{
  "source_file": "0930",
  "source_platform": "Medium",
  "author": "Riccardo Andreoni",
  "title_listed": "Build a Convolutional Neural Network from Scratch using Numpy",
  "keywords": "",
  "domain": "图像处理",
  "content_type": "深度技术解析",
  "is_paid_flag": "否",
  "url": "https://towardsdatascience.com/build-a-convolutional-neural-network-from-scratch-using-numpy-139cbbf3c45e/",
  "publish_date": "2023-11-23",
  "corpus_stage": "",
  "notes": "",
  "id": "59baf80239ce",
  "collected_at": "2026-10-02T03:14:49",
  "title": "Build a Convolutional Neural Network from Scratch using Numpy | Towards Data Science",
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It also helps in adverstising on the website.\n\n\n  * Cookie\nedgebucket\n  * Duration\nsession\n  * Description\nReddit sets this cookie to save the information about a log-on Reddit user, for the purpose of advertisement recommendations and updating the content.\n\n\n  * Cookie\ndid\n  * Duration\n1 year\n  * Description\nArbor sets this cookie to show targeted ads to site visitors.This cookie expires after 2 months or 1 year.\n\n\nUncategorised\nOther uncategorised cookies are those that are being analysed and have not been classified into a category as yet.\nNo cookies to display.\nReject All Save My Preferences Accept All\nToggle Search[](https://towardsdatascience.com/)Toggle Mobile Navigation\n[](https://towardsdatascience.com/)\nPublish AI, ML & data-science insights to a global community of data professionals.\nSign in[Submit an Article](https://write.insightmediagroup.io)\n[Latest](https://towardsdatascience.com/latest/)[Editor's Picks](https://towardsdatascience.com/tag/editors-pick/)[Newsletter](https://towardsdatascience.com/tag/the-variable/)[ShipAI](https://towardsdatascience.com/shipai/)\n* * *\n[Write For TDS](https://towardsdatascience.com/submissions/)\n[LinkedIn](https://www.linkedin.com/company/towards-data-science/)[X](https://x.com/TDataScience)Toggle Search\n[Data Science](https://towardsdatascience.com/category/data-science/)\n# Build a Convolutional Neural Network from Scratch using Numpy\nMaster Computer Vision by building a CNN from scratch all by yourself.\n[Riccardo Andreoni](https://towardsdatascience.com/author/riccardo.andreoni/)\nNovember 23, 20238 min read\n### Master computer vision by building a CNN from scratch all by yourself\n![These colored windows remind me of the layers of CNNs and their filters. Image source: unsplash.com.](../images/000.jpeg)\nThese colored windows remind me of the layers of CNNs and their filters. Image source: [unsplash.com](https://unsplash.com/photos/photo-of-green-leafed-plants-inside-building-CuEvrPd3NYc).\nAs **Computer Vision** applications are now present everywhere in our daily lives, it is fundamental for every Data Science practitioner to **understand their functioning principles** and **familiarize themselves with them**.\nIn [this article](https://towardsdatascience.com/building-a-deep-neural-network-from-scratch-using-numpy-4f28a1df157a), I built a **Deep Neural Network** without relying on popular modern deep learning libraries like Tensorflow, Pytorch, and Keras. I then classified images of handwritten digits with it. While the achieved results didn't reach state-of-the-art levels, they were nevertheless satisfactory. Now, I want to take a **further step** in developing a [**Convolutional Neural Network**](https://en.wikipedia.org/wiki/Convolutional_neural_network) (CNN) using only the Python library [**Numpy**](https://numpy.org/).\nPython deep learning libraries, like the ones mentioned above, are **extremely powerful tools**. However, as a downside, they shield Data Science practitioners from understanding the low-level functioning principles of Neural Networks. This is especially true with CNNs, as their **processes are less intuitive** compared with the classical fully connected networks. The only way to address this issue is to get our hands dirty and implement CNNs ourselves: this is the motivation behind this task.\nLearn this step by step with the interactive [AI and Data Scientist](https://roadmap.sh/ai-data-scientist), [Computer Science](https://roadmap.sh/computer-science) and [Machine Learning](https://roadmap.sh/machine-learning) roadmaps.\nThis article is intended as a practical, **hands-on guide** rather than a comprehensive guide of CNN functioning principles. As a consequence, the theoretical part is concise and mostly serves the understanding of the practical section. For this reason, you will find an exhaustive list of resources at the end of this post. I warmly invite you to check them out!\n## Convolutional Neural Networks\nConvolutional Neural Networks use a specific architecture and operations that make them **well-suited for tasks related to images** , such as image classification, object localization, image segmentation, and more. Their design roughly **mirrors the human visual cortex** , where each biological neuron responds to only a small portion of the visual field. Moreover, higher-level neurons react to the outputs of lower-level neurons.\nWhile classical fully connected networks can handle image-related tasks, their effectiveness degrades significantly when applied to medium or large images due to the large number of parameters they require. For instance, a 200x200 pixel image contains 40,000 pixels, and if the first layer of the network has 1,000 units, it results in 40 million weights just for that layer. This challenge is highly alleviated by CNNs as they implement **partially connected layers** and **weight sharing**.\nThe **main components** of a Convolutional Neural Network are:\n  * **Convolutional layers**\n  * **Pooling layers**\n\n\n### Convolutional Layer\nA convolutional layer consists of a **set of filters** , also known as **kernels**. When applied to the input of the layer, these filters **modify the original images** in specific ways.\nA filter can be described as a matrix, whose elements values define the king of modification applied to the original image. For instance, a 3x3 kernel, like the following one, highlights the vertical edges of the image:\n![](../images/body-8b6d3e7184477b6fc724.gif)\nThis kernel instead accentuates the horizontal edges:\n![](../images/body-b644f3df1d780b12956d.gif)\n![Source: Wikipedia.](../images/001.png)\nSource: [Wikipedia](https://en.wikipedia.org/wiki/Kernel_\\(image_processing\\)).\nIt is important to note that the values of the elements of these kernels are not manually chosen but are parameters that the network learns during the training process.\nThe main function of convolutions is to isolate and highlight the different features present in the image. Later on, dense layers will use these features.\n### Pooling Layer\nPooling layers are **more simple than convolutional layers**. Their purpose is to minimize the computational load and memory usage of the network. They achieve this task by **downsizing the input image's dimensions**. Reducing the dimension results in a reduction of the number of parameters that the CNN has to learn.\nPooling layers also employ a kernel, typically of dimension 2x2, to aggregate a section of the input image into a single value. For example, a 2x2 max pooling kernel extracts 4 pixels from the input image and outputs only the pixel with the maximum value.\n## Python Implementation\nYou can find all the code shown in this section in [my GitHub repository](https://github.com/andreoniriccardo/CNN-from-scratch).\n> [**GitHub - andreoniriccardo/CNN-from-scratch: Convolutional Neural Network from scratch**](https://github.com/andreoniriccardo/CNN-from-scratch)\nThe concept behind this implementation consists of creating **Python classes** that represent the **convolutional and max pooling layers**. Furthermore, as this CNN will be applied to the famous open-source [MNIST](https://datahub.io/machine-learning/mnist_784#readme) dataset, I also create a specific class for the Softmax layer.\nWithin each class, I define the methods that perform the forward propagation and backpropagation steps.\nAs a final step, the layers are appended into a list to build the final Convolutional Neural Network.\n### Convolutional Layer Implementation\nThe code defining a **convolutional layer** is the following:\npython\n\n```\nclass ConvolutionLayer:\n    def __init__(self, kernel_num, kernel_size):\n        self.kernel_num = kernel_num\n        self.kernel_size = kernel_size        \n        self.kernels = np.random.randn(kernel_num, kernel_size, kernel_size) / (kernel_size**2)\n\n    def patches_generator(self, image):\n        image_h, image_w = image.shape\n        self.image = image\n        for h in range(image_h-self.kernel_size+1):\n            for w in range(image_w-self.kernel_size+1):\n                patch = image[h:(h+self.kernel_size), w:(w+self.kernel_size)]\n                yield patch, h, w\n\n    def forward_prop(self, image):\n        image_h, image_w = image.shape\n        convolution_output = np.zeros((image_h-self.kernel_size+1, image_w-self.kernel_size+1, self.kernel_num))\n        for patch, h, w in self.patches_generator(image):\n            convolution_output[h,w] = np.sum(patch*self.kernels, axis=(1,2))\n        return convolution_output\n\n    def back_prop(self, dE_dY, alpha):\n        dE_dk = np.zeros(self.kernels.shape)\n        for patch, h, w in self.patches_generator(self.image):\n            for f in range(self.kernel_num):\n                dE_dk[f] += patch * dE_dY[h, w, f]\n        self.kernels -= alpha*dE_dk\n        return dE_dk\n```\n\nThe constructor of the `**ConvolutionLayer**`class takes as inputs the number of kernels of the convolutional layer and their size. I assume to use only squared kernels of size`**kernel_size**`by`**kernel_size**`**.**\nLater, I generate random filters of shape `**(kernel_num, kernel_size, kernel_size)**`and, for normalization, I divide each element by the squared kernel size.\nThe `**patches_generator()**`method is a generator. It yields the portions of the images on which to perform each convolution step.\nThe `**forward_prop()**`method carries out the convolution for each patch generated by the method above.\nFinally, the `**back_prop()**`method is responsible for computing the gradient of the loss function with respect to each layer's weight. It also updates the weights' values correspondingly. Note that the loss function mentioned here is not the global loss of the network. Instead, it consists of the loss function passed by the max pooling layer to the previous convolutional layer.\nTo demonstrate the actual effect of this class, I created an instance of the `**ConvolutionLayer**`with 32 filters, each of size 3x3. Then I apply the forward propagation method on an image, resulting in an output consisting of 32 slightly smaller images.\nThe initial input image has size 28x28 pixels and is depicted below:\n![Image by the author.](../images/002.png)\nImage by the author.\nOnce I applied the `**forward_prop()**`method of the convolutional layer, I obtain 32 images of size 26x26 pixels. One of them is the following:\n![Image by the author.](../images/003.png)\nImage by the author.\nAs you can see, the image has been reduced in size, and the clarity of the handwritten digit is worse. It is important to note that this operation was carried out by a filter containing random values, and therefore, it does not accurately represent the actual step performed by a trained CNN. Still, you can grasp the idea of how these convolutions yield smaller images where the distinctive features of the object are isolated.\n### Max Pooling Layer Implementation\nI used Numpy to define the Max Pooling layer class as follows:\npython\n\n```\nclass MaxPoolingLayer:\n    def __init__(self, kernel_size):\n        self.kernel_size = kernel_size\n\n    def patches_generator(self, image):\n        output_h = image.shape[0] // self.kernel_size\n        output_w = image.shape[1] // self.kernel_size\n        self.image = image\n\n        for h in range(output_h):\n            for w in range(output_w):\n                patch = image[(h*self.kernel_size):(h*self.kernel_size+self.kernel_size), (w*self.kernel_size):(w*self.kernel_size+self.kernel_size)]\n                yield patch, h, w\n\n    def forward_prop(self, image):\n        image_h, image_w, num_kernels = image.shape\n        max_pooling_output = np.zeros((image_h//self.kernel_size, image_w//self.kernel_size, num_kernels))\n        for patch, h, w in self.patches_generator(image):\n            max_pooling_output[h,w] = np.amax(patch, axis=(0,1))\n        return max_pooling_output\n\n    def back_prop(self, dE_dY):\n        dE_dk = np.zeros(self.image.shape)\n        for patch,h,w in self.patches_generator(self.image):\n            image_h, image_w, num_kernels = patch.shape\n            max_val = np.amax(patch, axis=(0,1))\n\n            for idx_h in range(image_h):\n                for idx_w in range(image_w):\n                    for idx_k in range(num_kernels):\n                        if patch[idx_h,idx_w,idx_k] == max_val[idx_k]:\n                            dE_dk[h*self.kernel_size+idx_h, w*self.kernel_size+idx_w, idx_k] = dE_dY[h,w,idx_k]\n            return dE_dk\n```\n\nThe constructor method only assigns the kernel size value. The following methods operate similarly to the ones defined for the convolutional layer, with the main difference being that the `**back_prop()**`method doesn't update any weight values. In fact, the pooling layer**doesnt' rely on weights** to perform the aggregation.\n### Softmax Layer Implementation\nFinally, I define the **Softmax layer**. It has the objective of **flattening the output volume** obtained from the final max pooling layer. The Softmax layer outputs 10 values, which can be interpreted as the probability of an image corresponding to the 0-to-9 digits.\nThe implementation has the same structure of the ones seen above:\npython\n\n```\nclass SoftmaxLayer:\n    def __init__(self, input_units, output_units):\n        self.weight = np.random.randn(input_units, output_units)/input_units\n        self.bias = np.zeros(output_units)\n\n    def forward_prop(self, image):\n        self.original_shape = image.shape\n        image_flattened = image.flatten()\n        self.flattened_input = image_flattened\n        first_output = np.dot(image_flattened, self.weight) + self.bias\n        self.output = first_output\n        softmax_output = np.exp(first_output) / np.sum(np.exp(first_output), axis=0)\n        return softmax_output\n\n    def back_prop(self, dE_dY, alpha):\n        for i, gradient in enumerate(dE_dY):\n            if gradient == 0:\n                continue\n            transformation_eq = np.exp(self.output)\n            S_total = np.sum(transformation_eq)\n\n            dY_dZ = -transformation_eq[i]*transformation_eq / (S_total**2)\n            dY_dZ[i] = transformation_eq[i]*(S_total - transformation_eq[i]) / (S_total**2)\n\n            dZ_dw = self.flattened_input\n            dZ_db = 1\n            dZ_dX = self.weight\n\n            dE_dZ = gradient * dY_dZ\n\n            dE_dw = dZ_dw[np.newaxis].T @ dE_dZ[np.newaxis]\n            dE_db = dE_dZ * dZ_db\n            dE_dX = dZ_dX @ dE_dZ\n\n            self.weight -= alpha*dE_dw\n            self.bias -= alpha*dE_db\n\n            return dE_dX.reshape(self.original_shape)\n```\n\n![Image by the author.](../images/004.png)\nImage by the author.\n## Conclusions\nIn this post, we saw a **theoretical introduction** to the fundamental CNN architectural elements such as convolutional and pooling layers. I am positive that the step-by-step Python implementation will provide you a **practical understanding** of how these theoretical concepts can be translated into code.\nI invite you to clone the [GitHub repository](https://github.com/andreoniriccardo/CNN-from-scratch) containing the code and play with the `**main.py**`script. Of course, this network doesn't achieve state-of-the-art performances, as it is not built for this objective, but nevertheless reaches a**96% accuracy** after a few epochs.\nFinally, in order to expand your knowledge about CNN and computer vision, I suggest checking some of the resources listed below.\n* * *\nIf you liked this story, consider following me to be notified of my upcoming projects and articles!\n## References\n  * [\"Deep Learning\" by Ian Goodfellow, Yoshua Bengio, and Aaron Courville](https://www.deeplearningbook.org/)\n  * [\"Hands-On Machine Learning with Scikit-Learn, Keras, and TensorFlow\" by Aurélien Géron](https://www.oreilly.com/library/view/hands-on-machine-learning/9781492032632/)\n  * [\"ImageNet Classification with Deep Convolutional Neural Networks\" by Alex Krizhevsky, Ilya Sutskever, and Geoffrey Hinton](https://proceedings.neurips.cc/paper/4824-imagenet-classification-with-deep-convolutional-neural-networks.pdf)\n  * [\"Very Deep Convolutional Networks for Large-Scale Image Recognition\" by Karen Simonyan and Andrew Zisserman (VGGNet)](https://arxiv.org/abs/1409.1556)\n  * [\"Python Machine Learning\" by Sebastian Raschka and Vahid Mirjalili](https://www.amazon.com/Python-Machine-Learning-scikit-learn-TensorFlow/dp/1789955750)\n  * [\"Convolutional Neural Networks in Python: Master Data Science and Machine Learning with Modern Deep Learning in Python, Theano, and TensorFlow\" by Jason Brownlee](https://zlibrary-asia.se/book/2740843/d253d5?dsource=recommend)\n  * [\"Hands-On Convolutional Neural Networks with TensorFlow 2\" by Alex Gotev](https://www.amazon.com/Hands-Neural-Networks-TensorFlow-2-0/dp/1789615550)\n\n\n## Register Now to Continue Reading\n### Unlock Towards Data Science with a Free Account\nAll articles on Towards Data Science are available to registered users. 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    },
    {
      "type": "external",
      "href": "https://roadmap.sh/machine-learning"
    },
    {
      "type": "external",
      "href": "https://en.wikipedia.org/wiki/Kernel_(image_processing)"
    },
    {
      "type": "external",
      "href": "https://github.com/andreoniriccardo/CNN-from-scratch"
    },
    {
      "type": "external",
      "href": "https://datahub.io/machine-learning/mnist_784"
    },
    {
      "type": "external",
      "href": "https://www.deeplearningbook.org/"
    },
    {
      "type": "external",
      "href": "https://www.oreilly.com/library/view/hands-on-machine-learning/9781492032632/"
    },
    {
      "type": "external",
      "href": "https://proceedings.neurips.cc/paper/4824-imagenet-classification-with-deep-convolutional-neural-networks.pdf"
    },
    {
      "type": "external",
      "href": "https://arxiv.org/abs/1409.1556"
    },
    {
      "type": "external",
      "href": "https://www.amazon.com/Python-Machine-Learning-scikit-learn-TensorFlow/dp/1789955750"
    },
    {
      "type": "external",
      "href": "https://zlibrary-asia.se/book/2740843/d253d5?dsource=recommend"
    },
    {
      "type": "external",
      "href": "https://www.amazon.com/Hands-Neural-Networks-TensorFlow-2-0/dp/1789615550"
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      "href": "https://www.facebook.com/sharer/sharer.php?title=Build+a+Convolutional+Neural+Network+from+Scratch+using+Numpy&u=https%3A%2F%2Ftowardsdatascience.com%2Fbuild-a-convolutional-neural-network-from-scratch-using-numpy-139cbbf3c45e%2F"
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It also helps in adverstising on the website.\n\n\n  * Cookie\nedgebucket\n  * Duration\nsession\n  * Description\nReddit sets this cookie to save the information about a log-on Reddit user, for the purpose of advertisement recommendations and updating the content.\n\n\n  * Cookie\ndid\n  * Duration\n1 year\n  * Description\nArbor sets this cookie to show targeted ads to site visitors.This cookie expires after 2 months or 1 year.\n\n\nUncategorised\nOther uncategorised cookies are those that are being analysed and have not been classified into a category as yet.\nNo cookies to display.\nReject All Save My Preferences Accept All\nToggle Search[](https://towardsdatascience.com/)Toggle Mobile Navigation\n[](https://towardsdatascience.com/)\nPublish AI, ML & data-science insights to a global community of data professionals.\nSign in[Submit an Article](https://write.insightmediagroup.io)\n[Latest](https://towardsdatascience.com/latest/)[Editor's Picks](https://towardsdatascience.com/tag/editors-pick/)[Newsletter](https://towardsdatascience.com/tag/the-variable/)[ShipAI](https://towardsdatascience.com/shipai/)\n* * *\n[Write For TDS](https://towardsdatascience.com/submissions/)\n[LinkedIn](https://www.linkedin.com/company/towards-data-science/)[X](https://x.com/TDataScience)Toggle Search\n[Data Science](https://towardsdatascience.com/category/data-science/)\n# Build a Convolutional Neural Network from Scratch using Numpy\nMaster Computer Vision by building a CNN from scratch all by yourself.\n[Riccardo Andreoni](https://towardsdatascience.com/author/riccardo.andreoni/)\nNovember 23, 20238 min read\n### Master computer vision by building a CNN from scratch all by yourself\n![These colored windows remind me of the layers of CNNs and their filters. Image source: unsplash.com.](https://assets.insightmediagroup.io/media/wp-content/uploads/2023/11/13er5aXKjkMxJZhmiYsfVWA.jpeg)\nThese colored windows remind me of the layers of CNNs and their filters. Image source: [unsplash.com](https://unsplash.com/photos/photo-of-green-leafed-plants-inside-building-CuEvrPd3NYc).\nAs **Computer Vision** applications are now present everywhere in our daily lives, it is fundamental for every Data Science practitioner to **understand their functioning principles** and **familiarize themselves with them**.\nIn [this article](https://towardsdatascience.com/building-a-deep-neural-network-from-scratch-using-numpy-4f28a1df157a), I built a **Deep Neural Network** without relying on popular modern deep learning libraries like Tensorflow, Pytorch, and Keras. I then classified images of handwritten digits with it. While the achieved results didn't reach state-of-the-art levels, they were nevertheless satisfactory. Now, I want to take a **further step** in developing a [**Convolutional Neural Network**](https://en.wikipedia.org/wiki/Convolutional_neural_network) (CNN) using only the Python library [**Numpy**](https://numpy.org/).\nPython deep learning libraries, like the ones mentioned above, are **extremely powerful tools**. However, as a downside, they shield Data Science practitioners from understanding the low-level functioning principles of Neural Networks. This is especially true with CNNs, as their **processes are less intuitive** compared with the classical fully connected networks. The only way to address this issue is to get our hands dirty and implement CNNs ourselves: this is the motivation behind this task.\nLearn this step by step with the interactive [AI and Data Scientist](https://roadmap.sh/ai-data-scientist), [Computer Science](https://roadmap.sh/computer-science) and [Machine Learning](https://roadmap.sh/machine-learning) roadmaps.\nThis article is intended as a practical, **hands-on guide** rather than a comprehensive guide of CNN functioning principles. As a consequence, the theoretical part is concise and mostly serves the understanding of the practical section. For this reason, you will find an exhaustive list of resources at the end of this post. I warmly invite you to check them out!\n## Convolutional Neural Networks\nConvolutional Neural Networks use a specific architecture and operations that make them **well-suited for tasks related to images** , such as image classification, object localization, image segmentation, and more. Their design roughly **mirrors the human visual cortex** , where each biological neuron responds to only a small portion of the visual field. Moreover, higher-level neurons react to the outputs of lower-level neurons.\nWhile classical fully connected networks can handle image-related tasks, their effectiveness degrades significantly when applied to medium or large images due to the large number of parameters they require. For instance, a 200x200 pixel image contains 40,000 pixels, and if the first layer of the network has 1,000 units, it results in 40 million weights just for that layer. This challenge is highly alleviated by CNNs as they implement **partially connected layers** and **weight sharing**.\nThe **main components** of a Convolutional Neural Network are:\n  * **Convolutional layers**\n  * **Pooling layers**\n\n\n### Convolutional Layer\nA convolutional layer consists of a **set of filters** , also known as **kernels**. When applied to the input of the layer, these filters **modify the original images** in specific ways.\nA filter can be described as a matrix, whose elements values define the king of modification applied to the original image. For instance, a 3x3 kernel, like the following one, highlights the vertical edges of the image:\n![](https://assets.insightmediagroup.io/media/wp-content/uploads/2023/11/12V5YIujyr_f9k7RrQqeqqw.gif)\nThis kernel instead accentuates the horizontal edges:\n![](https://assets.insightmediagroup.io/media/wp-content/uploads/2023/11/1CVIbKfPNMTRu8FKJgdXMYg.gif)\n![Source: Wikipedia.](https://assets.insightmediagroup.io/media/wp-content/uploads/2023/11/15k6xQ2RoZrqyJYct6F7TtA.png)\nSource: [Wikipedia](https://en.wikipedia.org/wiki/Kernel_\\(image_processing\\)).\nIt is important to note that the values of the elements of these kernels are not manually chosen but are parameters that the network learns during the training process.\nThe main function of convolutions is to isolate and highlight the different features present in the image. Later on, dense layers will use these features.\n### Pooling Layer\nPooling layers are **more simple than convolutional layers**. Their purpose is to minimize the computational load and memory usage of the network. They achieve this task by **downsizing the input image's dimensions**. Reducing the dimension results in a reduction of the number of parameters that the CNN has to learn.\nPooling layers also employ a kernel, typically of dimension 2x2, to aggregate a section of the input image into a single value. For example, a 2x2 max pooling kernel extracts 4 pixels from the input image and outputs only the pixel with the maximum value.\n## Python Implementation\nYou can find all the code shown in this section in [my GitHub repository](https://github.com/andreoniriccardo/CNN-from-scratch).\n> [**GitHub - andreoniriccardo/CNN-from-scratch: Convolutional Neural Network from scratch**](https://github.com/andreoniriccardo/CNN-from-scratch)\nThe concept behind this implementation consists of creating **Python classes** that represent the **convolutional and max pooling layers**. Furthermore, as this CNN will be applied to the famous open-source [MNIST](https://datahub.io/machine-learning/mnist_784#readme) dataset, I also create a specific class for the Softmax layer.\nWithin each class, I define the methods that perform the forward propagation and backpropagation steps.\nAs a final step, the layers are appended into a list to build the final Convolutional Neural Network.\n### Convolutional Layer Implementation\nThe code defining a **convolutional layer** is the following:\npython\n\n```\nclass ConvolutionLayer:\n    def __init__(self, kernel_num, kernel_size):\n        self.kernel_num = kernel_num\n        self.kernel_size = kernel_size        \n        self.kernels = np.random.randn(kernel_num, kernel_size, kernel_size) / (kernel_size**2)\n\n    def patches_generator(self, image):\n        image_h, image_w = image.shape\n        self.image = image\n        for h in range(image_h-self.kernel_size+1):\n            for w in range(image_w-self.kernel_size+1):\n                patch = image[h:(h+self.kernel_size), w:(w+self.kernel_size)]\n                yield patch, h, w\n\n    def forward_prop(self, image):\n        image_h, image_w = image.shape\n        convolution_output = np.zeros((image_h-self.kernel_size+1, image_w-self.kernel_size+1, self.kernel_num))\n        for patch, h, w in self.patches_generator(image):\n            convolution_output[h,w] = np.sum(patch*self.kernels, axis=(1,2))\n        return convolution_output\n\n    def back_prop(self, dE_dY, alpha):\n        dE_dk = np.zeros(self.kernels.shape)\n        for patch, h, w in self.patches_generator(self.image):\n            for f in range(self.kernel_num):\n                dE_dk[f] += patch * dE_dY[h, w, f]\n        self.kernels -= alpha*dE_dk\n        return dE_dk\n```\n\nThe constructor of the `**ConvolutionLayer**`class takes as inputs the number of kernels of the convolutional layer and their size. I assume to use only squared kernels of size`**kernel_size**`by`**kernel_size**`**.**\nLater, I generate random filters of shape `**(kernel_num, kernel_size, kernel_size)**`and, for normalization, I divide each element by the squared kernel size.\nThe `**patches_generator()**`method is a generator. It yields the portions of the images on which to perform each convolution step.\nThe `**forward_prop()**`method carries out the convolution for each patch generated by the method above.\nFinally, the `**back_prop()**`method is responsible for computing the gradient of the loss function with respect to each layer's weight. It also updates the weights' values correspondingly. Note that the loss function mentioned here is not the global loss of the network. Instead, it consists of the loss function passed by the max pooling layer to the previous convolutional layer.\nTo demonstrate the actual effect of this class, I created an instance of the `**ConvolutionLayer**`with 32 filters, each of size 3x3. Then I apply the forward propagation method on an image, resulting in an output consisting of 32 slightly smaller images.\nThe initial input image has size 28x28 pixels and is depicted below:\n![Image by the author.](https://assets.insightmediagroup.io/media/wp-content/uploads/2023/11/11SNR0WdkkFc1jSs55AoeMQ.png)\nImage by the author.\nOnce I applied the `**forward_prop()**`method of the convolutional layer, I obtain 32 images of size 26x26 pixels. One of them is the following:\n![Image by the author.](https://assets.insightmediagroup.io/media/wp-content/uploads/2023/11/1oEej3MiJmdAiifLrYRuQUg.png)\nImage by the author.\nAs you can see, the image has been reduced in size, and the clarity of the handwritten digit is worse. It is important to note that this operation was carried out by a filter containing random values, and therefore, it does not accurately represent the actual step performed by a trained CNN. Still, you can grasp the idea of how these convolutions yield smaller images where the distinctive features of the object are isolated.\n### Max Pooling Layer Implementation\nI used Numpy to define the Max Pooling layer class as follows:\npython\n\n```\nclass MaxPoolingLayer:\n    def __init__(self, kernel_size):\n        self.kernel_size = kernel_size\n\n    def patches_generator(self, image):\n        output_h = image.shape[0] // self.kernel_size\n        output_w = image.shape[1] // self.kernel_size\n        self.image = image\n\n        for h in range(output_h):\n            for w in range(output_w):\n                patch = image[(h*self.kernel_size):(h*self.kernel_size+self.kernel_size), (w*self.kernel_size):(w*self.kernel_size+self.kernel_size)]\n                yield patch, h, w\n\n    def forward_prop(self, image):\n        image_h, image_w, num_kernels = image.shape\n        max_pooling_output = np.zeros((image_h//self.kernel_size, image_w//self.kernel_size, num_kernels))\n        for patch, h, w in self.patches_generator(image):\n            max_pooling_output[h,w] = np.amax(patch, axis=(0,1))\n        return max_pooling_output\n\n    def back_prop(self, dE_dY):\n        dE_dk = np.zeros(self.image.shape)\n        for patch,h,w in self.patches_generator(self.image):\n            image_h, image_w, num_kernels = patch.shape\n            max_val = np.amax(patch, axis=(0,1))\n\n            for idx_h in range(image_h):\n                for idx_w in range(image_w):\n                    for idx_k in range(num_kernels):\n                        if patch[idx_h,idx_w,idx_k] == max_val[idx_k]:\n                            dE_dk[h*self.kernel_size+idx_h, w*self.kernel_size+idx_w, idx_k] = dE_dY[h,w,idx_k]\n            return dE_dk\n```\n\nThe constructor method only assigns the kernel size value. The following methods operate similarly to the ones defined for the convolutional layer, with the main difference being that the `**back_prop()**`method doesn't update any weight values. In fact, the pooling layer**doesnt' rely on weights** to perform the aggregation.\n### Softmax Layer Implementation\nFinally, I define the **Softmax layer**. It has the objective of **flattening the output volume** obtained from the final max pooling layer. The Softmax layer outputs 10 values, which can be interpreted as the probability of an image corresponding to the 0-to-9 digits.\nThe implementation has the same structure of the ones seen above:\npython\n\n```\nclass SoftmaxLayer:\n    def __init__(self, input_units, output_units):\n        self.weight = np.random.randn(input_units, output_units)/input_units\n        self.bias = np.zeros(output_units)\n\n    def forward_prop(self, image):\n        self.original_shape = image.shape\n        image_flattened = image.flatten()\n        self.flattened_input = image_flattened\n        first_output = np.dot(image_flattened, self.weight) + self.bias\n        self.output = first_output\n        softmax_output = np.exp(first_output) / np.sum(np.exp(first_output), axis=0)\n        return softmax_output\n\n    def back_prop(self, dE_dY, alpha):\n        for i, gradient in enumerate(dE_dY):\n            if gradient == 0:\n                continue\n            transformation_eq = np.exp(self.output)\n            S_total = np.sum(transformation_eq)\n\n            dY_dZ = -transformation_eq[i]*transformation_eq / (S_total**2)\n            dY_dZ[i] = transformation_eq[i]*(S_total - transformation_eq[i]) / (S_total**2)\n\n            dZ_dw = self.flattened_input\n            dZ_db = 1\n            dZ_dX = self.weight\n\n            dE_dZ = gradient * dY_dZ\n\n            dE_dw = dZ_dw[np.newaxis].T @ dE_dZ[np.newaxis]\n            dE_db = dE_dZ * dZ_db\n            dE_dX = dZ_dX @ dE_dZ\n\n            self.weight -= alpha*dE_dw\n            self.bias -= alpha*dE_db\n\n            return dE_dX.reshape(self.original_shape)\n```\n\n![Image by the author.](https://assets.insightmediagroup.io/media/wp-content/uploads/2023/11/1qTqfAZ4_OxzUXW2ZJ_04Dg.png)\nImage by the author.\n## Conclusions\nIn this post, we saw a **theoretical introduction** to the fundamental CNN architectural elements such as convolutional and pooling layers. I am positive that the step-by-step Python implementation will provide you a **practical understanding** of how these theoretical concepts can be translated into code.\nI invite you to clone the [GitHub repository](https://github.com/andreoniriccardo/CNN-from-scratch) containing the code and play with the `**main.py**`script. Of course, this network doesn't achieve state-of-the-art performances, as it is not built for this objective, but nevertheless reaches a**96% accuracy** after a few epochs.\nFinally, in order to expand your knowledge about CNN and computer vision, I suggest checking some of the resources listed below.\n* * *\nIf you liked this story, consider following me to be notified of my upcoming projects and articles!\n## References\n  * [\"Deep Learning\" by Ian Goodfellow, Yoshua Bengio, and Aaron Courville](https://www.deeplearningbook.org/)\n  * [\"Hands-On Machine Learning with Scikit-Learn, Keras, and TensorFlow\" by Aurélien Géron](https://www.oreilly.com/library/view/hands-on-machine-learning/9781492032632/)\n  * [\"ImageNet Classification with Deep Convolutional Neural Networks\" by Alex Krizhevsky, Ilya Sutskever, and Geoffrey Hinton](https://proceedings.neurips.cc/paper/4824-imagenet-classification-with-deep-convolutional-neural-networks.pdf)\n  * [\"Very Deep Convolutional Networks for Large-Scale Image Recognition\" by Karen Simonyan and Andrew Zisserman (VGGNet)](https://arxiv.org/abs/1409.1556)\n  * [\"Python Machine Learning\" by Sebastian Raschka and Vahid Mirjalili](https://www.amazon.com/Python-Machine-Learning-scikit-learn-TensorFlow/dp/1789955750)\n  * [\"Convolutional Neural Networks in Python: Master Data Science and Machine Learning with Modern Deep Learning in Python, Theano, and TensorFlow\" by Jason Brownlee](https://zlibrary-asia.se/book/2740843/d253d5?dsource=recommend)\n  * [\"Hands-On Convolutional Neural Networks with TensorFlow 2\" by Alex Gotev](https://www.amazon.com/Hands-Neural-Networks-TensorFlow-2-0/dp/1789615550)\n\n\n## Register Now to Continue Reading\n### Unlock Towards Data Science with a Free Account\nAll articles on Towards Data Science are available to registered users. Create a free account today to access this article and thousands more.\nRegister for freeSign in to your account\n* * *\n![](https://assets.insightmediagroup.io/media/1787657095428_xaezyc.jpeg)\nWritten By\n## [Riccardo Andreoni](https://towardsdatascience.com/author/riccardo.andreoni/)\n[See all from Riccardo Andreoni](https://towardsdatascience.com/author/riccardo.andreoni/)\n[Convolutional Network](https://towardsdatascience.com/tag/convolutional-network/)[Data Science](https://towardsdatascience.com/tag/data-science/)[Hands On Tutorials](https://towardsdatascience.com/tag/hands-on-tutorials/)[Machine Learning](https://towardsdatascience.com/tag/machine-learning/)[Programming](https://towardsdatascience.com/tag/programming/)\nShare This Article\n  * [](https://www.facebook.com/sharer/sharer.php?u=https%3A%2F%2Ftowardsdatascience.com%2Fbuild-a-convolutional-neural-network-from-scratch-using-numpy-139cbbf3c45e%2F&title=Build%20a%20Convolutional%20Neural%20Network%20from%20Scratch%20using%20Numpy)\n  * [](https://www.linkedin.com/shareArticle?mini=true&url=https%3A%2F%2Ftowardsdatascience.com%2Fbuild-a-convolutional-neural-network-from-scratch-using-numpy-139cbbf3c45e%2F&title=Build%20a%20Convolutional%20Neural%20Network%20from%20Scratch%20using%20Numpy)\n  * [](https://x.com/share?url=https%3A%2F%2Ftowardsdatascience.com%2Fbuild-a-convolutional-neural-network-from-scratch-using-numpy-139cbbf3c45e%2F&text=Build%20a%20Convolutional%20Neural%20Network%20from%20Scratch%20using%20Numpy)\n\n\nTowards Data Science is a community publication. 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