How to find inflection point in python?

Question

I have a histogram of an image in RGB which represents the three curves of the three components R, G and B. I want to find the inflection points of each curve. I used the second derivative to find them but I can't, the second derivative does not cancel its returns null. So how can I find the inflection point? Is there any other method to find them?

import os, cv2, random
import numpy as np
import matplotlib.pyplot as plt
import math
from sympy import *

image = cv2.imread('C:/Users/Xers/Desktop/img.jpg')

CHANNELS = ['r', 'g', 'b']

for i, channel in enumerate( CHANNELS ):
    

  histogram = cv2.calcHist([image], [i], None, [256], [0,256])

  histogram = cv2.GaussianBlur( histogram, (5,5), 0)

  plt.plot(histogram, color = channel)

     
  x= plt.xlim([0,256])
  y = plt.ylim([0, 24000])


  derivative1= np.diff(histogram, axis=0)
  derivative2= np.diff(derivative1, axis=0)

  inf_point = np.where ( derivative2 == 0)[0]
  print(inf_point)
plt.show()

Mugwump

you mean inflection point? – Epidemic

Are you asking a maths question? Welcome to SO please take the time to take the tour and readHow to Ask and the other links found on that page. – Youngyoungblood

@Youngyoungblood The question does not seem to be about math, but more about how to put that math into code. – Addia

@Epidemic yes inflection point – Mugwump

derivative2= np.diff(derivative1, axis=0).astype(int) – Transducer

@Transducer i get insignificant points for exemple i get 103 and 184 for the red curve and this in not the inflection point. – Mugwump

Answer 1

There are two issues of numerical nature with your code:

• the data does not seem to be continuous enough to rely on the second derivative computed from two subsequent np.diff() applications
• even if it were, the chances of it being exactly 0 are very slim

To address the first point, you should smooth your histogram (e.g. using a uniform or Gaussian filter on the histogram itself).

To solve the second point, instead of looking for == 0, look for positive-to-negative (and viceversa) switching point.

---

To give you some minimal example of a possible approach:

import numpy as np
import matplotlib.pyplot as plt
from scipy.ndimage import gaussian_filter1d


np.random.seed(0)

# generate noisy data
raw = np.cumsum(np.random.normal(5, 100, 1000))
raw /= np.max(raw)

# smooth
smooth = gaussian_filter1d(raw, 100)

# compute second derivative
smooth_d2 = np.gradient(np.gradient(smooth))

# find switching points
infls = np.where(np.diff(np.sign(smooth_d2)))[0]

# plot results
plt.plot(raw, label='Noisy Data')
plt.plot(smooth, label='Smoothed Data')
plt.plot(smooth_d2 / np.max(smooth_d2), label='Second Derivative (scaled)')
for i, infl in enumerate(infls, 1):
    plt.axvline(x=infl, color='k', label=f'Inflection Point {i}')
plt.legend(bbox_to_anchor=(1.55, 1.0))

Addia

i used a gaussian filter on the histogram using cv2.GaussianBlur and how i can find this switching point please? – Mugwump

@Mugwump Then use a wider sigma, the data is still too noisy. And look for ` >= 0` with non-consecutive indices or something to get the switching point. – Addia

i used the max of gaussian filter is (7,7) and it's still too noisy – Mugwump

@Mugwump I updated the answer with some minimal code to show that the approach I described should work well on sufficiently smoothed data. – Addia

thank you so much i have a question why you use 100 in gaussian_filter1d? – Mugwump

@Mugwump To have enough smoothing. It should be ~2x the size of the noisy peaks you want to overlook – Addia

how i can know the size of the noisy peaks please? – Mugwump

@Mugwump I just looked at the plot. – Addia

Answer 2

I used the code provided by nook2, but scaling the second derivate to a different range (also it will work for any input data, you don't have to change it every time)

import numpy as np
import matplotlib.pyplot as plt
from scipy.ndimage import gaussian_filter1d

np.random.seed(0)

# generate noisy data
raw = np.cumsum(np.random.normal(5, 100, 1000))
raw /= np.max(raw)

# smooth
smooth = gaussian_filter1d(raw, 100)

# compute second derivative
smooth_d2 = np.gradient(np.gradient(smooth))

# find switching points
infls = np.where(np.diff(np.sign(smooth_d2)))[0]

# plot results
plt.plot(raw, label='Noisy Data')
plt.plot(smooth, label='Smoothed Data')
plt.plot(np.max(smooth)*(smooth_d2)/(np.max(smooth_d2)-np.min(smooth_d2)) , label='Second Derivative (scaled)')
for i, infl in enumerate(infls, 1):
    plt.axvline(x=infl, color='k', label=f'Inflection Point {i}')
plt.legend(bbox_to_anchor=(1.55, 1.0))

Pycnometer