
│ 2│ ◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼│
│ 1│ ◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼│
└──┴──────────────────┘

The function is padded on the left with zero.

Let’s dilate the function f(x) by the so-called unit-ball structuring element \(g(x) = 1\) over the range :\({-1 \leq x \leq 1}\), 0 otherwise:

function_dilated = Function(function)
function_dilated.dilate(unit_ball)
print function_dilated.plot(plot_type='umbra')

Result:

┌──┬──────────────────┐
│11│                  │
│10│      ◼◼◼         │
│ 9│  ◼◼◼◼◼◼◼◼◼   ◼◼◼ │
│ 8│  ◼◼◼◼◼◼◼◼◼◼  ◼◼◼ │
│ 7│◼◼◼◼◼◼◼◼◼◼◼◼ ◼◼◼◼ │
│ 6│◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼│
│ 5│◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼│
│ 4│◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼│
│ 3│◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼│
│ 2│◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼│
│ 1│◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼│
└──┴──────────────────┘

Plot with the original function as mask:

print function_dilated.plot(plot_type='umbra', mask=function)

Result:

┌──┬──────────────────┐
│11│                  │
│10│      ●◼●         │
│ 9│  ●◼◼●●◼●◼●   ●◼● │
│ 8│  ●◼◼●●◼◼◼◼●  ●◼● │
│ 7│●◼●◼◼●●◼◼◼◼● ●◼◼● │
│ 6│●◼◼◼◼●●◼◼◼◼◼●●◼◼●◼│
│ 5│◼◼◼◼◼●◼◼◼◼◼◼●◼◼◼●◼│
│ 4│◼◼◼◼◼◼◼◼◼◼◼◼●◼◼◼●◼│
│ 3│◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼●◼│
│ 2│◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼│
│ 1│◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼│
└──┴──────────────────┘

Recompute the dilation using the point wise maximum of the set of translated corresponding to the structuring element:

function_dilated = Function(function)
for offset in -1, 1:
    function_dilated.pointwise_max(function.clone().translate(offset))
print function_dilated.plot(plot_type='umbra', mask=function)

Result:

┌──┬──────────────────┐
│11│                  │
│10│      ●◼●         │
│ 9│  ●◼◼●●◼●◼●   ●◼● │
│ 8│  ●◼◼●●◼◼◼◼●  ●◼● │
│ 7│●◼●◼◼●●◼◼◼◼● ●◼◼● │
│ 6│●◼◼◼◼●●◼◼◼◼◼●●◼◼●◼│
│ 5│◼◼◼◼◼●◼◼◼◼◼◼●◼◼◼●◼│
│ 4│◼◼◼◼◼◼◼◼◼◼◼◼●◼◼◼●◼│
│ 3│◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼●◼│
│ 2│◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼│
│ 1│◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼│
└──┴──────────────────┘

Erode by the unit ball:

function_eroded = Function(function)
function_eroded.erode(unit_ball)
print function_eroded.plot(plot_type='umbra', mask=function)

Result:

┌──┬──────────────────┐
│11│                  │
│10│       ◻          │
│ 9│   ◻◻  ◻ ◻     ◻  │
│ 8│   ◻◻  ◻◼◼◻    ◻  │
│ 7│ ◻ ◻◻  ◻◼◼◻   ◻◻  │
│ 6│ ◻◼◼◻  ◻◼◼◼◻  ◻◻ ◻│
│ 5│◼◼◼◼◻ ◻◼◼◼◼◻ ◻◼◻ ◻│
│ 4│◼◼◼◼◼◼◼◼◼◼◼◻ ◻◼◻ ◻│
│ 3│◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◻ ◻│
│ 2│◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼│
│ 1│◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼◼│
└──┴──────────────────┘