Deep Dreams of a CNN trained on MNIST data – III – catching dream patterns at smaller length scales

In the first two articles of this series on Convolutional Neural Networks [CNNs] and "Deep Dream" pictures

Deep Dreams of a CNN trained on MNIST data – II – some code for pattern carving
Deep Dreams of a CNN trained on MNIST data – I – a first approach based on one selected map of a convolutional layer

I introduced the concept of CNN-based "pattern carving" and a related algorithm; it consists of a small number of iterations over of a sequence of image manipulation steps (downscaling to a resolution the CNN can handle, detection and amplification of a map triggering pattern by a CNN, upscaling to the original size and normalization). The included pattern detection and amplification algorithm is nothing else than the OIP-algorithm, which I discussed in another article series on CNNs in this blog. The OIP-algorithm is able to create a pattern, which triggers a chosen CNN map optimally, out of a chaotic fluctuation background. The difference is that we apply such an OIP-algorithms now to a structured input image - the basis for a "dream". The "carving algorithm" is just a simplifying variation of more advanced Deep Dream algorithms; it can be combined with simple CNNs trained on gray and low-resolution images.

In the last article I provided some code which creates dream like pictures with the help of carving by a CNN trained on MNIST data. Nice, but ... In the original theory of "Deep Dreaming" people from Google and others applied their algorithms to a cascade of so called "octaves": Octaves represent the structures within an image at different levels of resolution. Unfortunately, at first sight, such an approach seems to be beyond the capabilities of our MNIST CNN, because the latter works on a fixed and very coarse resolution scale of 28x28 pixels, only.

As we cannot work on different scales of resolution directly: Can we handle pattern detection and amplification on different length scales in a different way? Can we somehow extend the carving process to smaller length scales and a possible detection of smaller map-triggering patterns within the input image?
The answer in short is: Yes, we can.
We "just" have to replace working with "octaves" by looking at sub-segments of the images - and apply our "carving" algorithm with up and down-scaling to these sub-segments as well as to the full image during an iteration process. We shall test the effects of such an approach in this article for a single selected sub-area of the input image, before we apply it more thoroughly to different input images in further articles.

A first step towards Deep Dreams "dreaming" detail structures of a chosen image ...

We again use the image of a bunch of roses, which we worked with in the last articles, as a starting point for the creation of a Deep Dream picture. The easiest way to define sub-areas of our (quadratic) input image is to divide it into 4 (quadratic) adjacent sub-segments or sub-images, each with half of the side length of the original image. We thus get two rows, each with two neighboring sub-areas of the size 280x280 px. Then we could apply the carving algorithm to one selected or to all of the sub-segments. But how do we combine such an approach with an overall treatment of the full image? The answer is that we have to work in parallel on both length-scales involved. We can do this within each cycle of down- and up-scaling according to the following scheme:

  • Step 1: Pick the input image IMG at the full working size (here 560 x 560 px) - and reshape it into a tensor suitable for Tensorflow 2 [TF2] functions.
  • Step 2: Down- and upscale the input image "IMG" (560x560 => 28x28 => 560x560) - with information loss
  • Step 3: Calculate the difference between the re-up-scaled image to the original image as a tensor for later detail corrections.
  • Step 4: Subdivide IMG into 4 quadrants (each with a size of of 280x280 px). Save the quadrants as sub-images S_IMG_1, .... S_IMG_4 - and create respective tensors.
  • Step 5: For all sub-images: Down- and up-scale the sub-images S_IMG_m (280x280 => 28x28 => 280x280) - with information loss.
  • Step 6: For all sub-images: Determine the differences between the re-upscaled sub-images and the original sub-images as tensors for later detail corrections.
  • Step 7: Loop A [around 4 to 6 iterations]
    • Step LA-1: Pick the down-scaled image IMG_d (of size 28x28 px) as the present input image for the CNN and the OIP-analysis
    • Step LA-2: Apply the OIP-algorithm for N epochs (20 < N < 40) to the downscaled image IMG_d (28x28)
    • Step LA-3: Upscale the resulting changed version of image IMG_d to the original IMG- size by bicubic interpolation => IMG_u
    • Step LA-4: Add the (constant) correction tensor for details to IMG_u.
    • (Step LA-5: Loop B over Sub-Images)
      • Step LB-1: Cut out the area corresponding to sub-image S_IMG_m from the changed full image IMG_u. Use it as Sub_IMG_m during the following steps.
      • Step LB-2: Downscale the present sub-image Sub_IMG_m to the size suitable for the CNN (here: 28x28) by bicubic interpolation => Sub_IMG_M_d.
      • Step LB-3: Apply the OIP-algorithm for N epochs (20 < N < 40) to the downscaled sub-image (28x28) Sub_IMG_m_d
      • Step LB-4: Upscale the changed Sub_IMG_m_d to the original size of Sub_Img_m by bicubic interpolation => Sub_IMG_m_u
      • Step LB-5: Add the (constant) correction tensor to the tensor for the upscaled sub-image Sub_IMG_m_u
      • Step LB-6: Replace the sub-image region of the upscaled full image IMG_u with the (corrected) sub-image Sub_IMG_m_u
      • Step LB-7: Downscale the new full image IMG_u again (to 28x28 => IMG_d) and use the resulting IMG_d for the next iteration of Loop A

The Loop B has been placed in brackets because we are going to apply the suggested technique only to a single one of the 4 sub-image quadrants in this article.

Code fragments for the sub-image preparation

The following code for a Jupyter cell prepares the quadrants of the original image IMG - here of a bunch of roses. The code is straight forward and easy to understand.

# ****************************
# Work with sub-Images  
# ************************

%matplotlib inline

import PIL
from PIL import Image, ImageOps

fig_size = plt.rcParams["figure.figsize"]
fig_size[0] = 16
fig_size[1] = 20
fig_3 = plt.figure(3)
ax3_1_1 = fig_3.add_subplot(541)
ax3_1_2 = fig_3.add_subplot(542)
ax3_1_3 = fig_3.add_subplot(543)
ax3_1_4 = fig_3.add_subplot(544)
ax3_2_1 = fig_3.add_subplot(545)
ax3_2_2 = fig_3.add_subplot(546)
ax3_2_3 = fig_3.add_subplot(547)
ax3_2_4 = fig_3.add_subplot(548)
ax3_3_1 = fig_3.add_subplot(5,4,9)
ax3_3_2 = fig_3.add_subplot(5,4,10)
ax3_3_3 = fig_3.add_subplot(5,4,11)
ax3_3_4 = fig_3.add_subplot(5,4,12)
ax3_4_1 = fig_3.add_subplot(5,4,13)
ax3_4_2 = fig_3.add_subplot(5,4,14)
ax3_4_3 = fig_3.add_subplot(5,4,15)
ax3_4_4 = fig_3.add_subplot(5,4,16)
ax3_5_1 = fig_3.add_subplot(5,4,17)
ax3_5_2 = fig_3.add_subplot(5,4,18)
ax3_5_3 = fig_3.add_subplot(5,4,19)
ax3_5_4 = fig_3.add_subplot(5,4,20)

# size to work with 
# ******************
img_wk_size = 560 

# bring the orig img down to (560, 560) 
# ***************************************
imgvc = Image.open("rosen_orig_farbe.jpg")
imgvc_wk_size = imgvc.resize((img_wk_size,img_wk_size), resample=PIL.Image.BICUBIC)
# Change to np arrays 
ay_picc = np.array(imgvc_wk_size)
print("ay_picc.shape = ", ay_picc.shape)
print("r = ", ay_picc[0,0,0],    " g = ", ay_picc[0,0,1],    " b = " , ay_picc[0,0,2] )
print("r = ", ay_picc[200,0,0],  " g = ", ay_picc[200,0,1],  " b = " , ay_picc[200,0,2] )

# Turn color to gray 
#Red * 0.3 + Green * 0.59 + Blue * 0.11
#Red * 0.2126 + Green * 0.7152 + Blue * 0.0722
#Red * 0.299 + Green * 0.587 + Blue * 0.114

ay_picc_g = ( 0.299  * ay_picc[:,:,0] + 0.587  * ay_picc[:,:,1] + 0.114  * ay_picc[:,:,2] )  
ay_picc_g = ay_picc_g.astype('float32') 
t_picc_g  = ay_picc_g.reshape((1, img_wk_size, img_wk_size, 1))
t_picc_g  = tf.image.per_image_standardization(t_picc_g)

# downsize to (28,28)  
t_picc_g_28 = tf.image.resize(t_picc_g, [28,28], method="bicubic", antialias=True)
t_picc_g_28 = tf.image.per_image_standardization(t_picc_g_28)

# get the correction for the full image 
t_picc_g_wk_size = tf.image.resize(t_picc_g_28, [img_wk_size,img_wk_size], method="bicubic", antialias=True)
t_picc_g_wk_size = tf.image.per_image_standardization(t_picc_g_wk_size)

t_picc_g_wk_size_corr = t_picc_g - t_picc_g_wk_size
t_picc_g_wk_size_re   = t_picc_g_wk_size + t_picc_g_wk_size_corr


# Display wk_size orig images  
ax3_1_1.imshow(imgvc_wk_size)
ax3_1_2.imshow(t_picc_g[0,:,:,0], cmap=plt.cm.gray)
ax3_1_3.imshow(t_picc_g_28[0,:,:,0], cmap=plt.cm.gray)
ax3_1_4.imshow(t_picc_g_wk_size_re[0,:,:,0], cmap=plt.cm.gray)


# Split in 4 sub-images 
# ***********************

half_wk_size  = int(img_wk_size / 2)  

t_picc_g_1  = t_picc_g[:, 0:half_wk_size, 0:half_wk_size, :]
t_picc_g_2  = t_picc_g[:, 0:half_wk_size, half_wk_size:img_wk_size, :]
t_picc_g_3  = t_picc_g[:, half_wk_size:img_wk_size, 0:half_wk_size, :]
t_picc_g_4  = t_picc_g[:, half_wk_size:img_wk_size, half_wk_size:img_wk_size, :]

# Display wk_size orig images  
ax3_2_1.imshow(t_picc_g_1[0,:,:,0], cmap=plt.cm.gray)
ax3_2_2.imshow(t_picc_g_2[0,:,:,0], cmap=plt.cm.gray)
ax3_2_3.imshow(t_picc_g_3[0,:,:,0], cmap=plt.cm.gray)
ax3_2_4.imshow(t_picc_g_4[0,:,:,0], cmap=plt.cm.gray)


# Downscale sub-images 
t_picc_g_1_28 = tf.image.resize(t_picc_g_1, [28,28], method="bicubic", antialias=True)
t_picc_g_2_28 = tf.image.resize(t_picc_g_2, [28,28], method="bicubic", antialias=True)
t_picc_g_3_28 = tf.image.resize(t_picc_g_3, [28,28], method="bicubic", antialias=True)
t_picc_g_4_28 = tf.image.resize(t_picc_g_4, [28,28], method="bicubic", antialias=True)


# Display downscales sub-images  
ax3_3_1.imshow(t_picc_g_1_28[0,:,:,0], cmap=plt.cm.gray)
ax3_3_2.imshow(t_picc_g_2_28[0,:,:,0], cmap=plt.cm.gray)
ax3_3_3.imshow(t_picc_g_3_28[0,:,:,0], cmap=plt.cm.gray)
ax3_3_4.imshow(t_picc_g_4_28[0,:,:,0], cmap=plt.cm.gray)


# get correction values for upsizing 
t_picc_g_1_wk_half = tf.image.resize(t_picc_g_1_28, [half_wk_size,half_wk_size], method="bicubic", antialias=True)
t_picc_g_1_wk_half = tf.image.per_image_standardization(t_picc_g_1_wk_half)
t_picc_g_2_wk_half = tf.image.resize(t_picc_g_2_28, [half_wk_size,half_wk_size], method="bicubic", antialias=True)
t_picc_g_2_wk_half = tf.image.per_image_standardization(t_picc_g_2_wk_half)
t_picc_g_3_wk_half = tf.image.resize(t_picc_g_3_28, [half_wk_size,half_wk_size], method="bicubic", antialias=True)
t_picc_g_3_wk_half = tf.image.per_image_standardization(t_picc_g_3_wk_half)
t_picc_g_4_wk_half = tf.image.resize(t_picc_g_4_28, [half_wk_size,half_wk_size], method="bicubic", antialias=True)
t_picc_g_4_wk_half = tf.image.per_image_standardization(t_picc_g_4_wk_half)
 
    
t_picc_g_1_corr =  t_picc_g_1 - t_picc_g_1_wk_half   
t_picc_g_2_corr =  t_picc_g_2 - t_picc_g_2_wk_half   
t_picc_g_3_corr =  t_picc_g_3 - t_picc_g_3_wk_half   
t_picc_g_4_corr =  t_picc_g_4 - t_picc_g_4_wk_half   

t_picc_g_1_re   = t_picc_g_1_wk_half + t_picc_g_1_corr 
t_picc_g_2_re   = t_picc_g_2_wk_half + t_picc_g_2_corr 
t_picc_g_3_re   = t_picc_g_3_wk_half + t_picc_g_3_corr 
t_picc_g_4_re   = t_picc_g_4_wk_half + t_picc_g_4_corr 

print(t_picc_g_1_re.shape)

# Display downscales sub-images  
ax3_4_1.imshow(t_picc_g_1_re[0,:,:,0], cmap=plt.cm.gray)
ax3_4_2.imshow(t_picc_g_2_re[0,:,:,0], cmap=plt.cm.gray)
ax3_4_3.imshow(t_picc_g_3_re[0,:,:,0], cmap=plt.cm.gray)
ax3_4_4.imshow(t_picc_g_4_re[0,:,:,0], cmap=plt.cm.gray)

ay_img_comp = np.zeros((img_wk_size,img_wk_size))
ay_t_comp   = ay_img_comp.reshape((1,img_wk_size, img_wk_size,1))
t_pic_comp  = ay_t_comp
t_pic_comp[0, 0:half_wk_size, 0:half_wk_size, 0]                       = t_picc_g_1_re[0, :, :, 0]
t_pic_comp[0, 0:half_wk_size, half_wk_size:img_wk_size, 0]             = t_picc_g_2_re[0, :, :, 0]
t_pic_comp[0, half_wk_size:img_wk_size, 0:half_wk_size, 0]             = t_picc_g_3_re[0, :, :, 0]
t_pic_comp[0, half_wk_size:img_wk_size, half_wk_size:img_wk_size, 0]   = t_picc_g_4_re[0, :, :, 0]

ax3_5_1.imshow(t_picc_g[0,:,:,0], cmap=plt.cm.gray)
ax3_5_2.imshow(t_pic_comp[0,:,:,0], cmap=plt.cm.gray)

 
Note the computation of the correction tensors; we prove their effectiveness below by displaying respective results for the full image, cut out and re-upscaled, corrected quadrants and replaced areas of the original image.

The results in the prepared image frames look like:

Good!

Some code for the carving algorithm - applied to the full image AND a single selected sub-image

Now, for a first test, we concentrate on the sub-image in the lower-right corner - and apply the above algorithm. A corresponding Jupyter cell code is given below:

# *************************************************************************
# OIP analysis on sub-image tiles (to be used after the previous cell)
# **************************************************************************
# Note: To be applied after previous cell !!!
# ******

#interactive plotting - will be used in a future version when we deal with "octaves" 
#%matplotlib notebook 
#plt.ion()

%matplotlib inline

# preparation of figure frames 
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
fig_size = plt.rcParams["figure.figsize"]
fig_size[0] = 12
fig_size[1] = 12
fig_4 = plt.figure(4)
ax4_1_1 = fig_4.add_subplot(441)
ax4_1_2 = fig_4.add_subplot(442)
ax4_1_3 = fig_4.add_subplot(443)
ax4_1_4 = fig_4.add_subplot(444)

ax4_2_1 = fig_4.add_subplot(445)
ax4_2_2 = fig_4.add_subplot(446)
ax4_2_3 = fig_4.add_subplot(447)
ax4_2_4 = fig_4.add_subplot(448)

ax4_3_1 = fig_4.add_subplot(4,4,9)
ax4_3_2 = fig_4.add_subplot(4,4,10)
ax4_3_3 = fig_4.add_subplot(4,4,11)
ax4_3_4 = fig_4.add_subplot(4,4,12)

ax4_4_1 = fig_4.add_subplot(4,4,13)
ax4_4_2 = fig_4.add_subplot(4,4,14)
ax4_4_3 = fig_4.add_subplot(4,4,15)
ax4_4_4 = fig_4.add_subplot(4,4,16)

fig_size = plt.rcParams["figure.figsize"]
fig_size[0] = 12
fig_size[1] = 6
fig_5 = plt.figure(5)
axa5_1 = fig_5.add_subplot(241)
axa5_2 = fig_5.add_subplot(242)
axa5_3 = fig_5.add_subplot(243)
axa5_4 = fig_5.add_subplot(244)
axa5_5 = fig_5.add_subplot(245)
axa5_6 = fig_5.add_subplot(246)
axa5_7 = fig_5.add_subplot(247)
axa5_8 = fig_5.add_subplot(248)
li_axa5 = [axa5_1, axa5_2, axa5_3, axa5_4, axa5_5, axa5_6, axa5_7, axa5_8]

# Some general OIP run parameters 
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
n_epochs  = 30          # should be divisible by 5  
n_steps   = 6           # number of intermediate reports 
epsilon   = 0.01        # step size for gradient correction  
conv_criterion = 2.e-4  # criterion for a potential stop of optimization 

# image width parameters
# ~~~~~~~~~~~~~~~~~~~~~
img_wk_size   = 560    # must be identical to the last cell   
half_wk_size  = int(img_wk_size / 2) 

# min / max values of the input image 
#~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
# Note : to be used for contrast enhancement 
min1 = tf.reduce_min(t_picc_g)
max1 = tf.reduce_max(t_picc_g)
# parameters to deal with the spectral distribution of gray values
spect_dist = min(abs(min1), abs(max1))
spect_fact   = 0.85
height_fact  = 1.1

# Set the gray downscaled input image as a startng point for the iteration loop 
# ~~~~~~~~~~~~~----------~~~~--------------~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
MyOIP._initial_inp_img_data = t_picc_g_28

# ************
# Main Loop 
# ***********

num_main_iterate = 4
# map_index_main = -1       # take a cost function for a full layer 
map_index_main = 56         # map-index to be used for the OIP analysis of the full image  
map_index_sub  = 56         # map-index to be used for the OIP analysis of the sub-images   


for j in range(num_main_iterate):
   
    # ******************************************************
    # deal with the full (downscaled) input image 
    # ******************************************************
    
    # Apply OIP-algorithm to the whole downscaled image  
    # ~~~~~~~~~~~~~~~~~~~-----------~~~~~~~~~~~~~~~~~~~~~~~
    map_index_main_oip = map_index_main   # map-index we are interested in 

    # Perform the OIP analysis 
    MyOIP._derive_OIP(map_index = map_index_oip, 
                      n_epochs = n_epochs, n_steps = n_steps, 
                      epsilon = epsilon , conv_criterion = conv_criterion, 
                      b_stop_with_convergence=False,
                      b_print=False,
                      li_axa = li_axa5,
                      ax1_1 = ax4_1_1, ax1_2 = ax4_1_2)

    # display the modified downscaled image  (without and with contrast)
    # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    t_oip_g_28  = MyOIP._inp_img_data
    ay_oip_g_28 = t_oip_g_28[0,:,:,0].numpy()
    ay_oip_g_28_cont = MyOIP._transform_tensor_to_img(T_img=t_oip_g_28[0,:,:,0], centre_move=0.33, fact=1.0)
    ax4_1_3.imshow(ay_oip_g_28_cont, cmap=plt.cm.gray)
    ax4_1_4.imshow(t_picc_g_28[0, :, :, 0], cmap=plt.cm.gray)
    
    # rescale to 560 and re-add details via the correction tensor  
    # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    t_oip_g_wk_size        = tf.image.resize(t_oip_g_28, [img_wk_size,img_wk_size], 
                                             method="bicubic", antialias=True)
    t_oip_g_wk_size_re     = t_oip_g_wk_size + t_picc_g_wk_size_corr 
    
    # standardize to get an intermediate image 
    # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    # t_oip_g_loop           = t_oip_g_wk_size_re.numpy()
    # t_oip_g_wk_size_re_std = tf.image.per_image_standardization(t_oip_g_wk_size_re)
    t_oip_g_wk_size_re_std = tf.image.per_image_standardization(t_oip_g_wk_size_re)
    t_oip_g_loop           = t_oip_g_wk_size_re_std.numpy()
    
    # contrast required as the reduction of irrelvant pixels has smoothed out the gray scale beneath 
    # ~~~~~~~~~~~~~~~~~
    # the added details at pattern areas = high level of whitened socket + relative small detail variation    
    t_oip_g_wk_size_re_std_plt = tf.clip_by_value(height_fact*t_oip_g_wk_size_re_std, -spect_dist*spect_fact, spect_dist*spect_fact)
    ax4_2_1.imshow(t_oip_g_wk_size[0,:,:,0], cmap=plt.cm.gray)
    ax4_2_2.imshow(t_oip_g_wk_size_re_std[0,:,:,0], cmap=plt.cm.gray)
    ax4_2_3.imshow(t_oip_g_wk_size_re_std_plt[0,:,:,0], cmap=plt.cm.gray)
    ax4_2_4.imshow(t_picc_g[0,:,:,0], cmap=plt.cm.gray)
   

    # *************************************************************
    # deal with a chosen sub-image - in this version: "t_oip_4_g"
    # **************************************************************
   
    # Cut out and downscale a sub-image region 
    # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    t_oip_4_g     = t_oip_g_loop[:, half_wk_size:img_wk_size, half_wk_size:img_wk_size, :]
    t_oip_4_g_28  = tf.image.resize(t_oip_4_g, [28,28], method="bicubic", antialias=True)
    
    # use the downscaled sub-image as an input image to the OIP-algorithm 
    # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    MyOIP._initial_inp_img_data = t_oip_4_g_28
    
    
    # Perform the OIP analysis on the sub-image
    # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    map_index_sub_oip  = map_index_sub   # we may vary this in later versions 
    
    MyOIP._derive_OIP(map_index = map_index_sub_oip, 
                      n_epochs = n_epochs, n_steps = n_steps, 
                      epsilon = epsilon , conv_criterion = conv_criterion, 
                      b_stop_with_convergence=False,
                      b_print=False,
                      li_axa = li_axa5,
                      ax1_1 = ax4_3_1, ax1_2 = ax4_3_2)
    
    
    # display the modified sub-image without and with contrast  
    # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    t_oip_4_g_28  = MyOIP._inp_img_data
    ay_oip_4_g_28 = t_oip_4_g_28[0,:,:,0].numpy()
    ay_oip_4_g_28_cont = MyOIP._transform_tensor_to_img(T_img=t_oip_4_g_28[0,:,:,0], centre_move=0.33, fact=1.0)
    ax4_3_3.imshow(ay_oip_4_g_28_cont, cmap=plt.cm.gray)
    ax4_3_4.imshow(t_oip_4_g[0, :, :, 0], cmap=plt.cm.gray)
    
    # upscaling of the present sub-image 
    # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    t_pic_4_g_wk_half = tf.image.resize(t_oip_4_g_28, [half_wk_size,half_wk_size], method="bicubic", antialias=True)
    
    # add the detail correction to the manipulated upscaled sub-image
    # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    t_pic_4_g_wk_half = t_pic_4_g_wk_half + t_picc_g_4_corr
    #t_pic_4_g_wk_half = tf.image.per_image_standardization(t_pic_4_g_wk_half)
    ax4_4_1.imshow(t_pic_4_g_wk_half[0, :, :, 0], cmap=plt.cm.gray)
    ax4_4_2.imshow(t_oip_4_g[0, :, :, 0], cmap=plt.cm.gray)
    
    # Overwrite the related region in the full image with the new sub-image  
    # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    t_oip_g_loop[0, half_wk_size:img_wk_size, half_wk_size:img_wk_size, 0]  = t_pic_4_g_wk_half[0, :, :, 0]
    t_oip_g_loop_std = tf.image.per_image_standardization(t_oip_g_loop)
    
    #ax4_4_3.imshow(t_oip_g_loop[0, :, :, 0], cmap=plt.cm.gray)   
    ax4_4_3.imshow(t_oip_g_loop_std[0, :, :, 0], cmap=plt.cm.gray)   
    ay_oip_g_loop_std_plt = tf.clip_by_value(height_fact*t_oip_g_loop_std, -spect_dist*spect_fact, spect_dist*spect_fact)
    ax4_4_4.imshow(ay_oip_g_loop_std_plt[0, :, :, 0], cmap=plt.cm.gray)
    
    # Downscale the resulting new full image and feed it into into the next iteration of the main loop
    # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    MyOIP._initial_inp_img_data = tf.image.resize(t_oip_g_loop_std, [28,28], method="bicubic", antialias=True)
   

 

The result of these operations after 4 iterations is displayed in the following image:

We recognize again the worm-like shape resulting for map 56, which we have seen in the last article, already. (By the way: Our CNN's map 56 originally is strongly triggered for the shapes of handwritten 9-digits and plays a significant role in classifying respective images correctly). But there is a new feature which appeared in the lower right part of the image - a kind of wheel or two closely neighbored 9-like shapes.

The following code compares the final result with the original image:

fig_size = plt.rcParams["figure.figsize"]
fig_size[0] = 12
fig_size[1] = 12
fig_7X = plt.figure(7)
ax1_7X_1 = fig_7X.add_subplot(221)
ax1_7X_2 = fig_7X.add_subplot(222)
ax1_7X_3 = fig_7X.add_subplot(223)
ax1_7X_4 = fig_7X.add_subplot(224)

ay_pic_full_cont = MyOIP._transform_tensor_to_img(T_img=t_oip_g_loop_std[0, :, :, 0], centre_move=0.46, fact=0.8)

ax1_7X_1.imshow(imgvc_wk_size)
ax1_7X_2.imshow(t_picc_g[0,:,:,0], cmap=plt.cm.gray)
ax1_7X_3.imshow(ay_oip_g_loop_std_plt[0, :, :, 0], cmap=plt.cm.gray)
ax1_7X_4.imshow(ay_pic_full_cont, cmap=plt.cm.gray)

Giving:

Those who do not like the relative strong contrast may do their own experiments with other parameters for contrast enhancement.

You see: Simplifying prejudices about the world, which get or got manifested in inflexible neural networks (or brains ?), may lead to strange, if not bad dreams. May also some politicians learn from this ... hmmm, just joking, as this requires both an openness and capacity for learning ... can't await Jan, 21st ...

Conclusion

Our "carving algorithm", which corresponds to an amplification of traces of map-related patterns that a CNN detects in an input image, can also be used for sub-image-areas and their pixel information. We can therefore use even very limited CNNs trained on low resolution MNIST images to create a "Deep MNIST Dream" out of a chosen arbitrary input image. At least in principle.

Addendum 06.01.2021: There is, however, a pitfall - our CNN (for MNIST or other image samples) may have been trained and work on standardized image/pixel data, only. But even if the overall input image may have been standardized before operating on it in the sense of the algorithm discussed in this article, we should take into account that a sub-image area may contain pixel data which are far from a standardized distribution. So, our corrections for details may require more than just the calculation of a difference term. We may have to reverse intermediate standardization steps, too. We shall take care of this in forthcoming code changes.

In the next article we are going to extend our algorithm to multiple cascaded sub-areas of our image - and we vary the maps used for different sub-areas at the same time.

 

A simple CNN for the MNIST dataset – VIII – filters and features – Python code to visualize patterns which activate a map strongly

Our series about a simple CNN trained on the MNIST data turns back to some coding.

A simple CNN for the MNIST dataset – VII – outline of steps to visualize image patterns which trigger filter maps
A simple CNN for the MNIST dataset – VI – classification by activation patterns and the role of the CNN’s MLP part
A simple CNN for the MNIST dataset – V – about the difference of activation patterns and features
A simple CNN for the MNIST dataset – IV – Visualizing the activation output of convolutional layers and maps
A simple CNN for the MNIST dataset – III – inclusion of a learning-rate scheduler, momentum and a L2-regularizer
A simple CNN for the MNIST datasets – II – building the CNN with Keras and a first test
A simple CNN for the MNIST datasets – I – CNN basics

In the last article I discussed an optimization algorithm which should be able to create images of pixel patterns which trigger selected feature maps of a CNN strongly. In this article we shall focus on the required code elements. I again want to emphasize that I apply and modify some basic ideas which I read in a book of F. Chollet and in a contribution of a guy called Mohamed to a discussion at kaggle.com (see my last article for references). A careful reader will notice differences not only with respect to coding; there are major differences regarding the normalization of intermediate data and the construction of input images. To motivate the latter point I first want to point out that OIPs are idealized technical abstractions and that not all maps may react to purely statistical data on short length scales.

Images of OIPs which trigger feature maps are artificial technical abstractions!

In the last articles I made an explicit distinction between two types of patterns which we can analyze in the context of CNNs:

  • FCP: A pattern which emerges within and across activation maps of a chosen (deep) convolutional layer due to filter operations which the CNN applied to a specific input image.
  • OIP: A pattern which is present within the pixel value distribution of an input image and to which a CNN map reacts strongly.

Regarding OIPs there are some points which we should keep in mind:

  • We do something artificial when we create an image of an elementary OIP pattern to which a chosen feature map reacts. Such an OIP is already an abstraction in the sense that it reflects an idealized pattern - i.e. a specific geometrical correlation between pixel values of an input image which passes a very specific filter combinations. We forget about all other figurative elements of the input image which may trigger other maps.
  • There is an additional subtle point regarding our present approach to OIP-visualization:
    Our algorithm - if it works - will lead to OIP images which trigger a map's neurons maximally on average. What does "average" mean with respect to the image area? A map always covers the whole input image area. Now let us assume that a filter combination of lower layers reacts to an elementary pattern limited in size and located somewhere on the input image. But some filters or filter combinations may be capable of detecting such a pattern at multiple locations of an input image.
    One example would be a crossing of two relatively thin lines. Such a crossing could appear at many places in an input image. In fact, a trained CNN has seen several thousand images of handwritten "4"s where the crossing of the horizontal and the vertical line actually appeared at many different locations and may have learned about this when establishing filters. Under such conditions it is obvious that a map gets optimally activated if a relatively small elementary pattern appears multiple times within the image which our algorithm artificially constructs out of initial random data.
    So our algorithm will with a high probability lead to OIP images which consist of a combination or a superposition of elementary patterns at multiple locations. In this sense an OIP image constructed due to the rule of a maximum average activation is another type of idealization. In a real MNIST image the re-occurrence of elementary patterns may not be present at all. Therefore, we should be careful and not directly associate the visualization of a pattern created by our algorithm with an elementary OIP or "feature".
  • The last argument can in some cases also be reverted: There may be unique large scale patterns which can only be detected by filters of higher (i.e. deeper) convolutional levels which filter coarsely and with respect to relatively large areas of the image. In our approach such unique patterns may only appear in OIP images constructed for maps of the deepest convolutional layer.

Independence of the statistical data of the input image?

In the last article I showed you already some interesting patterns for certain maps which emerged from randomly varying pixel values in an input image. The fluctuations were forced into patterns by the discussed optimization loop. An example of the resulting evolution of the pixel values is shown below: With more and more steps of the optimization loop an OIP-pattern emerges out of the initial "chaos".

Images were taken at optimization steps 0, 9, 18, 36, 72, 144, 288, 599 of a loop. Convergence is given by a change of the loss values between two different steps divided by the present loss value :
3.6/41 => 3.9/76 => 3.3/143 => 2.3/240 => 0.8/346 => 0.15/398 => 0.03/418

As we work with gradient ascent in OIP detection a lower loss means a lower activation of the map.

If we change the wavelength of the initial input fluctuations we get a somewhat, though not fundamentally different result (actually with a lower loss value of 381):

This gives us confidence in the general usability of the method. However, I would like to point out that during your own experiments you may also experience the contrary:

For some maps and for some initial statistical input data varying at short length scales, only, the optimization process will not converge. It will not even start to do so. Instead you may experience a zero activation of the selected map during all steps of the optimization for a given random input.

You should not be too surprised by this fact. Our CNN was optimized to react to patterns present in written digits. As digits have specific elements (features?) as straight lines, bows, circles, line-crossings, etc., we should expect that not all input will trigger the activation of a selected map which reacts on pixel value variations at relatively large length scales. Therefore, it is helpful to be able to vary the statistical input pattern at different length scales when you start your hunt for a nice visualization of an OIP and/or elementary feature.

All in all we cannot exclude a dependency on the statistical initial input image fluctuations. Our algorithm will find a maximum with respect to the specific input data fluctuations presented to him. Due to this point we should always be aware of the fact that the visualizations produced by our algorithm will probably mark a local maximum in the multidimensional parameter or representation space - not a global one. But also a local maximum may reveal significant sub-structures a specific filter combination is sensitive to.

Libraries

To build a suitable code we need to import some libraries, which you first must install into your virtual Python environment:

  
import numpy as np
import scipy
import time 
import sys 
import math

from sklearn.preprocessing import StandardScaler
import tensorflow as tf
from tensorflow import keras as K
from tensorflow.python.keras import backend as B 
from tensorflow.keras import models
from tensorflow.keras import layers
from tensorflow.keras import regularizers
from tensorflow.keras import optimizers
from tensorflow.keras.optimizers import schedules
from tensorflow.keras.utils import to_categorical
from tensorflow.keras.datasets import mnist

from tensorflow.python.client import device_lib

import matplotlib as mpl
from matplotlib import pyplot as plt
from matplotlib.colors import ListedColormap
import matplotlib.patches as mpat 

import os 
from os import path as path

 

Basic code elements to construct OIP patterns from statistical input image data

To develop some code for OIP visualizations we follow the outline of steps discussed in the last article. To encapsulate the required functionality we set up a Python class. The following code fragment gives an overview about some variables which we are going to use. In the comments and explanations I sometimes used the word "reference" to mark a difference between

  • addresses to some intermediate Tensorflow 2 [TF2] objects handled by a "watch()"-method of TF2's GradientTape-object for eager calculation of gradients
  • and eventual return objects (of a function based on keras' backend) filled with values which we can directly use during the optimization iterations for corrections.

This is only a reminder of complex internal TF2 background operations based on complex layer models; I do not want to stress any difference in the sense of pointers and objects. Actually, I do not even know the precise programming patterns used behind TF2's watch()-mechanism; but according to the documentation it basically records all operations involving the "watched" objects. The objective is the ability to produce gradient values of defined functions with respect to any variable changes instantaneously in eager execution later on.

 
# class to produce images of OIPs for a chosen CNN-map
# *******************************************************
class My_OIP:
    '''
    Version 0.2, 01.09.2020
    ~~~~~~~~~~~~~~~~~~~~~~~~~
    This class allows for the creation and the display of OIP-patterns 
    to which a selected map of a CNN-model reacts   
    
    Functions:
    ~~~~~~~~~~
    1) _load_cnn_model()             => load cnn-model
    2) _build_oip_model()            => build an oip-model to create OIP-images
    3) _setup_gradient_tape()        => Implements TF2 GradientTape to watch input data
                                        for eager gradient calculation
    4) _oip_strat_0_optimization_loop():
                                     => Method implementing a simple strategy to create OIP-images 
                                        based on superposition of random data on large distance scales
    5) _oip_strat_1_optimization_loop():
       (NOT YET DEVELOPED)           => Method implementing a complex strategy to create OIP-images 
                                        based on partially evolved intermediate image 
                                        getting enriched by small scale fluctuations
    6) _derive_OIP():                => Method used externally to start the creation of 
                                        an OIP for a chosen map 
    7) _build_initial_img_data()     => Method to construct an initial image based on 
                                        a superposition by random date on 4 different length scales 
    
    
    Requirements / Preconditions:
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
    In the present version 
    * a CNN-model is assumed which works with standardized (!) input images,
    * a CNN-Modell trained on MNIST data is assumed ,
    * exactly 4 length scales for random data fluctations are used
      to compose initial statistical image data 
      (roughly with a factor of 2 between them) 
      
    
    '''
    def __init__(self, cnn_model_file = 'cnn_best.h5', 
                 layer_name = 'Conv2D_3', map_index = 0, 
                 img_dim = 28, 
                 b_build_oip_model = True  
                ): 
        '''
        Input: 
        ~~~~~~
            cnn_model_file:     Name of a file containing  a full CNN-model
                                can later be overwritten by _load_cnn_model()
            layer_name:         We can define a layer name we are interested in already when starting; s. below
            map_index:          We may define a map we are interested in already when starting; s. below
            img_dim:            The dimension of the assumed quadratic images 
        
        Major internal variables:
        **************************
            _cnn_model:             A reference to the CNN model object
            _layer_name:            The name of convolutional layer 
                                    (can be overwritten by method _build_oip_model() ) 
            _map_index:             index of the map in the layer's output array 
                                    (can later be overwritten by other methods) 
            _r_cnn_inputs:          A reference to the input tensor of the CNN model (here: 1 image - NOT a batch of images)
            _layer_output:          Tensor with all maps of a certain layer
           
            _oip_submodel:          A new model connecting the input of the cnn-model with a certain map
            
            _tape:                  An instance of TF2's GradientTape-object 
                                    Watches input, output, loss of a model 
                                    and calculates gradients in TF2 eager mode 
            _r_oip_outputs:         A reference to the output of the new OIP-model 
            _r_oip_grads:           Reference to gradient tensors for the new OIP-model 
            _r_oip_loss:            Reference to a loss defined by operations on the OIP-output  
            _val_oip_loss:          Reference to a loss defined by operations on the OIP-output  
            
            _iterate:               Keras backend function to invoke the new OIP-model 
                                    and calculate both loss and gradient values (in TF2 eager mode) 
                                    This is the function to be used in the optimization loop for OIPs
            
            Parameters controlling the optimization loop:
            ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
            _oip_strategy:          0, 1 - There are two strategies to evolve OIP patterns out of statistical data - only the first strategy is supported in this version  
                                    0: Simple superposition of fluctuations at different length scales
                                    1: Evolution over partially evolved images based on longer scale variations 
                                       enriched with fluctuations on shorter length scales 
                                    Both strategies can be combined with a precursor calculation 
            
            
            _b_oip_precursor:       True/False - Use a precursor analysis of long range variations 
                                    regarding loss => search for optimum variations for a given map
                                    (Some initial input images do not trigger a map at all or 
                                    sub-optimally => We test out multiple initial fluctuation patterns). 
            
            _ay_epochs:             A list of 4 optimization epochs to be used whilst 
                                    evolving the img data via strategy 1 and intermediate images 
            _n_epochs:              Number of optimization epochs to be used with strategy 0 
            
            _n_steps:               Defines at how many intermediate points we show images and report 
                                    on the optimization process 
            
            _epsilon:               Factor to control the amount of correction imposed by 
                                    the gradient values of the OIP-model 
            
            Input image data of the OIP-model and references to it 
            ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
            _initial_precursor_img  The initial image to start a precursor optimization with.
                                    Would normally be an image of only long range fluctuations. 
            _precursor_image:       The evolved image updated during the precursor loop 
            
            _initial_inp_img_data:  A tensor representing the data of the input image 
            _inp_img_data:          A tensor representing the data of the input img during optimization  
            _img_dim:               We assume quadratic images to work with 
                                    with dimension _img_dim along each axis
                                    For the time being we only support MNIST images 
            
            Parameters controlling the composition of random initial image data 
            ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
            _li_dim_steps:          A list of the intermediate dimensions for random data;
                                    these data are smoothly scaled to the image dimensions 
            _ay_facts:              A Numpy array of 4 factors to control the amount of 
                                    contribution of the statistical variations 
                                    on the 4 length scales to the initial image
                                   
        '''    
        
        # Input data and variable initializations
        # ****************************************
        
        # the model 
        self._cnn_model_file = cnn_model_file
        self._cnn_model      = None 
        
        # the layer 
        self._layer_name = layer_name
        # the map 
        self._map_index  = map_index
        
        # References to objects and the OIP sub-model
        # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
        self._r_cnn_inputs  = None # reference to input of the CNN_model
                                   # also used in the OIP-model  
        
        self._layer_output  = None
        self._oip_submodel  = None
        
        self._tape          = None # TF2 GradientTape variable
        # some "references"
        self._r_oip_outputs = None # output of the OIP-submodel to be watched 
        self._r_oip_grads   = None # gradients determined by GradientTape   
        self._r_oip_loss    = None 
        # respective values
        self._val_oip_grads = None
        self._val_oip_loss  = None
        
        # The Keras function to produce concrete outputs of the new OIP-model  
        self._iterate       = None
        
        # The strategy to produce an OIP pattern out of statistical input images 
        # ~~~~~~~~~~~~~~~~~~~~~~~~~--------~~~~~~
        # 0: Simple superposition of fluctuations at different length scales 
        # 1: Move over 4 interediate images - partially optimized 
        self._oip_strategy = 0
        
        # Parameters controlling the OIP-optimization process 
        # ~~~~~~~~~~~~~~~~~~~~~~~~~--------~~~~~~
        # Use a precursor analysis ? 
        self._b_oip_precursor = False
       
        # number of epochs for optimization strategy 1
        self._ay_epochs    = np.array((20, 40, 80, 400))
        len_epochs         = len(self._ay_epochs)

        # number of epochs for optimization strategy 0
        self._n_epochs     = self._ay_epochs[len_epochs-1]   
        self._n_steps      = 7   # divides the number of n_epochs into n_steps 
                                 # to produce intermediate outputs
        
        # size of corrections by gradients
        self._epsilon       = 0.01 # step-size for gradient correction
        
        
        # Input images and references to it 
        # ~~~~~~~~
        # precursor image
        self._initial_precursor_img = None
        self._precursor_img         = None
        # The evetually used input image - a superposition of initial random fluctuations
        self._initial_inp_img_data  = None  # The initial data constructed 
        self._inp_img_data          = None  # The data used and varied for optimization 
        # image dimension
        self._img_dim               = img_dim   # = 28 => MNIST images for the time being 
        
        
        # Parameters controlling the setup of an initial image 
        # ~~~~~~~~~~~~~~~~~~~~~~~~~--------~~~~~~~~~~~~~~~~~~~
        # The length scales for initial input fluctuations
        self._li_dim_steps = ( (3, 3), (7,7), (14,14), (28,28) )
        # Parameters for fluctuations  - used both in strategy 0 and strategy 1  
        self._ay_facts     = np.array( (0.5, 0.5, 0.5, 0.5) )
        
        
        # ********************************************************
        # Model setup - load the cnn-model and build the oip-model
        # ************
        
        if path.isfile(self._cnn_model_file): 
            # We trigger the initial load of a model 
            self._load_cnn_model(file_of_cnn_model = self._cnn_model_file, b_print_cnn_model = True)
            # We trigger the build of a new sub-model based on the CNN model used for OIP search 
            self._build_oip_model(layer_name = self._layer_name, b_print_oip_model = True ) 
        else:
            print("<\nWarning: The standard file " + self._cnn_model_file + 
                  " for the cnn-model could not be found!\n " + 
                  " Please use method _load_cnn_model() to load a valid model")
            
        return
 

 
The purpose of most of the variables will become clearer as we walk though the class's methods below.

Loading the original trained CNN model

Let us say we have a trained CNN-model with all final weight parameters for node-connections saved in some h5-file (see the 4th article of this series for more info). We then can load the CNN-model and derive sub-models from its layer elements. The following method performs the loading task for us:

 
    #
    # Method to load a specific CNN model
    # **********************************
    def _load_cnn_model(self, file_of_cnn_model='cnn_best.h5', b_print_cnn_model=True ):
        
        # Check existence of the file
        if not path.isfile(self._cnn_model_file): 
            print("<\nWarning: The file " + file_of_cnn_model + 
                  " for the cnn-model could not be found!\n" + 
                  "Please change the parameter \"file_of_cnn_model\"" + 
                  " to load a valid model")

        # load the CNN model 
        self._cnn_model_file = file_of_cnn_model
        self._cnn_model = models.load_model(self._cnn_model_file)

        # Inform about the model and its file file 7
        # ~~~~~~~~~~~~~~~~~~~~~~~~~~~
        print("Used file to load a ´ model = ", self._cnn_model_file)
        # we print out the models structure
        if b_print_cnn_model:
            print("Structure of the loaded CNN-model:\n")
            self._cnn_model.summary()

        # handle/references to the models input => more precise the input image 
        # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
        #    Note: As we only have one image instead of a batch 
        #    we pick only the first tensor element!
        #    The inputs will be needed for buildng the oip-model 
        self._r_cnn_inputs = self._cnn_model.inputs[0]  # !!! We have a btach with just ONE image 
        
        # print out the shape - it should be known fro the original cnn-model
        print("shape of cnn-model inputs = ", self._r_cnn_inputs.shape)
        
        return

 
Actually, I used this function already in the class' "__init__()"-method - provided the standard file for the last CNN exists. (In a more advanced version you would in addition check that the name of the standard CNN-model meets your expectations.)

The code is straightforward. You see that the structure of the original CNN-model is printed out if requested by the user.

Note also that we assigned the first element of the input tensor of the CNN-model, i.e. a single image tensor, to the variable "self._r_cnn_inputs". This tensor will become a major ingredient in a new Keras model which we are going to build in a minute and which we shall use to calculate gradient components of a loss function with respect to all pixel values of the input image. The gradient's component values will in turn be used during gradient ascent to correct the pixel values. Repeated corrections should lead to a systematic approach of a maximum of our loss function, which describes the map's activation. (Remember: Such a maximum may depend on the input image fluctuations).

Build a new Keras model based on the input tensor and a chosen layer

The next method is more interesting. We need to build a new Keras layer "model" based on the input layer and a deeper layer of the original CNN-model. (We already used the same kind of "trick" when we tried to visualize the activation output of a convolutional layer of the CNN.)

 
    #
    # Method to construct a model to optimize input for OIP-detection 
    # ***************************************
    def _build_oip_model(self, layer_name = 'Conv2D_3', b_print_oip_model=True ): 
        '''
        We need a Conv layer to build a working model for input image optimization. 
        We get the Conv layer by the layer's name. 
        The new model connects the first input element of the CNN to 
        the output maps of the named Conv layer CNN 
        '''
        # free some RAM - hopefully 
        del self._oip_submodel
        
        self._layer_name = layer_name
        if self._cnn_model == None: 
            print("Error: cnn_model not yet defined.")
            sys.exit()
        # We build a new model based ion the model inputs and the output 
        self._layer_output = self._cnn_model.get_layer(self._layer_name).output
        
        # We do not acre at the moment about the composition of the input 
        # We trust in that we handle only one image - and not a batch
        model_name = "mod_oip__" + layer_name 
        self._oip_submodel = models.Model( [self._r_cnn_inputs], [self._layer_output], name = model_name)                                    
        
        # We print out the oip model structure
        if b_print_oip_model:
            print("Structure of the constructed OIP-sub-model:\n")
            self._oip_submodel.summary()
        return

 
We use the tensor for a single input image and the output of layer (= a collection of maps) of the original CNN-model as the definition elements of the new Keras model.

TF2's GradientTape() - watch the change of variables which have an impact on model gradients and the loss function

What do we have so far? We have defined a new Keras model connecting input and output data of a layer of the original model. TF2 can determine related gradients by the node connections defined in the original model. However, we cannot rely on a graph analysis by Tensorflow as we were used to with TF1. TF2 uses eager mode - i.e. it calculates gradients directly. What does "directly" mean? Well - as soon as changes to variables occur which have an impact on the gradient values. This in turn means that "something" has to watch out for such changes. TF2 offers a special object for this purpose: tf.GradientTape. See:
https://www.tensorflow.org/guide/eager
https://www.tensorflow.org / api_docs/python/tf/GradientTape

So, as a next step, we set up a method to take care of "GradientTape()".

    #
    # Method to watch gradients 
    # *************************
    def _setup_gradient_tape(self):
        '''
        For TF2 eager execution we need to watch input changes and trigger gradient evaluation
        '''   
        # Working with TF2 GradientTape
        self._tape = None
        
        # Watch out for input, output variables with respect to gradient chnages
        with tf.GradientTape() as self._tape: 
            # Input
            # ~~~~~~~
            self._tape.watch(self._r_cnn_inputs)
            # Output 
            # ~~~~~~~
            self._r_oip_outputs = self._oip_submodel(self._r_cnn_inputs)
            
            # Loss 
            # ~~~~~~~
            self._r_oip_loss = tf.reduce_mean(self._r_oip_outputs[0, :, :, self._map_index])
            print(self._r_oip_loss)
            print("shape of oip_loss = ", self._r_oip_loss.shape)

 
Note that the loss definition included in the code fragment is specific for a chosen map. This implies that we have to call this method every time we chose a different map for which we want to create OIP visualizations.

The advantage of the above code element is that "_tape()" can produce gradient values for the relation of the input data and loss data of a model automatically whenever we change the input image data. Gradient values can be called by

  self._r_oip_grads  = self._tape.gradient(self._r_oip_loss, self._r_cnn_inputs)

Gradient ascent

As already discussed in my last article we apply a gradient ascent method to our "loss" function whose outcome rises with the activation of the neurons of a map. The following code sets up a method which first calls "_setup_gradient_tape()" and afterwards applies a normalization to the gradient values which "_tape()" produces. It then defines a convenience function and eventually calls a method which runs the optimization loop.

    #        
    # Method to derive OIP from a given initial input image
    # ********************
    def _derive_OIP(self, map_index = 1, n_epochs = None, n_steps = 4, 
                          epsilon = 0.01, 
                          conv_criterion = 5.e-4, 
                          b_stop_with_convergence = False ):
        '''
        V0.3, 31.08.2020
        This method starts the process of producing an OIP of statistical input image data
        
        Requirements:    An initial input image with statistical fluctuations of pixel values 
        -------------    must have been created. 
        
        Restrictions:    This version only supports the most simple strategy - "strategy 0":  
        -------------    Optimize in one loop - starting from a superposition of fluctuations
                         No precursor, no intermediate evolution of input image data 
        
        Input:
        ------
        map_index:       We can chose a map here       (overwrites previous settings)
        n_epochs:        Number of optimization steps  (overwrites previous settings) 
        n_steps:         Defines number of intervals (with length n_epochs/ n_steps) for reporting
                         This number also sets a requirement for providing n_step external axis-frames 
                         to display intermediate images of the emerging OIP  
                         => see _oip_strat_0_optimization_loop()
        epsilon:         Size for corrections by gradient values
        conv_criterion:  A small threshold number for (difference of loss-values / present loss value )
        b_stop_with_convergence: 
                         Boolean which decides whether we stop a loop if the conv-criterion is fulfilled
                         
        '''
        self._map_index = map_index
        self._n_epochs  = n_epochs   
        self._n_steps   = n_steps
        self._epsilon   = epsilon
        
        # number of epochs for optimization strategy 0 
        # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
        if n_epochs == None:
            len_epochs = len(self._ay_epochs)
            self._n_epochs   = self._ay_epochs[len_epochs-1]
        else: 
            self._n_epochs = n_epochs
            
        # Rest some variables  
        self._val_oip_grads = None
        self._val_oip_loss  = None 
        self._iterate       = None 
        
        # Setup the TF2 GradientTape watch
        # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
        self._setup_gradient_tape()
        print("GradientTape watch activated ")
    
        # Gradient handling - so far we only deal with addresses 
        # ~~~~~~~~~~~~~~~~~~
        self._r_oip_grads  = self._tape.gradient(self._r_oip_loss, self._r_cnn_inputs)
        #print("shape of grads = ", self._r_oip_grads.shape)
        
        # normalization of the gradient 
        self._r_oip_grads /= (B.sqrt(B.mean(B.square(self._r_oip_grads))) + 1.e-7)
        #grads = tf.image.per_image_standardization(grads)
        
        # define an abstract recallable Keras function 
        # producing loss and gradients for corrected img data 
        # the first list of addresses points to the input data, the last to the output data 
        self._iterate = B.function( [self._r_cnn_inputs], [self._r_oip_loss, self._r_oip_grads] )
        
        # get the initial image into a variable for optimization 
        self._inp_img_data = None
        self._inp_img_data = self._initial_inp_img_data
        
        # Optimization loop for strategy 0 
        # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
        if self._oip_strategy == 0: 
            self._oip_strat_0_optimization_loop( conv_criterion = conv_criterion, 
                                                b_stop_with_convergence = b_stop_with_convergence )
    

 
Gradient value normalization is done here with respect to the L2-norm of the gradient. I.e., we adjust the length of the gradient to a unit length. Why does such a normalization help us with respect to convergence? You remember from a previous article series about MLPs that we had to care about a reasonable balance of an adaptive application of gradient values and a systematic reduction of the learning rate when approaching the global minimum of a cost function. The various available adaptive gradient methods care about speed and a proper deceleration of the steps by which we move across the cost hyperplane. When we approached a minimum we needed to care about overshooting. In our present gradient ascent case we have no such adaptive fine-tuning available. What we can do, however, is to control the length of the gradient vector such that changes remain of the same order as long as we do not change the step-size factor (epsilon). I.e.:

  • We do not wildly accelerate our path on the loss hyperplane when some local pixel areas want to drive us into a certain direction.
  • We reduce the chance of creating pixel values out of normal boundaries.

The last point fits well with a situation where the CNN has been trained on normalized or standardizes input images. Due to the normalization the size of pixel value corrections now depends significantly on the factor "epsilon". We should choose it small enough with respect to the pixel values.

Another interesting statement in the above code fragment is

        self._iterate = B.function( [self._r_cnn_inputs], [self._r_oip_loss, self._r_oip_grads] )

Here we use the Keras Backend to define a convenience function which relates input data with dependent outputs, whose calculations we previously defined by suitable statements. The list which is used as the first parameter of this function "_iterate()" defines the input variables, the list used as a second parameter defines the output variables which will internally be calculated via the GradientTape-functionality defined before. The "_iterate()"-function makes it much easier for us to build the optimization loop.

The optimization loop for the construction of images that visualize OIPs and features

The optimization loop must systematically correct the pixel values to approach a maximum of the loss function. The following method "_oip_strat_0_optimization_loop()" does this job for us. (The "0" in the method's name refers to a first simple approach.)

    #        
    # Method to optimize an emerging OIP out of statistical input image data 
    # (simple strategy - just optimize, no precursor, no intermediate pattern evolution 
    # ********************************
    def _oip_strat_0_optimization_loop(self, conv_criterion = 5.e-4, 
                                            b_stop_with_convergence = False ):
        
        '''
        V 0.2, 28.08.2020 
        
        Purpose: 
        This method controls the optimization loop for OIP reconstruction of an initial 
        input image with a statistical distribution of pixel values. 
        It also provides intermediate output in the form of printed data and images.
        
        Parameter: 
        conv_criterion:  A small threshold number for (difference of loss-values / present loss value )
        b_stop_with_convergence: 
                         Booelan which decides whether we stop a loop if the conv-criterion is fulfilled
        
        
        This method produces some intermediate output during the optimization loop in form of images.
        It uses an external grid of plot frames and their axes-objects. The addresses of the 
        axis-objects must provided by an external list "li_axa[]".  
        We need a seqence of >= n_steps axis-frames length(li_axa) >= n_steps).    
        With Jupyter the grid for plots can externally be provided by statements like 
        
        # figure
        # -----------
        #sizing
        fig_size = plt.rcParams["figure.figsize"]
        fig_size[0] = 16
        fig_size[1] = 8
        fig_a = plt.figure()
        axa_1 = fig_a.add_subplot(241)
        axa_2 = fig_a.add_subplot(242)
        axa_3 = fig_a.add_subplot(243)
        axa_4 = fig_a.add_subplot(244)
        axa_5 = fig_a.add_subplot(245)
        axa_6 = fig_a.add_subplot(246)
        axa_7 = fig_a.add_subplot(247)
        axa_8 = fig_a.add_subplot(248)
        li_axa = [axa_1, axa_2, axa_3, axa_4, axa_5, axa_6, axa_7, axa_8]
        
        '''
        
        # Check that we really have an input image tensor
        if ( (self._inp_img_data == None) or 
             (self._inp_img_data.shape[1] != self._img_dim) or 
             (self._inp_img_data.shape[2] != self._img_dim) ) :
            print("There is no initial input image or it does not fit dimension requirements (28,28)")
            sys.exit()
        
        # Print some information
        print("*************\nStart of optimization loop\n*************")
        print("Strategy: Simple initial mixture of long and short range variations")
        print("Number of epochs = ", self._n_epochs)
        print("Epsilon =  ", self._epsilon)
        print("*************")
        
        # some initial value
        loss_old = 0.0
       
        # Preparation of intermediate reporting / img printing
        # --------------------------------------
        # Number of intermediate reporting points during the loop 
        steps = math.ceil(self._n_epochs / self._n_steps )
        # Logarithmic spacing of steps (most things happen initially)
        n_el = math.floor(self._n_epochs / 2**(self._n_steps) ) 
        li_int = []
        for j in range(self._n_steps):
            li_int.append(n_el*2**j)
        print("li_int = ", li_int)
        # A counter for intermediate reporting  
        n_rep = 0
        # Array for intermediate image data 
        li_imgs = np.zeros((self._img_dim, self._img_dim, 1), dtype=np.float32)
        
        # Convergence? - A list for steps meeting the convergence criterion
        # ~~~~~~~~~~~~
        li_conv = []
        
        
        # optimization loop 
        # *******************
        # A counter for steps with zero loss and gradient values 
        n_zeros = 0
        
        for j in range(self._n_epochs):
            
            # Get output values of our Keras iteration function 
            # ~~~~~~~~~~~~~~~~~~~
            self._val_oip_loss, self._val_oip_grads = self._iterate([self._inp_img_data])
            
            # loss difference to last step - shuold steadily become smaller 
            loss_diff = self._val_oip_loss - loss_old 
            #print("loss_val = ", loss_val)
            #print("loss_diff = ", loss_diff)
            loss_old = self._val_oip_loss
            
            if j > 10 and (loss_diff/(self._val_oip_loss + 1.-7)) < conv_criterion:
                li_conv.append(j)
                lenc = len(li_conv)
                # print("conv - step = ", j)
                # stop only after the criterion has been met in 4 successive steps
                if b_stop_with_convergence and lenc > 5 and li_conv[-4] == j-4:
                    return
            
            grads_val     = self._val_oip_grads
            # the gradients average value 
            avg_grads_val = (tf.reduce_mean(grads_val)).numpy()
            
            # Check if our map reacts at all
            # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
            if self._val_oip_loss == 0.0 and avg_grads_val == 0.0:
                print( "0-values, j= ", j, 
                       " loss = ", self._val_oip_loss, " avg_loss = ", avg_grads_val )
                n_zeros += 1
            
            if n_zeros > 10: 
                print("More than 10 times zero values - Try a different initial random distribution of pixel values")
                return
            
            # gradient ascent => Correction of the input image data 
            # ~~~~~~~~~~~~~~~
            self._inp_img_data += grads_val * self._epsilon
            
            # Standardize the corrected image - we won't get a convergence otherwise 
            # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
            self._inp_img_data = tf.image.per_image_standardization(self._inp_img_data)
            
            # Some output at intermediate points 
            #     Note: We us logarithmic intervals because most changes 
            #           appear in the intial third of the loop's range  
            if (j == 0) or (j in li_int) or (j == self._n_epochs-1) :
                # print some info 
                print("\nstep " + str(j) + " finalized")
                #print("Shape of grads = ", grads_val.shape)
                print("present loss_val = ", self._val_oip_loss)
                print("loss_diff = ", loss_diff)
                # display the intermediate image data in an external grid 
                imgn = self._inp_img_data[0,:,:,0].numpy()
                #print("Shape of intermediate img = ", imgn.shape)
                li_axa[n_rep].imshow(imgn, cmap=plt.cm.viridis)
                # counter
                n_rep += 1
         
        return

 
The code is easy to understand: We use our convenience function "self._iterate()" to produce actual loss and gradient values within the loop. Then we change the pixel values of our input image and feed the changed image back into the loop. Note the positive sign of the correction! All complicated gradient calculations are automatically and "eagerly" done internally thanks to "GradientTape".

We said above that we limited the gradient values. How big is the size of the resulting corrections compared to the image data? If and when we use standardized image data and scale our gradient to unit length size then the relative size of the changes are of the order of the step size "epsilon" for the correction. In our case we set epsilon to 5.e-4.

The careful reader has noticed that I standardized the image data after the correction with the (normalized) gradient. Well, this step proved to be absolutely necessary to get convergence in the sense that we really approach an extremum of the cost function. Reasons are:

  • My CNN was trained on standardized MNIST input images.
  • We did not include any normalization layers into our CNN.
  • Without counter-measures our normalized gradient values would eventually drive unlimited activation values.

The last point deserves some explanation:
We used the ReLU-function as the activation function of nodes in the inner layers of our CNN. For positive input values ReLU actually is a linear function. Now let us assume that we have found a rather optimal input pattern which via a succession of basically linear operations drives an activation pattern of a map's neurons. What happens if we just add constant small values to each pixel per iteration step? The output values after the sequence of linear transformations will just grow! With our method and the ReLU activation function we walk around a surface until we reach a linear ramp and climb it up. Without compensatory steps we will not find a real maximum because there is none. The situation is very different at the convolutional layers than at the eventual output layer of the CNN's MLP-part.

You may ask yourself why we experienced nothing of this during the classification training? First answer: We did not optimize input data but weights during the training. Second answer: During training we did NOT maximize potentially unbound activation values but minimized a cost function based on output values of the last a MLP-layer. And these values were produced by a sigmoid function! The sigmoid function limits any input to the range ]0, +1[. In addition, the cost function (categorial_crossentropy) is designed to be convex for deviations of limited calculated values from a limited target vector.

The consequence is that we have to limit the values of the (corrected) input data and the related gradients in our present optimization procedure at the same time! This is done by the standardization of the image data. Remember that the correction values are around of the relative order of 5.e-4. In the end this is the order of the fluctuations which are unavoidable in the final OIP image; but now we have a chance to converge to a related small region around a real maximum.

The last block in the code deals with intermediate output - not only printed data on the loss function but also in form of intermediate images of the hopefully emerging pattern. These images can be provided in an external grid of figures in e.g. a Jupyter environment. The important point is that we define a suitable number of Matplotlib's axis-objects and deliver their addresses via an external array "li_axa[]". I am well aware of that the plotting solution coded here is a very basic one and it requires some ahead planning of the user - feel free to program it in a better way.

Initial input image data - with variations on different length scales

We lack just one further ingredient: We need a method to construct an input image with statistical data. I have discussed already that it may be helpful to vary data on different length scales. A very simple approach to tackle this problem manually is the following one:

We use squares - each with a different small and limited number of cells; e.g. a (4x4)-, a (7x7)-, a (14x14)- and (28x28)-square. Note that (28x28) is the original size of the MNIST images. For other samples we may need different sizes of such squares and more of them. We then fill the cells with random numbers between [-1, 1]. We smoothly scale the variations of the smaller squares up to the real input image size; here: (28x28). We can do this by applying a bicubic interpolation to the data. In the end we add up all random data and normalize or standardize the resulting distribution of pixel values. See the code below for details:

    # 
    # Method to build an initial image from a superposition of random data on different length scales 
    # ***********************************
    def _build_initial_img_data( self, 
                                 strategy = 0, 
                                 li_epochs    = (20, 50, 100, 400), 
                                 li_facts     = (0.5, 0.5, 0.5, 0.5),
                                 li_dim_steps = ( (3,3), (7,7), (14,14), (28,28) ), 
                                 b_smoothing = False):
        
        '''
        V0.2, 31.08.2020
        Purpose:
        ~~~~~~~~
        This method constructs an initial input image with a statistical distribution of pixel-values.
        We use 4 length scales to mix fluctuations with different "wave-length" by a simple  
        approach: 
        We fill four squares with a different number of cells below the number of pixels 
        in each dimension of the real input image; e.g. (4x4), (7x7, (14x14), (28,28) <= (28,28). 
        We fill the cells with random numbers in [-1.0, 1.]. We smootly scale the resulting pattern 
        up to (28,28) (or what ever the input image dimensions are) by bicubic interpolations 
        and eventually add up all values. As a final step we standardize the pixel value distribution.          
        
        Limitations
        ~~~~~~~~~~~
        This version works with 4 length scales. it only supports a simple strategy for 
        evolving OIP patterns. 
        '''

        self._oip_strategy = strategy
        self._ay_facts     = np.array(li_facts)
        self._ay_epochs    = np.array(li_epochs)
        
        self._li_dim_steps = li_dim_steps
        
        fluct_data = None

        
        # Strategy 0: Simple superposition of random patterns at 4 different wave-length
        # ~~~~~~~~~~
        if self._oip_strategy == 0:
            
            dim_1_1 = self._li_dim_steps[0][0] 
            dim_1_2 = self._li_dim_steps[0][1] 
            dim_2_1 = self._li_dim_steps[1][0] 
            dim_2_2 = self._li_dim_steps[1][1] 
            dim_3_1 = self._li_dim_steps[2][0] 
            dim_3_2 = self._li_dim_steps[2][1] 
            dim_4_1 = self._li_dim_steps[3][0] 
            dim_4_2 = self._li_dim_steps[3][1] 
            
            fact1 = self._ay_facts[0]
            fact2 = self._ay_facts[1]
            fact3 = self._ay_facts[2]
            fact4 = self._ay_facts[3]
            
            # print some parameter information
            # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
            print("\nInitial image composition - strategy 0:\n Superposition of 4 different wavelength patterns")
            print("Parameters:\n", 
                 fact1, " => (" + str(dim_1_1) +", " + str(dim_1_2) + ") :: ", 
                 fact2, " => (" + str(dim_2_1) +", " + str(dim_2_2) + ") :: ", 
                 fact3, " => (" + str(dim_3_1) +", " + str(dim_3_2) + ") :: ", 
                 fact4, " => (" + str(dim_4_1) +", " + str(dim_4_2) + ")" 
                 )
            
            # fluctuations
            fluct1 =  2.0 * ( np.random.random((1, dim_1_1, dim_1_2, 1)) - 0.5 ) 
            fluct2 =  2.0 * ( np.random.random((1, dim_2_1, dim_2_2, 1)) - 0.5 ) 
            fluct3 =  2.0 * ( np.random.random((1, dim_3_1, dim_3_2, 1)) - 0.5 ) 
            fluct4 =  2.0 * ( np.random.random((1, dim_4_1, dim_4_2, 1)) - 0.5 ) 

            # Scaling with bicubic interpolation to the required image size
            fluct1_scale = tf.image.resize(fluct1, [28,28], method="bicubic", antialias=True)
            fluct2_scale = tf.image.resize(fluct2, [28,28], method="bicubic", antialias=True)
            fluct3_scale = tf.image.resize(fluct3, [28,28], method="bicubic", antialias=True)
            fluct4_scale = fluct4

            # superposition
            fluct_data = fact1*fluct1_scale + fact2*fluct2_scale + fact3*fluct3_scale + fact4*fluct4_scale
        
        
        # get the standardized plus smoothed and unsmoothed image 
        # ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
        #    TF2 provides a function performing standardization of image data function
        fluct_data_unsmoothed = tf.image.per_image_standardization(fluct_data) 
        fluct_data_smoothed   = tf.image.per_image_standardization(
                                    tf.image.resize( fluct_data, [28,28], 
                                                     method="bicubic", antialias=True) )

        if b_smoothing: 
            self._initial_inp_img_data = fluct_data_smoothed
        else:
            self._initial_inp_img_data = fluct_data_unsmoothed

        # There should be no difference 
        img_init_unsmoothed = fluct_data_unsmoothed[0,:,:,0].numpy()
        img_init_smoothed   = fluct_data_smoothed[0,:,:,0].numpy()
        
        ax1_1.imshow(img_init_unsmoothed, cmap=plt.cm.viridis)
        ax1_2.imshow(img_init_smoothed, cmap=plt.cm.viridis)

        print("Initial images plotted")
        
        return self._initial_inp_img_data    

 
The factors fact1, fact2 and fact3 determine the relative amplitudes of the fluctuations at the different length scales. Thus the user is e.g. able to suppress short-scale fluctuations completely.

I only took exactly four squares to simulate fluctuation on different length scales. A better code would make the number of squares and length scales parameterizable. Or it would work with a Fourier series right from the start. I was too lazy for such elaborated things. The plots again require the definition of some external Matplotlib figures with axis-objects. You can provide a suitable figure in a Jupyter cell.

Conclusion

In this article we have build a simple class to create OIPs for a specific CNN map out of an input image with a random distribution of pixel values. The algorithm should have made it clear that this a constructive work performed during iteration:

  • We start from the "detection" of slight traces of a pattern in the initial statistical pixel value distribution; the
    pattern actually is a statistical pixel correlation which triggers the chosen map,
  • then we amplify the recognized pattern elements
  • and suppress pixel values which are not relevant into a homogeneous background.

So the headline of this article is a bit misleading: We do not only "detect" a map related OIP; we also "create" it.

Our new Python class makes use of a given trained CNN-model and follows the outline of steps discussed in a previous article. The class has many limitations - in its present version it is only usable for MNIST images and the user has to know a lot about internals. However, I hope that my readers nevertheless have understood the basic ingredients. From there it is only a small step towards are more general and more capable version.

I have also underlined in this article that the images produced by the coded methods may only represent local maxima of the loss function for average map activation and idealized patterns composed of re-occuring elementary sub-structures. In the next article

A simple CNN for the MNIST dataset – IX – filter visualization at a convolutional layer

I am going to apply the code to most of the maps of the highest, i.e. inner-most convolutional layer of my trained CNN. We shall discover a whole zoo of simple and complex input image patterns. But we shall also confirm the suspicion that our optimization algorithm for finding an OIP for a specific map does not react to each and every kind of initial statistical pixel value fluctuation presented to the CNN.